research article exponential stability of neutral stochastic...

Research ArticleExponential Stability of Neutral StochasticFunctional Differential Equations with Two-Time-ScaleMarkovian Switching

Junhao Hu and Zhiying Xu

College of Mathematics and Statistics South-Central University for Nationalities Wuhan 430074 China

Correspondence should be addressed to Junhao Hu junhaohu74gmailcom

Received 22 December 2013 Accepted 16 January 2014 Published 16 March 2014

Academic Editor Weihai Zhang

Copyright copy 2014 J Hu and Z Xu This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We develop exponential stability of neutral stochastic functional differential equations with two-time-scale Markovian switchingmodeled by a continuous-time Markov chain which has a large state space To overcome the computational effort and thecomplexity we split the large-scale system into several classes and lump the states in each class into one class by the different statesof changes of the subsystems then we give a limit system to effectively ldquoreplacerdquo the large-scale system Under suitable conditionsusing the stability of the limit system as a bridge the desired asymptotic properties of the large-scale system with Brownian motionand Poisson jump are obtained by utilizing perturbed Lyapunov function methods and Razumikhin-type criteria Two examplesare provided to demonstrate our results

1 Introduction

In many practical dynamical systems such as neural net-works computer aided design population ecology chemicalprocess simulation and automatic control stochastic differ-ential equations represent the class of important dynamics(see [1ndash4]) During the recent several years the asymptoticproperties of neutral stochastic functional differential equa-tions have been investigated by many authors (see [5ndash14])Mao [10 11] gave the exponential stability of neutral stochasticfunctional differential equations by using the Razumikhin-type theorems Zhou and Hu [14] used the same argumentto discuss the exponential stability in 119901th moment of neu-tral stochastic functional differential equations and neutralstochastic functional differential equations with Markovianswitching Wu et al [13] examined the almost sure robuststability of nonlinear neutral stochastic functional differen-tial equations with infinite delay including the exponentialstability and the polynomial stability Song and Shen [12]investigated the asymptotic behavior of neutral stochasticfunctional differential equations under the more generalconditions than the classical linear growth condition Chenet al [5] considered the exponential stability in mean square

moment of mild solution for impulsive neutral stochasticpartial functional differential equations by employing theinequality technique The attraction and quasi-invariant setsof neutral stochastic partial functional differential equationswere also studied in the recent paper [9]

In this paper we will consider neutral stochastic func-tional differential equations with two-time-scale Markovianswitching modeled by a continuous-time Markov chainwhich has a large state space The computational effort andthe complexity become a real concern To overcome thedifficulties we have devoted much effort to the modelingand analysis of such systems in which one of the mainideas is to split a large-scale system into several classes andlump the states in each class into one state (see [3 15ndash21])Khasminskii et al for the first time established the asymptoticproperties of the Markov chain 119903

120576

(sdot) by introducing a smallparameter 120576 gt 0 (see [22]) Yin and Zhang developed themethod in their book [4] that a complicated system can bereplaced by the corresponding limit system that has a muchsimpler structure Motivated by the papers [16 21] undersuitable conditions using the stability of the limit system asa bridge we will study the exponential stability of neutralstochastic functional differential equations with Brownian

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 907982 15 pageshttpdxdoiorg1011552014907982

2 Mathematical Problems in Engineering

motion and Poisson jump by utilizing perturbed Lyapunovfunction methods and Razumikhin-type criteria

The remainder of this paper is organized as follows InSection 2 we introduce some notations and notions neededin our investigation In Section 3 we state our main resultsthat is exponential stability of neutral stochastic functionaldifferential equations with two-time-scaleMarkovian switch-ingThe exponential stability for neutral stochastic functionaldifferential equations driven by pure jumps is also discussedin Section 4 Finally two examples are presented to justify andillustrate applications of the theory in Section 5

2 Preliminaries

Throughout this paper unless otherwise specified let(ΩF F

119905119905ge0

P) be a complete probability space with afiltration F

119905119905ge0

satisfying the usual conditions (ie it isincreasing and right continuous and F

0contains all P-

null sets) Let 119882(119905) = (1198821(119905) 119882

119898(119905))

119879 be an 119898-dimensional Brownian motion defined on the probabilityspace For 120591 gt 0 let 119862([minus120591 0]R119899

) denote the family ofcontinuous functions 120593 from [minus120591 0] to R119899 with norm 120593 =

supminus120591le120579le0

|120593(120579)| where | sdot | is the Euclidean norm in R119899If 119860 is a vector or matrix its transpose is denoted by 119860

119879while its trace norm is denoted by |119860| = radictrace(119860119879119860)Denote by 119862

119887

F0([minus120591 0]R119899

) the family of all F0measurable

and bounded 119862([minus120591 0]R119899

)-valued random variables For119901 gt 0 and 119905 ge 0 denote by 119871

119901

F119905([minus120591 0]R119899

) the familyof allF

119905-measurable119862([minus120591 0]R119899

)-valued random variables120601 = 120601(120579) minus120591 le 120579 le 0 such that sup

minus120591le120579le0119864|120601(120579)|

119901

lt infinWe will denote the indicator function of a set 119866 by 119868

119866

Consider an 119899-dimensional neutral stochastic functionaldifferential equation with Markovian switching as follows

119889 [119909 (119905) minus 119863 (119909119905 119903 (119905))]

= 119891 (119909119905 119905 119903 (119905)) 119889119905 + 119892 (119909

119905 119905 119903 (119905)) 119889119908 (119905)

(1)

on 119905 ge 0 with initial data 1199090

= 120585 isin 119862([minus120591 0]R119899

) and 119909119905=

119909(119905 + 120579) minus120591 le 120579 le 0 which is regarded as a 119862([minus120591 0]R119899

)-valued stochastic process Moreover 119891 119862([minus120591 0]R119899

) times

R+

times S rarr R119899 119892 119862([minus120591 0]R119899

) times R+

times S rarr R119899times119898119863 119862([minus120591 0]R119899

) times S rarr R119899Let 119903(119905) (119905 ge 0) be a right-continuous Markov chain on

the probability space taking values in a finite state space S =

1 2 119872 with generator Γ = (120574119894119895)119872times119872

given by

P 119903 (119905 + Δ) = 119895 | 119903 (119905) = 119894 = 120574119894119895Δ + ∘ (Δ) if 119894 = 119895

1 + 120574119894119894Δ + ∘ (Δ) if 119894 = 119895

(2)

where Δ gt 0 Here 120574119894119895

ge 0 is the transition rate from 119894 to 119895 if119894 = 119895 while 120574

119894119894= minussum

119894 = 119895120574119894119895

We assume the Markov 119903(sdot) is independent of the Brown-ian motion 119882(sdot) It is well known that almost every samplepath 119903(sdot) is a right-continuous step function with finitenumber of simple jumps in any finite subinterval of R

+=

[0infin) As a standing hypothesis we assume that the Markov

chain is irreducible This is equivalent to the condition thatfor any 119894 119895 isin S we can find 119894

1 1198942 119894

119896isin S such that

1205741198941198941

12057411989411198942

sdot sdot sdot 120574119894119896119895

gt 0 (3)

Then Γ always has an eigenvalue 0 The algebraic interpreta-tion of irreducibility is rank (Γ) = 119872minus1 Under this conditiontheMarkov chain has a unique stationary distribution120587Γ = 0subject to sum

119872

119895=1120587119895

= 1 and 120587119895

gt 0 for all 119895 isin S For a real-valued function 120590(sdot) defined on S we define

Γ120590 (sdot) (119894) = sum

119895isinS

120574119894119895120590 (119895) = sum

119895 = 119894

120574119894119895(120590 (119895) minus 120590 (119894)) (4)

for each 119894 isin SLet 119862

21

(R119899

times R+

times SR+) denote the family of all

nonnegative functions 119881(119909 119905 119894) on R119899

times R+

times S which arecontinuously twice differentiable in 119909 and once differentiablein 119905 If 119881(119909 119905 119894) isin 119862

21

(R119899

times R+times SR

+) define an operator

L119881 from 119862([minus120591 0]R119899

) times R+times S to R by

L119881 (120593 119905 119894) = 119881119905(120593 (0) minus 119863 (120593 119894) 119905 119894)

+ 119881119909(120593 (0) minus 119863 (120593 119894) 119905 119894) 119891 (120593 119905 119894)

+1

2trace [119892

119879

(120593 119905 119894)

times 119881119909119909

(120593 (0) minus 119863 (120593 119894) 119905 119894)

times 119892 (120593 119905 119894) ]

+

119897

sum

119895=1

120574119894119895119881 (120593 (0) minus 119863 (120593 119895) 119905 119895)

(5)

where

120593 isin 119862 ([minus120591 0] R119899

) 119881119905=

120597119881 (119909 119905 119894)

120597119905

119881119909= (

120597119881 (119909 119905 119894)

1205971199091

120597119881 (119909 119905 119894)

1205971199092

120597119881 (119909 119905 119894)

120597119909119899

)

119881119909119909

= (1205972

119881(119909 119905 119894)

120597119909119894120597119909

119895

)

119899times119899

(6)

For a parameter 120576 gt 0 we rewrite the Markov chain 119903(119905)

as 119903120576

(119905) and the generator Γ as Γ120576 Γ120576 is given by

Γ120576

=1

120576Γ + Γ (7)

where Γ120576 represents the fast varying motions and Γ rep-resents the slowly changing dynamics Set Γ

120576

= (120574120576

119894119895)119872times119872

Γ = (120574

119894119895)119872times119872

and Γ = (120574119894119895)119872times119872

For the sake of simplicitysuppose that

S = S1

cup S2

cup sdot sdot sdot cup S119897

(8)

with S119896

= 1199041198961

119904119896119872119896

119872 = 1198721+ 119872

2+ sdot sdot sdot + 119872

119897 and

Γ = diag (Γ1

Γ119897

) (9)

Mathematical Problems in Engineering 3

where Γ119896 is a generator of aMarkov chain taking values inS119896

for every 119896 isin 1 119897We give the first assumption as follows

Assumption 1 For each 119896 isin 1 119897 Γ119896 is irreducible

In order to emphasize the effect of the fast switching (1)can be given by

119889 [119909120576

minus 119863 (119909120576

119905 119903

120576

(119905))]

= 119891 (119909120576

119905 119905 119903

120576

(119905)) 119889119905 + 119892 (119909120576

119905 119905 119903

120576

(119905)) 119889119908 (119905)

119909120576

0= 120585 isin 119862 ([minus120591 0] R

119899

) 119903120576

= 1199030

(10)

To assure the existence and uniqueness of the solution wegive the following standard assumptions

Assumption 2 (local Lipschitz condition) For each integer120572 ge 1 there exists a constant 119871

120572gt 0 such that

1003816100381610038161003816119891 (120593 119905 119894) minus 119891 (120601 119905 119894)1003816100381610038161003816 or

1003816100381610038161003816119892 (120593 119905 119894) minus 119892 (120601 119905 119894)1003816100381610038161003816

le 119871120572

1003817100381710038171003817120593 minus 12060110038171003817100381710038172

(11)

for all 119894 isin S 119905 ge 0 and those 120593 120601 isin 119862([minus120591 0]R119899

) with 120593 or

120601 le 120572 and 119891(0 119905 119894) equiv 0 119892(0 119905 119894) equiv 0

Assumption 3 (linear growth condition) There is an 119871 gt 0for any 120593 isin 119862([minus120591 0]R119899

) 119905 ge 0 119894 isin S such that

1003816100381610038161003816119891 (120593 119905 119894)10038161003816100381610038162

or1003816100381610038161003816119892 (120593 119905 119894)

10038161003816100381610038162

le 119871 (1 +1003817100381710038171003817120593

10038171003817100381710038172

) (12)

Assumption 4 For all 119894 isin S and those 120593 120601 isin 119862([minus120591 0]R119899

)there is a constant 0 lt 120581 lt 1 such that

1003816100381610038161003816119863 (120593 119894) minus 119863 (120601 119894)1003816100381610038161003816 le 120581

1003817100381710038171003817120593 minus 12060110038171003817100381710038172

119863 (0 119894) equiv 0

(13)

Under Assumptions 2 3 and 4 (10) has a unique solutiondenoted by 119909

120576120585119894

(119905) on 119905 ge 0 where 119909120576120585119894 is dependent on the

initial value (120585 119894) (see [8]) Moreover for every 119901 gt 0 andany compact subset 119866 of 119862([minus120591 0]R119899

) there is a positiveconstant 119867 which is independent of 120576 such that

sup(120585119894)isin119866timesS

119864[ supminus120591le119904le119905

10038161003816100381610038161003816119909120576120585119894

(119904)10038161003816100381610038161003816

119901

] le 119867 119905 ge 0 (14)

Since the state space of the Markov chain is large it is toocomplicated to deal with directlyWe need to analyse the limitequation of (10) To continue make all the states in each S119896

into a single state and define an aggregated process 119903120576

(sdot) as

119903120576

(119905) = 119896 if 119903120576

(119905) isin S119896

(15)

Denote the state space of 119903120576(119905) by S = 1 119897 the stationarydistribution Γ

119896 by 120583119896

= (120583119896

1 120583

119896

119872119896

) isin R1times119872119896 and 120583 =

diag(1205831

120583119897

) isin R119897times119872 Define

Γ = (120574119894119895)119897times119897

= 120583Γ1 (16)

with 1 = diag(11198721

1119872119897

) and 1119872119896

= (1 1)119879

isin R119872119896times1119896 = 1 119897 It has been known that 119903120576(sdot) converges weakly to119903(sdot) as 120576 rarr 0 where 119903(sdot) is a continuous-time Markov chainwith generator Γ and state space S (see [4]) Define

119863(120593 119896) =

119872119896

sum

119895=1

120583119896

119895119863(120593 119904

119896119895)

119891 (120593 119905 119896) =

119872119896

sum

119895=1

120583119896

119895119891 (120593 119905 119904

119896119895)

119892 (120593 119905 119896) 119892119879

(120593 119905 119896) =

119872119896

sum

119895=1

120583119896

119895119892 (120593 119905 119904

119896119895) 119892

119879

(120593 119905 119904119896119895)

(17)

for each 119904119896119895

isin S119896 with 119896 isin 1 119897 and 119895 isin 1 119872119896 It is

easy to know that119863(120593 119896)119891(120593 119905 119896) and119892(120593 119905 119896) are the lim-its with respect to the stationary distribution of the Markovchain Consider that for any 120593 = 0 119892(120593 119905 119904

119896119895)119892

119879

(120593 119905 119904119896119895)

are nonnegative definite matrices so we denote its ldquosquarerootrdquo of 119892(120593 119905 119904

119896119895)119892

119879

(120593 119905 119904119896119895) by 119892(120593 119905 119896) For degenerate

diffusions we can see the argument in [23]The limit equation of (10) is defined as follows

119889 [120593 (0) minus 119863 (120593 119903 (119905))]

= 119891 (120593 119905 119903 (119905)) 119889119905 + 119892 (120593 119905 119903 (119905)) 119889119908 (119905)

1199090= 120585 119903 = 119903

0

(18)

3 Exponential Stability of NSFDE withTwo-Time-Scale Markovian Switching

In this section we establish the Razumikhin-type theoremon the exponential stability for (10) Denote by 119862

119901

(R119899

times

R+times SR

+) the family of nonnegative real-valued functions

defined on R119899

times R+

times S that are 119901-times continuouslydifferentiable with respect to 119909 At the same time we needanother assumption and a lemma with respect to 119881(119909 119905 119894) isin

119862119901

(R119899

times R+times SR

+) for some 119901 ge 4

Assumption 5 For each 119896 isin S 119881(119909 119905 119894) rarr infin as |119909| rarr

infin Moreover 120597119901119881(119909 119905 119894) = 119874(1) 120597120580119881(119909 119905 119894)(|119909|120580

+ |119910|120580

) le

119870(|119909|119901

+|119910|119901

+1) for 1 le 120580 le 119901minus1 where 120597120580119881(119909 119905 119894)denotes the120580th derivative of 119881(119909 119905 119894) with respect to 119909 and 119874(119910) denotesthe function of 119910 satisfying sup

119910|119874(119910)|119910 lt infin

Lemma 6 Suppose that 119901 ge 1 there is a positive constant120581 isin (0 1) such that

E1003816100381610038161003816119863 (120593 119896)

1003816100381610038161003816119901

le 120581119901 supminus120591le120579le0

1198901205741205791003817100381710038171003817120593

1003817100381710038171003817119901

(120593 119894) isin 119871119901

F119905([minus120591 0] R

119899

) times S

(19)


Then for any 120585 isin 119871119901

F0([minus120591 0]R119899

) the solution for (10)satisfies

supminusinfinlt119904le119905

119890120574119904

E |119909 (119904)|119901

le

10038171003817100381710038171205851003817100381710038171003817119901

1 minus 120581orsup

0le119904le119905119890120574119904E

1003816100381610038161003816119909 (119904) minus 119863 (119909119904 119903 (119904))

1003816100381610038161003816119901

(1 minus 120581)119901

119905 ge 0

(20)

Proof Note the following elementary inequality

(119909 + 119910)119901

= (1 minus 1205811)1minus119901

(119909119901

+ 1205811

1minus119901

119910119901

)

forall119909 119910 ge 0 1205811gt 0

(21)

We have from condition (20) that for any 119905 ge 0

119890120574119905

E|119909 (119905)|119901

le 119890120574119905

[(1 minus 120581)1minus119901

E1003816100381610038161003816119909 (119905) minus 119863 (119909

119905 119903 (119905))

1003816100381610038161003816119901

+ 1205811minus119901

E1003816100381610038161003816119863 (119909

119905 119903 (119905))

1003816100381610038161003816119901

]

le (1 minus 120581)1minus119901

119890120574119905

E1003816100381610038161003816119909 (119905) minus 119863 (119909

119905 119903 (119905))

1003816100381610038161003816119901

+ 120581119890120574119905 supminus120591le120579le0

119890120574120579

E |119909 (119905 + 120579)|119901

le (1 minus 120581)1minus119901 sup

0le119904le119905

119890120574119904

E1003816100381610038161003816119909 (119904) minus 119863 (119909

119904 119903 (119904))

1003816100381610038161003816119901

+ 120581 supminus120591le120579le0

119890120574(119904+120579)

E |119909 (119904 + 120579)|119901


0le119904le119905

119890120574119904

E1003816100381610038161003816119909 (119904) minus 119863 (119909

119904 119903 (119904))

1003816100381610038161003816119901

+ 120581 supminusinfinlt119904le119905

119890120574119904

E |119909 (119904)|119901

(22)

Then


119890120574119904

E |119909 (119904)|119901

le [ supminus120591le120579le0

E |119909 (120579)|2

]

or [(1 minus 120581)1minus119901 sup

0le119904le119905

119890120574119904

E1003816100381610038161003816119909 (119904) minus 119863 (119909

119904 119903 (119904))

1003816100381610038161003816119901

+120581 supminusinfinlt119904le119905

119890120574119904

E |119909 (119904)|119901

]

(23)

Therefore the desired result holds

Theorem 7 Let Assumptions 1ndash4 hold and let 1198881 119888

2 120582 119901 be all

positive numbers and 119902 gt 1 Assume that there exists a function119881(119909 119905 119896) isin 119862

119901

(R119899

timesR+timesSR

+) satisfying Assumption 5 such

that

1198881|119909|

119901

le 119881 (119909 119905 119896) le 1198882|119909|

119901

(24)

for all (119909 119905 119896) isin R119899

times R+

times S 119905 ge 0 119896 isin S Consider thefollowing

E1003816100381610038161003816119863 (120593 119896)

1003816100381610038161003816119901

le 120581119901 supminus120591le120579le0

119890]1205791003817100381710038171003817120593

1003817100381710038171003817119901

120581 = max 1205811 120581

119896 120593 isin 119871

119901

F119905

(25)

for all 119905 ge 0 0 lt 120581120590lt 1 120590 = 1 119896 and

E [max119896isin

S

L119881 (120593 119905 119896)]

le minus120582E [max119896isin

S

119881 (120593 (0) minus 119863 (120593 119896) 119905 119896)]

(26)

provided 120593 = 120593(120579) minus120591 le 120579 le 0 isin 119871119901

F119905([minus120591 0]R119899

)satisfying

E [min119896isin

S

119881 (120593 (120579) 119905 + 120579 119896)]

lt 119902E [max119896isin

S

119881 (120593 (0) minus 119863 (120593 119896) 119905 119896)]

(27)

for all minus120591 le 120579 le 0 Then for all 120585 isin 119862119887

F0([minus120591 0]R119899

) 119905 ge 0

lim sup120576rarr0

E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

119890minus]119905

(28)

where

] = min1205741

120591log

119902

(11988821198881) (1 minus 120581)

119901 (29)

120574 being the root of the following equation

1198882

1198881

(1 minus 120581)119901

119890120574120591

= 120582 (30)

In other words the trivial solution of (10) is 119901th momentexponentially stable and the119901thmoment Lyapunov exponentis not greater than minus]

Proof Let

(120593 119905 119895) =

119897

sum

119896=1

119881 (120593 119905 119896) 119868119895isinS119896 = 119881 (120593 119905 119896) if 119895 isin S

119896

(31)

By the definition of we know that

(120593120576

119905 119903120576

(119905)) = 119881 (120593120576

119905 119903120576

(119905))

119872

sum

119894=1

120574119897119894 (120593 119905 119894) =

119872

sum

119894=1

120574119897119894

119897

sum

119896=1

119881 (120593 119905 119896) 119868119894isinS119896 = 0

(32)

Extend 119903(119905) to [minus120591 0] by setting 119903(119905) = 119903(0) Recalling thefacts that 119909(119905) is continuous for all minus120591 le 120579 le 0 and 119903(119905) is


right continuous it is easy to see that E119881(119909(119905) 119905 119903(119905)) is rightcontinuous on 119905 ge minus120591 Let 120574 isin (0 ]) be arbitrary and define

119880 (119905) = supminus120591le120579le0

[119890120574(119905+120579)E119881 (119909

120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ]

= supminus120591le120579le0

[119890120574(119905+120579)

E (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ]

(33)

for all 119905 ge 0 We claim that

119863+

119880 (119905) = lim supℎrarr0+

119880 (119905 + ℎ) minus 119880 (119905)

ℎle 0 forall119905 ge 0 (34)

Note that for each 119905 ge 0 either 119880(119905) gt 119890120574119905E119881(119909

120576

(119905) minus

119863(119909120576

119905 119903

120576

(119905)) 119905 119903120576

(119905)) or 119880(119905) = 119890120574119905E119881(119909

120576

(119905) minus 119863(119909120576

119905 119903

120576

(119905))

119905 119903120576

(119905))If 119880(119905) gt 119890

120574119905E119881(119909120576

(119905) minus 119863(119909120576

119905 119903

120576

(119905)) 119905 119903120576

(119905)) becauseE119881(119909(119905) 119905 119903(119905)) is right continuous on 119905 ge minus120591 it is easyto obtain that for all ℎ gt 0 sufficiently small 119880(119905) gt

119890120574(119905+ℎ)E119881(119909

120576

(119905 + ℎ) minus119863(119909120576

119905+ℎ 119903

120576

(119905 + ℎ)) 119905 + ℎ 119903120576

(119905 + ℎ)) hence119880(119905 + ℎ) le 119880(119905) and 119863

+

119880(119905) le 0If 119880(119905) = 119890

120574119905E119881(119909120576

(119905) minus 119863(119909120576

119905 119903

120576

(119905)) 119905 119903120576

(119905)) we have

119890120574(119905+120579)

E119881 (119909120576

(119905 + 120579) minus 119863 (119909119905+120579

119903120576

(119905 + 120579)) 119905 + 120579 119903120576

(119905 + 120579))

le 119890120574119905

E119881 (119909120576

(119905) minus 119863 (119909119905 119903

120576

(119905)) 119905 119903120576

(119905))

(35)

for all minus120591 le 120579 le 0Then

E119881 (119909120576

(119905 + 120579) minus 119863 (119909119905+120579

119903120576

(119905 + 120579)) 119905 + 120579 119903120576

(119905 + 120579))

le 119890minus120574120579

E119881 (119909120576

(119905) minus 119863 (119909119905 119903

120576

(119905)) 119905 119903120576

(119905))

le 119890120574120591

E119881 (119909120576

(119905) minus 119863 (119909119905 119903

120576

(119905)) 119905 119903120576

(119905))

(36)

for all minus120591 le 120579 le 0On the other hand by Lemma 6 we derive

119890120574(119905+120579)

E119881 (119909120576

(119905 + 120579) 119905 + 120579 119903120576

(119905 + 120579))

le 1198882119890120574(119905+120579)

E1003816100381610038161003816119909

120576

(119905 + 120579)1003816100381610038161003816119901

le 1198882(1 minus 120581)

119901 sup0le119904le119905

119890120574119904

E1003816100381610038161003816119909

120576

(119904) minus 119863 (119909120576

119904 119903 (119904))

1003816100381610038161003816119901

le1198882

1198881

(1 minus 120581)119901 sup0le119904le119905

119890120574119904

E119881 (119909120576

(119904) minus 119863 (119909120576

119904 119903 (119904)) 119904 119903 (119904))

le1198882

1198881

(1 minus 120581)119901

119890120574119905

E119881 (119909120576

(119905) minus 119863 (119909120576

119905 119903 (119905)) 119905 119903 (119905))

(37)

Then

E119881 (119909120576

(119905 + 120579) 119905 + 120579 119903120576

(119905 + 120579))

lt 119902E119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

(38)

where 119902 gt (11988821198881)(1 minus 120581)

119901

119890120574120591 that is 120574 lt (1120591)(log(119902(119888

21198881)

(1 minus 120581)119901

))Consequently there exists a sufficiently small 120576

0gt 0 such

that for any 120576 isin (0 1205760)

E [min119896isin

S

119881 (120593120576

(120579) 119905 + 120579 119896)]


S

119881 (120593120576

(0) minus 119863 (120593120576

119896) 119905 119896)]

(39)

for all minus120591 le 120579 le 0 Thus

E [max119896isin

S

L119881 (120593120576

119905 119903120576

(119905))]


S

119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(40)

which implies that

E [L119881 (120593120576

119905 119903120576

(119905))]

le minus120582E [119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(41)

By the condition of 120574 lt ] le 120582 we get

E [L119881 (120593120576

119905 119903120576

(119905))]

le minus120574E [119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(42)

Next we consider

119880 (119905 + ℎ) minus 119880 (119905)

= lim sup120576rarr0

[119890120574(119905+120579+ℎ)

E [119881 (119909120576

(119905 + 120579 + ℎ)

minus 119863 (119909120576

119905+120579+ℎ 119903

120576

(119905 + 120579 + ℎ))

119905 + 120579 + ℎ 119903120576

(119905 + 120579 + ℎ)) ]

minus 119890120574(119905+120579)

E [119881 (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579))]]


Eint

119905+120579+ℎ

119905+120579

119890120574119904

[L119881 (120593120576

119904 119903120576

(119904))

+ 120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904))

119904 119903120576

(119904))] 119889119904


Eint

119905+120579+ℎ

119905+120579

119890120574119904

[L (120593120576

119904 119903120576

(119904))

+ 120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904))

119904 119903120576

(119904)) ] 119889119904

(43)


By the definition of operatorL we have

L (120593120576

119905 119903120576

(119905))

= 119905(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

+ 119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119891 (120593120576

119905 119903120576

(119905))

+1

2trace [119892

119879

(120593120576

119905 119903120576

(119905))

times 119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) ]

+

119872

sum

119894=1

120574120576

119903120576(119905)119894

(120593120576

(0) minus 119863 (119909120576

119905 119894) 119905 119894)

= 119905(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

+ 119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119891 (120593120576

119905 119903120576

(119905))

+1

2trace [119892

119879

(120593120576

119905 119903120576

(119905))

times 119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) ]

+

119872

sum

119894=1

120574119903120576(119905)119894

(120593120576

(0) minus 119863 (119909120576

119905 119894) 119905 119894)

= 119881119905(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

+ 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119891 (120593120576

119905 119903120576

(119905))

+1

2trace [119892

119879

(120593120576

119905 119903120576

(119905))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) ]

+

119897

sum

119896=1

120574120576

119903120576(119905)119896

119881 (120593120576

(0) minus 119863 (119909120576

119905 119896) 119905 119896)

+ 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times [119891 (120593120576

119905 119903120576

(119905)) minus 119891 (120593120576

119905 119903120576

(119905))]

+1

2trace [119892

119879

(120593120576

119905 119903120576

(119905))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) minus 119892119879

(120593120576

119905 119903120576

(119905))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) ]

+

119872

sum

119894=1

120574119903120576(119905)119894

(120593120576

(0) minus 119863 (119909120576

119905 119894) 119905 119894)

minus

119897

sum

119896=1

120574119903120576(119905)119896

119881 (120593120576

(0) minus 119863 (119909120576

119905 119896) 119905 119896)

= L119881 (120593120576

119905 119903120576

(119905))

+ 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times [119891 (120593120576

119905 119903120576

(119905)) minus 119891 (120593120576

119905 119903120576

(119905))]

+1

2trace [119892

119879

(120593120576

119905 119903120576

(119905))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) minus 119892119879

(120593120576

119905 119903120576

(119905))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) ]

+

119872

sum

119894=1

120574119903120576(119905)119894

(120593120576

(0) minus 119863 (119909120576

119905 119894) 119905 119894)

minus

119897

sum

119896=1

120574119903120576(119905)119896

119881 (120593120576

(0) minus 119863 (119909120576

119905 119896) 119905 119896)

(44)

Therefore

119880 (119905 + ℎ) minus 119880 (119905)


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [L119881 (120593120576

119904 119903120576

(119904))

+120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))] 119889119904

+ lim sup120576rarr0

E

times int

119905+120579+ℎ

119905+120579

119890120574119905

119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times [119891 (120593120576

119904 119903120576

(119904)) minus 119891 (120593120576

119904 119903120576

(119904))] 119889119904

+1

2lim sup120576rarr0

E

times int

119905+120579+ℎ

119905+120579

119890120574119904


times trace [119892119879

(120593120576

119904 119903120576

(119904))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times 119892 (120593120576

119904 119903120576

(119904)) minus 119892119879

(120593120576

119904 119903120576

(119904))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times 119892 (120593120576

119904 119903120576

(119904)) ] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times (

119872

sum

119894=1

120574119903120576(119904)119894

(120593120576

(0) minus 119863 (119909120576

119904 119894) 119904 119894)

minus

119897

sum

119896=1

120574119903120576(119904)119896

119881 (120593120576

(0) minus 119863 (119909120576

119904 119896) 119904 119896))119889119904

= 1198681+ 119868

2+ 119868

3+ 119868

4

(45)

By the definition of 119891

119891 (120593120576

119905 119903120576

(119905)) minus 119891 (120593120576

119905 119903120576

(119905))

=

119897

sum

119896=1

119872119896

sum

119895=1

119891 (120593120576

119905 119904119896119895) times [119868

119903120576(119905)=119904119896119895

minus 120583119896

119895119868119903120576(119905)=119896

]

(46)

This implies that

lim120576rarr0

Eint

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times [119891 (120593120576

119904 119903120576

(119904)) minus 119891 (120593120576

119904 119903120576

(119904))] 119889119904

le lim120576rarr0

[E

100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times [119891 (120593120576

119904 119903120576

(119904)) minus 119891 (120593120576

119904 119903120576

(119904))] 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

]

12

= lim120576rarr0

[

[

E

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times

119897

sum

119896=1

119872119896

sum

119895=1

119891 (120593120576

119904 119904119896119895)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

] 119889119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

]

]

12

le lim120576rarr0

[

[

E

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119897

sum

119896=1

119872119896

sum

119895=1

119890120574119904

119871 (1 +1003817100381710038171003817120593

1003817100381710038171003817119901

)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

] 119889119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

]

]

12

(47)

By the argument of Lemma 714 in [4] the right side of theabove inequality is equivalent to 0 that is 119868

2= 0 Similarly

we can show that

1198683=

1

2lim sup120576rarr0

E

times int

119905+120579+ℎ

119905+120579

119890120574119904


(120593120576

119904 119903120576

(119904))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times 119892 (120593120576

119904 119903120576

(119904))

minus 119892119879

(120593120576

119904 119903120576

(119904))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times 119892 (120593120576

119904 119903120576

(119904)) ] 119889119904 = 0

(48)

By the definition of Γ and Γ we have

119872

sum

119894=1

120574119903120576(119905)119894

(120593120576

(0) minus 119863 (119909120576

119905 119894) 119905 119894)

= Γ (120593120576

(0) minus 119863 (119909120576

119905 119894) 119905 sdot) (119903

120576

(119905))

119897

sum

119896=1

120574119903120576(119905)119896

119881 (120593 (0) minus 119863 (119909120576

119905 119896) 119905 119896)

= Γ119881 (120593 (0) minus 119863 (119909120576

119905 119896) 119905 sdot) (119903

120576

(119905))

(49)

Hence

1198684= lim sup

120576rarr0

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times (

119872

sum

119894=1

120574119903120576(119904)119894

(120593120576

(0) minus 119863 (119909120576

119904 119894) 119904 119894)

minus

119897

sum

119896=1

120574119903120576(119904)119896

119881 (120593120576

(0) minus 119863 (119909120576

119904 119896) 119904 119896))119889119904



E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times (Γ (120593120576

(0) minus 119863 (119909120576

119904 119894) 119904 sdot) (119903

120576

(119904))

minus Γ119881 (120593120576

(0) minus 119863 (119909120576

119904 119896) 119904 sdot) (119904

119896119895)) 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

119897

sum

119896=1

119872119896

sum

119895=1

Γ (120593120576

(0) minus 119863 (119909120576

119904 119894) 119904 sdot) (119904

119896119895)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

] 119889119904

le lim sup120576rarr0

[

[

E

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

times

119897

sum

119896=1

119872119896

sum

119895=1

Γ (120593120576

(0) minus 119863 (119909120576

119904 119894) 119904 sdot) (119904

119896119895)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

]

]

12

(50)


4= 0 Therefore

119880 (119905 + ℎ) minus 119880 (119905)

= lim120576rarr0

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [L119881 (120593120576

119904 119903120576

(119904))

+120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))] 119889119904

le 0

(51)

That is

119880 (119905 + ℎ) le 119880 (119905) (52)

So 119880(119905 + ℎ) = 119880(119905) for all ℎ gt 0 sufficiently small and hence119863

+

119880(119905) = 0 Inequality (34) holds

It follows from (34) that 119880(119905) le 119880(0) for all 119905 ge 0 By thedefinition of 119880(119905)

lim sup120576rarr0

119890120574119905

E1003816100381610038161003816119909

120576

(119905) minus 119863 (119909120576

119905 119903

120576

(119905))1003816100381610038161003816119901

le 1198882lim sup120576rarr0

sup120591le120579le0

119890120574120579

E1003816100381610038161003816119909

120576

(120579) minus 119863 (119909120576

120579 119903

120576

(120579))1003816100381610038161003816

119901



(1 + 120581)119901minus1

times [E1003816100381610038161003816119909

120576

(120579)1003816100381610038161003816119901

+ 1205811minus119901

E1003816100381610038161003816119863 (119909

120576

120579 119903

120576

(120579))1003816100381610038161003816119901

]

le 1198882(1 + 120581)

11990110038171003817100381710038171205851003817100381710038171003817119901

119905 ge 0

(53)

By Lemma 6 we derive

lim sup120576rarr0

119890120574119905

E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

(54)

That is

lim sup120576rarr0

E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

119890minus120574119905

forall119905 ge 0 (55)

4 Neutral Stochastic Functional System withPure Jump

In this section we discuss the stability of the following neutralstochastic functional system with pure jump

119889 [120593120576

(0) minus 119863 (120593120576

119903120576

(119905))]

= 119891 (120593120576

119905 119903120576

(119905)) 119889119905 + intR119898

119887 (119909120576

119905minus 119905 119903

120576

(119905) 119911) (119889119905 119889119911)

1199090= 120585 isin 119862 ([minus120591 0] R

119899

) 119903 (0) isin S

(56)

where 119909120576

119905minus= lim

119904uarr119905119909120576

119904 119863 119862([minus120591 0]R119899

) times S rarr R119899 119887

119862([minus120591 0]R119899

) timesR+timesStimesR119898

rarr R119899times119898 We assume that eachcolumn 119887

(120575) of the 119899 times 119898 matrix 119887 = [119887119894119895] depends on 119911 only

through the 120575th coordinate 119911120575 that is

119887(120575)

(120593 119905 119894 119911) = 119887(120575)

(120593 119905 119894 119911120575)

119911 = (1199111 119911

119898) isin R

119898

119894 isin S

(57)

119873(119905 119911) is an119898-dimensional Poisson process and the compen-sated Poisson process is defined by

(119889119905 119889119911) = (1(119889119905 119889119911

1)

119898(119889119905 119889119911

119898))

= (1198731(119889119905 119889119911

1) minus 120582

1(119889119911

1) 119889119905 119873

119898(119889119905 119889119911

119898)

minus 120582119898

(119889119911119898) 119889119905)

(58)

where 119873120575 120575 = 1 119898 are independent one-dimensional

Poisson random measures with characteristic measure


120582120575 120575 = 1 119898 coming from 119898 independent one-

dimensional Poisson point processes The limit system of(56) is defined as follows

119889 [120593120576

(0) minus 119863 (119909120576

119905 119903

120576

(119905))]

= 119891 (119909120576

119905 119905 119903

120576

(119905)) 119889119905

+ intR119898

(119909120576

119905minus 119905 119903

120576

(119905) 119911) (119889119905 119889119911)

1199090= 120585 isin 119862 ([minus120591 0] R

119899

) 119903 (0) isin S

(59)

where 119909120576

119905minus= lim

119904uarr119905119909120576

119904and 119862([minus120591 0]R119899

)timesR+times StimesR119898

rarr

R119899times119898 Similar to the definition of 119891 we define

119863(120593 119896) =

119873119898

sum

119895=1

120583119896

119895119863(120593 119904

119896119895)

(120593 119905 119896 119911) =

119873119898

sum

119895=1

120583119896

119895119887 (120593 119905 119904

119896119895 119911)

(60)

for each 119904119896119895

isin S119896 with 119896 isin 1 119897 and 119895 isin 1 119873119898

To assure the existence and uniqueness of the solution of(59) we also give the following standard assumptions

Assumption 8 For any integer 120577 there is a constant 119871120577

gt 0such that

1003816100381610038161003816119891 (120593 119905 119894) minus 119891 (120601 119905 119894)1003816100381610038161003816

or

119898

sum

120575=1

intR

10038161003816100381610038161003816119887(120575)

(120593 119905 119894 119911120575) minus 119887

(120575)

(120601 119905 119894 119911120575)10038161003816100381610038161003816120582120575(119889119911

120575)

le 119871120577

1003817100381710038171003817120593 minus 12060110038171003817100381710038172

(61)

for all 119894 isin S and those120593 120601 isin 119862([minus120591 0]R119899

)with 120593or120601 le 120577and119891(0 119905 119894) equiv 0 119887(0 119905 119894 119911) equiv 0

Assumption 9 There is an 119871 gt 0 such that for any 120593 120601 isin

119862([minus120591 0]R119899

) 119894 isin S

1003816100381610038161003816119891 (120593 119905 119894)1003816100381610038161003816 or

119898

sum

120575=1

intR

10038161003816100381610038161003816119887(120575)

(120593 119905 119894 119911120575)10038161003816100381610038161003816120582120575(119889119911

120575)

le 119871 (1 +1003817100381710038171003817120593

10038171003817100381710038172

)

(62)



1003816100381610038161003816119863 (120593 119894) minus 119863 (120601 119894)1003816100381610038161003816 le 120581

1003817100381710038171003817120593 minus 12060110038171003817100381710038172

119863 (0 119894) equiv 0

(63)

Given that 119881 isin 119862119901

(R119899

times R+times SR


L119881 by

L119881 (120593 119905 119894)

= 119881119905(120593 (0) minus 119863 (120593 119894) 119905 119894)

+ 119881119909(120593 (0) minus 119863 (120593 119894) 119905 119894) 119891 (120593 119905 119894)

+

119873

sum

119895=1

120574119894119895119881 (120593 (0) minus 119863 (120593 119894) 119905 119895)

+ intR

119898

sum

120575=1

119881 (120593 (0) minus 119863 (120593 119894) + 119887(120575)

(120593 119905 120580 119911120575) 119905 120580)

minus 119881 (120593 (0) minus 119863 (120593 119894) 119905 119894)

minus 119881119909(120593 (0) minus 119863 (120593 119894) 119905 119894)

times 119887(120575)

(120593 119905 120580 119911120575) 120582

120575(119889119911

120575)

(64)

where

119881119909(120593 (0) minus 119863 (120593 119894) 119905 119894)

= (120597119881 (120593 (0) minus 119863 (120593 119894) 119905 119894)

1205971199091

120597119881 (120593 (0) minus 119863 (120593 119894) 119905 119894)

120597119909119898

)

(65)

Lemma 11 (see [20]) Let Assumptions 1 8 and 9 hold as120576 rarr 0 then (119909120576

(sdot) 119903120576

(sdot)) converges weakly to (119909(sdot) 119903(sdot)) in119863([0infin)R119899

times S) where 119863([0infin)R119899

times S) is the space offunctions defined on [0infin) that are right continuous and haveleft limits taking values in R119899

times S and are endowed with theSkorohod topology

Theorem 12 Let Assumptions 1 and 8ndash10 hold and let 1198881 119888

2

120582 119901 be all positive numbers and 119902 gt 1 Assume that thereexists a function 119881(119909 119905 119896) isin 119862

119901

(R119899

times R+times SR

+) satisfying

Assumption 5 such that

1198881|119909|

119901

le 119881 (119909 119905 119896) le 1198882|119909|

119901

119896 isin S (66)

for all (119909 119905 119896) isin R119899

times R+times S and 119905 ge 0 119896 isin S Consider the

following

E1003816100381610038161003816119863 (120593 119896)

1003816100381610038161003816119901

le 120581119901 supminus120591le120579le0

119890]120579 1003817100381710038171003817120593

1003817100381710038171003817

119901

120581 = max 1205811 120581

119896 120593 isin 119871

119901

F119905

(67)


E [max119896isin

S

L119881 (120593 119905 119896)] le minus120574E [max119896isin

S

119881 (120593 (0) minus 119863 (120593 119896))]

(68)



F119905([minus120591 0]R119899

)satisfying

E [min119896isin

S

119881 (119909 (119905 + 120579) 119905 + 120579 119896)]


S

119881 (120593 (0) minus 119863 (120593 119896) 119905 119894)] minus120591 le 120579 le 0

(69)

Then for all 120585 isin 119862([minus120591 0]R119899

) 119905 ge 0

lim sup120576rarr0

E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

119890minus]119905

(70)

where

] = min1205741

120591log

119902

(11988821198881) (1 minus 120581)

119901 (71)


1198882

1198881

(1 minus 120581)119901

119890120574120591

= 120582 (72)

Proof Define

(120593 119905 120588) =

119897

sum

119896=1

119881 (120593 119905 119896) 119868120588isinS119896 = 119881 (120593 119905 119896) if 120588 isin S

119896

(73)

Extend 119903(119905) to [minus120591 0] by setting 119903(119905) = 119903(0) thenE119881(119909(119905) 119905 119903(119905)) is right continuous on 119905 ge minus120591 Let 120574 isin (0 ])be arbitrary and define

119880 (119905)


[119890120574(119905+120579)

E119881 (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ]


[119890120574(119905+120579)

E (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ]

(74)


119863+

119880 (119905) = lim supℎrarr0+

119880 (119905 + ℎ) minus 119880 (119905)

ℎle 0 forall119905 ge 0 (75)

Similar to the proof of Theorem 7 we derive

E119881 (119909120576

(119905 + 120579) 119905 + 120579 119903120576

(119905 + 120579))

lt 119902E119881 (120593 (0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

(76)

for all minus120591 le 120579 le 0 where 119902 gt (11988821198881)(1 minus 120581)

119901

119890120574120591 that is 120574 lt

(1120591)(log(119902(11988821198881)(1 minus 120581)

119901

))

Thus

E [max119896isin

S

L119881 (120593120576

119905 119903120576

(119905))]


S

119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(77)

which implies that

E [L119881 (120593120576

119905 119903120576

(119905))]

le minus120582E [119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(78)


E [L119881 (120593120576

119905 119903120576

(119905))]

le minus120574E [119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(79)

We now consider

119880 (119905 + ℎ) minus 119880 (119905)


[119890120574(119905+120579+ℎ)

E

times [119881 (119909120576

(119905 + 120579 + ℎ)

minus 119863 (119909120576

119905+120579+ℎ 119903

120576

(119905 + 120579 + ℎ))

119905 + 120579 + ℎ 119903120576

(119905 + 120579 + ℎ)) ]

minus 119890120574(119905+120579)

E

times [119881 (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ] ]


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

[L119881 (120593120576

119904 119903120576

(119904))

+120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119905))] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [L (120593120576

119904 119903120576

(119904))

+ 120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119904 119903120576

(119904)) ] 119889119905

(80)


By the definition of the operator L we have

L (120593120576

119905 119903120576

(119905))

= 119905(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

+ 119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119891 (120593120576

119905 119903120576

(119905))

+

119898

sum

120575=1

intR

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905))

+ 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 119905 119903

120576

(119905))

minus (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

minus 119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 120582

120575(119889119911

120575)

+

119873

sum

119895=1

120574120576

119903120576(119905)119895

(120593120576

(0) minus 119863 (120593120576

119895) 119905 119895)

= L119881 (120593120576

119905 119903120576

(119905))

+ 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times [119891 (120593120576

119905 119903120576

(119905)) minus 119891 (120593120576

119905 119903120576

(119905))]

+

119898

sum

120575=1

intR

119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905))

+ 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 119905 119903

120576

(119905))

minus 119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905))

+ (120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 119905 119903

120576

(119905))

times 120582120575(119889119911

120575)

minus

119898

sum

120575=1

intR

119881119909(120593

120576

(0)

minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times (119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)

minus(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)) 120582

120575(119889119911

120575)

+

119873

sum

119895=1

120574119903120576(119905)119895

(120593120576

(0) minus 119863 (120593120576

119895) 119905 119895)

minus

119897

sum

119896=1

120574119903120576(119905)119896

(120593120576

(0) minus 119863 (120593120576

119896) 119905 119896)

(81)

This implies that

119880 (119905 + ℎ) minus 119880 (119905)


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [L119881 (120593120576

119904 119903120576

(119904))

+ 120574119881 (120593120576

(0)

minus119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times [119891 (120593120576

119904 119903120576

(119904)) minus 119891 (120593120576

119904 119903120576

(119904))] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [

119898

sum

120575=1

intR

[119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904))

+119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575) 119904 119903

120576

(119904))

minus 119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904))

+ (120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

119904 119903120576

(119904) )]

times120582120575(119889119911

120575) ] 119889119904

minus lim sup120576rarr0

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times

119898

sum

120575=1

intR

[119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times (119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

minus(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575))]

times120582120575(119889119911

120575) 119889119904



E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times (

119873

sum

119895=1

120574119903120576(119904)119895

(120593120576

(0) minus 119863 (120593120576

119895) 119904 119895)

minus

119897

sum

119896=1

120574119903120576(119904)119896

(120593120576

(0) minus 119863 (120593120576

119896) 119904 119896))119889119904

= 1198691+ 119869

2+ 119869

3+ 119869

4+ 119869

5

(82)

By the definition of

119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) minus

(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)

=

119897

sum

119894=1

119873119896

sum

119895=1

119887(120575)

(119909120576

119905minus 119905 119904

119896119895 119911

120575)

times [119868119903120576(119905)=119904119896119895

minus 120583119896

119895119868119903120576(119905)=119896

]

(83)

By Assumption 8 we have

1198694= lim sup

120576rarr0

119898

sum

120575=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times intR

[119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

minus (120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)]

times 120582120575(119889119911

120575) 119889119904


119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904))

119904 119903120576

(119904))

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)

times [119868119903120576(119904)=119904119896119895

minus120583119896

119895119868119903120576(119904)=119896

]

times 120582120575(119889119911

120575) 119889119904


119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

[E

100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904))

119904 119903120576

(119904))

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

times120582120575(119889119911

120575) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

]

12

(84)


4= 0 Similarly

by mean-value theorem we can show that there exists 120578(120575)

119905

which is between 120593120576

(0) minus 119863(120593120576

119903120576

(119905)) + 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)

and 120593120576

(0) minus 119863(120593120576

119903120576

(119905)) + (120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) such that

1198693= lim

120576rarr0

119898

sum

120575=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

intR

119881119909(120578

119904)

times [119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

minus(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)]

times 120582120575(119889119911

120575) 119889119904

= lim120576rarr0

119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120578

119904)

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

times 120582120575(119889119911

120575) 119889119904

le lim120576rarr0

119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

[E

100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120578

119905)

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)


times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

times 120582120575(119889119911

120575)119889119904

100381610038161003816100381610038161003816100381610038161003816

2

]

12

(85)

By the argument of Lemma 714 in [4] we have 1198693= 0 Similar

to the proof ofTheorem 7 we derive 1198692= 0 119869

5= 0Therefore

we arrive at119880 (119905 + ℎ) minus 119880 (119905)

= lim120576rarr0

Eint

119905+120579+ℎ

119905+120579

119890120574119904

times [L119881 (120593120576

119904 119903120576

(119904))

+120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))] 119889119904

le 0

(86)

Then

119880 (119905 + ℎ) le 119880 (119905) (87)

Similar to the proof of Theorem 7 we get

E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

119890minus]119905

(88)

The proof is therefore completed

5 Examples

We will give two examples to illustrate our theory

Example 1 Let 119903120576(sdot) be a Markov chain generated by Γ120576 given

in (14) with

Γ = (

minus1 0 1 0 0

1 minus2 1 0 0

2 1 minus3 0 0

0 0 0 minus1 1

0 0 0 1 minus1

) (89)

Γ = (

minus1 0 1 0 0

0 minus1 0 1 0

0 0 minus1 0 1

0 1 0 minus1 0

1 0 0 0 minus1

) (90)

The generator Γ is made up of two irreducible blocks by

(1205871

1205872

1205873)(

minus1 0 1

1 minus2 1

2 1 minus3

) = 0 (91)

and 1205871+ 120587

2+ 120587

3= 1 we get 1205831

= (58 18 14) In the sameway by

(1205874

1205875) (

minus1 1

1 minus1) = 0 (92)

and 1205874+ 120587

5= 1 we have 120583

2

= (12 12) So

Γ = 120583Γ1 = (

5

8

1

8

1

40 0

0 0 01

2

1

2

)

times (

minus1 0 1 0 0

0 minus1 0 1 0

0 0 minus1 0 1

0 1 0 minus1 0

1 0 0 0 minus1

)(

1 0

1 0

1 0

0 1

0 1

)

= (minus3

8

3

81 minus1

)

(93)

Consider a one-dimensional neutral stochastic functionaldifferential equation as follows

119889 [120593120576

(0) minus 119863 (120593120576

119903120576

(119905))]

= 119891 (120593120576

119903120576

(119905)) 119889119905 + 119892 (120593120576

119903120576

(119905)) 119889119908 (119905)

(94)

with

119863(120593 11990411) = minus06 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990412) = minus02 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990413) = minus04 int

0

minus1

120593 (120579) 119889120579

119891 (120593 11990411) = minus16120593 (0) minus 8 cos [120593 (0)]

119891 (120593 11990412) = 8120593 (0) + 4 cos [120593 (0)]

119891 (120593 11990413) = 16120593 (0)

119892 (120593 11990411) =

radic10

10int

0

minus1

120593 (120579) 119889120579 cos [int0

minus1

120593 (120579) 119889120579]

119892 (120593 11990412) = minus

radic2

2int

0

minus1

120593 (120579) 119889120579 sin [int

0

minus1

120593 (120579) 119889120579]

119892 (120593 11990413) =

radic3

2int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990421) = 05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990422) = 05 int

0

minus1

120593 (120579) 119889120579

119891 (120593 11990421) = minus2120593 (0) 119891 (120593 119904

22) = minus2120593 (0)


119892 (120593 11990421) =

int

0

minus1

120593 (120579) 119889120579 sin [int

0

minus1

120593 (120579) 119889120579]

4radic2

119892 (120593 11990422) =

int

0

minus1

120593 (120579) 119889120579 cos [int0

minus1

120593 (120579) 119889120579]

4radic2

(95)

For any 120593 isin 1198712

F119905([minus1 0]R) and 120581 = max06 02 04 = 06

applying the Holder inequality yields

E1003816100381610038161003816119863 (120593 119894)

10038161003816100381610038162

le 062 supminus1le120579le0

119890]120579E

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

le 0361003817100381710038171003817120593

10038171003817100381710038172

(96)

which implies condition (24) Then the limit equation is

119889 [120593 (0) minus 119863 (120593 119903 (119905))] = 119891 (120593 119903 (119905)) 119889119905 + 119892 (120593 119903 (119905)) 119889119908 (119905)

(97)

where 119903 is the Markov chain generated by Γ and

119863(120593 1) = minus05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 2) = 05 int

0

minus1

120593 (120579) 119889120579

119891 (120593 1) = minus5120593 (0) 119891 (120593 2) = minus2120593 (0)

119892 (120593 1) =1

2int

0

minus1

120593 (120579) 119889120579 119892 (120593 2) =1

4int

0

minus1

120593 (120579) 119889120579

(98)

We define 119881(119909 1) = 21199092 119881(119909 2) = 119909

2 And by simplecalculation we can get

L119881 (120593 1) le minus203

81205932

(0) +13

32

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

L119881 (120593 2) le minus5

21205932

(0) +13

16

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

(99)

Consequently

max119894=12

L119881 (120593 119894) le minus5

21205932

(0) +13

16

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

= minus5

4[max119894=12

119881 (119909 119894)] +13

16[min119894=12

119881 (119909 119894)]

(100)

It is easy to find a 119902 gt 1 such that 54minus1311990216 gt 0Thereforefor any 120601 isin 119871

2

F119905([minus1 0]R) satisfying E[min

119894isinS120601(120579)] le

119902E[max119894isinS120601(0)] on minus1 le 120579 le 0 (100) yields

E [max119894isinS

L119881 (120593 119894)] le minus (5

4minus

13119902

16)E [max

119894=12

119881 (119909 119894)] (101)

Hence byTheorem 7 the solution 119909120576

(119905) is mean square stablewhen 120576 is sufficiently small

Example 2 Let 119903120576(sdot) be a Markov chain generated by

Γ120576

=1

120576Γ + Γ =

1

120576(

minus2 0 2 0

1 minus2 0 1

0 2 minus2 0

0 1 1 minus2

) (102)

Here we set Γ = 0 By a similar way we get the stationarydistribution 120583 = (211 411 311 211)

Consider the following one-dimensional equation

119889 [120593120576

(0) minus 119863 (120593120576

119903120576

(119905))]

= 119891 (120593120576

119903120576

(119905)) 119889119905 + int

infin

0

120590 (119903120576

(119905) 119911) 119909120576

119905minus (119889119905 119889119911)

(103)

with

119863(120593 1) = minus09 int

0

minus1

120593 (120579) 119889120579

119863 (120593 2) = minus04 int

0

minus1

120593 (120579) 119889120579

119863 (120593 3) = minus05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 4) = minus03 int

0

minus1

120593 (120579) 119889120579

119891 (120593 1) = 2 sin [120593 (0)] 119891 (120593 2) = minus11

2120593 (0)

119891 (120593 3) = minus11

3120593 (0) 119891 (120593 4) = minus2 sin [120593 (0)]

(104)

Let

120572 (119911) =2

11120590 (1 119911) +

4

11120590 (2 119911) +

3

11120590 (3 119911) +

2

11120590 (4 119911)

int

infin

0

1205722

(119911) 120582 (119889119911) lt 2

(105)

For any 120593 isin 1198712

F119905([minus1 0]R) and 120581 = max09 04 05 03 =

09 applying the Holder inequality yields

E1003816100381610038161003816119863 (120593 119894)

10038161003816100381610038162


119890]120579E

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

le 0811003817100381710038171003817120593

10038171003817100381710038172

(106)


119889 [120593 (0) + 05 int

0

minus1

120593 (120579) 119889120579]

= minus3120593 (0) 119889119905 + int

infin

0

120572 (119911) 119909119905minus (119889119905 119889119911)

(107)


Let 119881(119909) = 1199092 then

L119881 (120593 119894) le minus61205932

(0) + int

infin

0

1205722

(119911) 120582 (119889119911)

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

(108)

We can find a 119902 gt 1 such that 6minus2119902 gt 0Therefore for any120601 isin

1198712


119894isinS120601(120579)] le 119902E[max

119894isinS120601(0)]

on minus1 le 120579 le 0 (108) yields

E [max119894isinS

L119881 (120593 119894)] le minus (6 minus 2119902)E [max119894=12

119881 (119909 119894)] (109)

Hence by Theorem 12 the solution 119909120576

(119905) is mean squarestable

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Thispaperwas supported by theNational Science Foundationof China with Grant no 61374085

References

[1] Y Shen and J Wang ldquoNoise-induced stabilization of therecurrent neural networks with mixed time-varying delays andMarkovian-switching parametersrdquo IEEETransactions onNeuralNetworks vol 18 no 6 pp 1857ndash1862 2007

[2] Y Shen and J Wang ldquoAlmost sure exponential stability ofrecurrent neural networks with Markovian switchingrdquo IEEETransactions on Neural Networks vol 20 no 5 pp 840ndash8552009

[3] H A Simon and A Ando ldquoAggregation of variables in dynamicsystemsrdquo Econometrica vol 29 pp 111ndash138 1961

[4] G G Yin and Q Zhang Continuous-Time Markov Chains andApplications A Singular Perturbations Approach Springer NewYork NY USA 1998

[5] H Chen C Zhu and Y Zhang ldquoA note on exponential stabilityfor impulsive neutral stochastic partial functional differentialequationsrdquo Applied Mathematics and Computation vol 227 pp139ndash147 2014

[6] G Hu and K Wang ldquoStability in distribution of neutralstochastic functional differential equations with Markovianswitchingrdquo Journal of Mathematical Analysis and Applicationsvol 385 no 2 pp 757ndash769 2012

[7] S Jankovic M Vasilova andM Krstic ldquoSome analytic approxi-mations for neutral stochastic functional differential equationsrdquoApplied Mathematics and Computation vol 217 no 8 pp 3615ndash3623 2010

[8] V Kolmanovskii N Koroleva T Maizenberg X Mao and AMatasov ldquoNeutral stochastic differential delay equations withMarkovian switchingrdquo Stochastic Analysis and Applications vol21 no 4 pp 819ndash847 2003

[9] D Li and D Xu ldquoAttracting and quasi-invariant sets ofstochastic neutral partial functional differential equationsrdquoActaMathematica Scientia B vol 33 no 2 pp 578ndash588 2013

[10] X Mao ldquoExponential stability in mean square of neutralstochastic differential-functional equationsrdquo Systems amp ControlLetters vol 26 no 4 pp 245ndash251 1995

[11] X Mao ldquoRazumikhin-type theorems on exponential stabilityof neutral stochastic functional-differential equationsrdquo SIAMJournal on Mathematical Analysis vol 28 no 2 pp 389ndash4011997

[12] Y Song and Y Shen ldquoNew criteria on asymptotic behavior ofneutral stochastic functional differential equationsrdquo Automat-ica vol 49 no 2 pp 626ndash632 2013

[13] F Wu S Hu and C Huang ldquoRobustness of general decaystability of nonlinear neutral stochastic functional differentialequations with infinite delayrdquo Systems amp Control Letters vol 59no 3-4 pp 195ndash202 2010

[14] S Zhou and S Hu ldquoRazumikhin-type theorems of neutralstochastic functional differential equationsrdquo Acta MathematicaScientia B vol 29 no 1 pp 181ndash190 2009

[15] G Badowski and G G Yin ldquoStability of hybrid dynamicsystems containing singularly perturbed random processesrdquoIEEE Transactions on Automatic Control vol 47 no 12 pp2021ndash2032 2002

[16] J Hu X Mao and C Yuan ldquoRazumikhin-type theorems onexponential stability of SDDEs containing singularly perturbedrandom processesrdquo Abstract and Applied Analysis vol 2013Article ID 854743 12 pages 2013

[17] A A Pervozvanskii and V G Gaitsgori Theory of SuboptimalDecisions Decomposition and Aggregation Kluwer AcademicDordrecht The Netherlands 1988

[18] F Wu G G Yin and L Y Wang ldquoStability of a pure randomdelay system with two-time-scale Markovian switchingrdquo Jour-nal of Differential Equations vol 253 no 3 pp 878ndash905 2012

[19] FWu G Yin and L YWang ldquoMoment exponential stability ofrandom delay systems with two-time-scale Markovian switch-ingrdquo Nonlinear Analysis Real World Applications vol 13 no 6pp 2476ndash2490 2012

[20] G Yin and H Yang ldquoTwo-time-scale jump-diffusion modelswith Markovian switching regimesrdquo Stochastics and StochasticsReports vol 76 no 2 pp 77ndash99 2004

[21] C Yuan andG Yin ldquoStability of hybrid stochastic delay systemswhose discrete components have a large state space a two-time-scale approachrdquo Journal of Mathematical Analysis andApplications vol 368 no 1 pp 103ndash119 2010

[22] R Z Khasminskii G Yin and Q Zhang ldquoAsymptotic expan-sions of singularly perturbed systems involving rapidly fluctu-ating Markov chainsrdquo SIAM Journal on Applied Mathematicsvol 56 no 1 pp 277ndash293 1996

[23] H J Kushner Approximation and Weak Convergence Methodsfor Random Processes with Applications to Stochastic SystemsTheory The MIT Press Cambridge Mass USA 1984

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


motion and Poisson jump by utilizing perturbed Lyapunovfunction methods and Razumikhin-type criteria

The remainder of this paper is organized as follows InSection 2 we introduce some notations and notions neededin our investigation In Section 3 we state our main resultsthat is exponential stability of neutral stochastic functionaldifferential equations with two-time-scaleMarkovian switch-ingThe exponential stability for neutral stochastic functionaldifferential equations driven by pure jumps is also discussedin Section 4 Finally two examples are presented to justify andillustrate applications of the theory in Section 5

2 Preliminaries

Throughout this paper unless otherwise specified let(ΩF F

119905119905ge0

P) be a complete probability space with afiltration F

119905119905ge0

satisfying the usual conditions (ie it isincreasing and right continuous and F

0contains all P-

null sets) Let 119882(119905) = (1198821(119905) 119882

119898(119905))

119879 be an 119898-dimensional Brownian motion defined on the probabilityspace For 120591 gt 0 let 119862([minus120591 0]R119899

) denote the family ofcontinuous functions 120593 from [minus120591 0] to R119899 with norm 120593 =


|120593(120579)| where | sdot | is the Euclidean norm in R119899If 119860 is a vector or matrix its transpose is denoted by 119860

119879while its trace norm is denoted by |119860| = radictrace(119860119879119860)Denote by 119862

119887

F0([minus120591 0]R119899

) the family of all F0measurable

and bounded 119862([minus120591 0]R119899

)-valued random variables For119901 gt 0 and 119905 ge 0 denote by 119871

119901

F119905([minus120591 0]R119899

) the familyof allF

119905-measurable119862([minus120591 0]R119899

)-valued random variables120601 = 120601(120579) minus120591 le 120579 le 0 such that sup

minus120591le120579le0119864|120601(120579)|

119901

lt infinWe will denote the indicator function of a set 119866 by 119868

119866

Consider an 119899-dimensional neutral stochastic functionaldifferential equation with Markovian switching as follows

119889 [119909 (119905) minus 119863 (119909119905 119903 (119905))]

= 119891 (119909119905 119905 119903 (119905)) 119889119905 + 119892 (119909

119905 119905 119903 (119905)) 119889119908 (119905)

(1)

on 119905 ge 0 with initial data 1199090

= 120585 isin 119862([minus120591 0]R119899

) and 119909119905=

119909(119905 + 120579) minus120591 le 120579 le 0 which is regarded as a 119862([minus120591 0]R119899

)-valued stochastic process Moreover 119891 119862([minus120591 0]R119899

) times

R+

times S rarr R119899 119892 119862([minus120591 0]R119899

) times R+

times S rarr R119899times119898119863 119862([minus120591 0]R119899

) times S rarr R119899Let 119903(119905) (119905 ge 0) be a right-continuous Markov chain on

the probability space taking values in a finite state space S =

1 2 119872 with generator Γ = (120574119894119895)119872times119872

given by

P 119903 (119905 + Δ) = 119895 | 119903 (119905) = 119894 = 120574119894119895Δ + ∘ (Δ) if 119894 = 119895

1 + 120574119894119894Δ + ∘ (Δ) if 119894 = 119895

(2)

where Δ gt 0 Here 120574119894119895

ge 0 is the transition rate from 119894 to 119895 if119894 = 119895 while 120574

119894119894= minussum

119894 = 119895120574119894119895

We assume the Markov 119903(sdot) is independent of the Brown-ian motion 119882(sdot) It is well known that almost every samplepath 119903(sdot) is a right-continuous step function with finitenumber of simple jumps in any finite subinterval of R

+=

[0infin) As a standing hypothesis we assume that the Markov

chain is irreducible This is equivalent to the condition thatfor any 119894 119895 isin S we can find 119894

1 1198942 119894

119896isin S such that

1205741198941198941

12057411989411198942

sdot sdot sdot 120574119894119896119895

gt 0 (3)

Then Γ always has an eigenvalue 0 The algebraic interpreta-tion of irreducibility is rank (Γ) = 119872minus1 Under this conditiontheMarkov chain has a unique stationary distribution120587Γ = 0subject to sum

119872

119895=1120587119895

= 1 and 120587119895

gt 0 for all 119895 isin S For a real-valued function 120590(sdot) defined on S we define

Γ120590 (sdot) (119894) = sum

119895isinS

120574119894119895120590 (119895) = sum

119895 = 119894

120574119894119895(120590 (119895) minus 120590 (119894)) (4)

for each 119894 isin SLet 119862

21

(R119899

times R+

times SR+) denote the family of all

nonnegative functions 119881(119909 119905 119894) on R119899

times R+

times S which arecontinuously twice differentiable in 119909 and once differentiablein 119905 If 119881(119909 119905 119894) isin 119862

21

(R119899

times R+times SR


L119881 from 119862([minus120591 0]R119899

) times R+times S to R by

L119881 (120593 119905 119894) = 119881119905(120593 (0) minus 119863 (120593 119894) 119905 119894)

+ 119881119909(120593 (0) minus 119863 (120593 119894) 119905 119894) 119891 (120593 119905 119894)

+1

2trace [119892

119879

(120593 119905 119894)

times 119881119909119909

(120593 (0) minus 119863 (120593 119894) 119905 119894)

times 119892 (120593 119905 119894) ]

+

119897

sum

119895=1

120574119894119895119881 (120593 (0) minus 119863 (120593 119895) 119905 119895)

(5)

where

120593 isin 119862 ([minus120591 0] R119899

) 119881119905=

120597119881 (119909 119905 119894)

120597119905

119881119909= (

120597119881 (119909 119905 119894)

1205971199091

120597119881 (119909 119905 119894)

1205971199092

120597119881 (119909 119905 119894)

120597119909119899

)

119881119909119909

= (1205972

119881(119909 119905 119894)

120597119909119894120597119909

119895

)

119899times119899

(6)

For a parameter 120576 gt 0 we rewrite the Markov chain 119903(119905)

as 119903120576

(119905) and the generator Γ as Γ120576 Γ120576 is given by

Γ120576

=1

120576Γ + Γ (7)

where Γ120576 represents the fast varying motions and Γ rep-resents the slowly changing dynamics Set Γ

120576

= (120574120576

119894119895)119872times119872

Γ = (120574

119894119895)119872times119872

and Γ = (120574119894119895)119872times119872

For the sake of simplicitysuppose that

S = S1

cup S2

cup sdot sdot sdot cup S119897

(8)

with S119896

= 1199041198961

119904119896119872119896

119872 = 1198721+ 119872

2+ sdot sdot sdot + 119872

119897 and

Γ = diag (Γ1

Γ119897

) (9)






119889 [119909120576

minus 119863 (119909120576

119905 119903

120576

(119905))]

= 119891 (119909120576

119905 119905 119903

120576

(119905)) 119889119905 + 119892 (119909120576

119905 119905 119903

120576

(119905)) 119889119908 (119905)

119909120576

0= 120585 isin 119862 ([minus120591 0] R

119899

) 119903120576

= 1199030

(10)




1003816100381610038161003816119891 (120593 119905 119894) minus 119891 (120601 119905 119894)1003816100381610038161003816 or

1003816100381610038161003816119892 (120593 119905 119894) minus 119892 (120601 119905 119894)1003816100381610038161003816

le 119871120572

1003817100381710038171003817120593 minus 12060110038171003817100381710038172

(11)


) with 120593 or




1003816100381610038161003816119891 (120593 119905 119894)10038161003816100381610038162

or1003816100381610038161003816119892 (120593 119905 119894)

10038161003816100381610038162

le 119871 (1 +1003817100381710038171003817120593

10038171003817100381710038172

) (12)



1003816100381610038161003816119863 (120593 119894) minus 119863 (120601 119894)1003816100381610038161003816 le 120581

1003817100381710038171003817120593 minus 12060110038171003817100381710038172

119863 (0 119894) equiv 0

(13)


120576120585119894




sup(120585119894)isin119866timesS

119864[ supminus120591le119904le119905

10038161003816100381610038161003816119909120576120585119894

(119904)10038161003816100381610038161003816

119901

] le 119867 119905 ge 0 (14)



(sdot) as

119903120576

(119905) = 119896 if 119903120576

(119905) isin S119896

(15)


119896 by 120583119896

= (120583119896

1 120583

119896

119872119896

) isin R1times119872119896 and 120583 =

diag(1205831

120583119897


Γ = (120574119894119895)119897times119897

= 120583Γ1 (16)

with 1 = diag(11198721

1119872119897

) and 1119872119896

= (1 1)119879


119863(120593 119896) =

119872119896

sum

119895=1

120583119896

119895119863(120593 119904

119896119895)

119891 (120593 119905 119896) =

119872119896

sum

119895=1

120583119896

119895119891 (120593 119905 119904

119896119895)

119892 (120593 119905 119896) 119892119879

(120593 119905 119896) =

119872119896

sum

119895=1

120583119896

119895119892 (120593 119905 119904

119896119895) 119892

119879

(120593 119905 119904119896119895)

(17)

for each 119904119896119895



119896119895)119892

119879

(120593 119905 119904119896119895)


119896119895)119892

119879



119889 [120593 (0) minus 119863 (120593 119903 (119905))]

= 119891 (120593 119905 119903 (119905)) 119889119905 + 119892 (120593 119905 119903 (119905)) 119889119908 (119905)

1199090= 120585 119903 = 119903

0

(18)



119901

(R119899

times

R+times SR


defined on R119899

times R+


119862119901

(R119899

times R+times SR

+) for some 119901 ge 4


infin Moreover 120597119901119881(119909 119905 119894) = 119874(1) 120597120580119881(119909 119905 119894)(|119909|120580

+ |119910|120580

) le

119870(|119909|119901

+|119910|119901


119910|119874(119910)|119910 lt infin


E1003816100381610038161003816119863 (120593 119896)

1003816100381610038161003816119901

le 120581119901 supminus120591le120579le0

1198901205741205791003817100381710038171003817120593

1003817100381710038171003817119901

(120593 119894) isin 119871119901

F119905([minus120591 0] R

119899

) times S

(19)


Then for any 120585 isin 119871119901

F0([minus120591 0]R119899



119890120574119904

E |119909 (119904)|119901

le

10038171003817100381710038171205851003817100381710038171003817119901

1 minus 120581orsup

0le119904le119905119890120574119904E

1003816100381610038161003816119909 (119904) minus 119863 (119909119904 119903 (119904))

1003816100381610038161003816119901

(1 minus 120581)119901

119905 ge 0

(20)


(119909 + 119910)119901

= (1 minus 1205811)1minus119901

(119909119901

+ 1205811

1minus119901

119910119901

)

forall119909 119910 ge 0 1205811gt 0

(21)


119890120574119905

E|119909 (119905)|119901

le 119890120574119905

[(1 minus 120581)1minus119901

E1003816100381610038161003816119909 (119905) minus 119863 (119909

119905 119903 (119905))

1003816100381610038161003816119901

+ 1205811minus119901

E1003816100381610038161003816119863 (119909

119905 119903 (119905))

1003816100381610038161003816119901

]

le (1 minus 120581)1minus119901

119890120574119905

E1003816100381610038161003816119909 (119905) minus 119863 (119909

119905 119903 (119905))

1003816100381610038161003816119901

+ 120581119890120574119905 supminus120591le120579le0

119890120574120579

E |119909 (119905 + 120579)|119901


0le119904le119905

119890120574119904

E1003816100381610038161003816119909 (119904) minus 119863 (119909

119904 119903 (119904))

1003816100381610038161003816119901

+ 120581 supminus120591le120579le0

119890120574(119904+120579)

E |119909 (119904 + 120579)|119901


0le119904le119905

119890120574119904

E1003816100381610038161003816119909 (119904) minus 119863 (119909

119904 119903 (119904))

1003816100381610038161003816119901


119890120574119904

E |119909 (119904)|119901

(22)

Then


119890120574119904

E |119909 (119904)|119901


E |119909 (120579)|2

]


0le119904le119905

119890120574119904

E1003816100381610038161003816119909 (119904) minus 119863 (119909

119904 119903 (119904))

1003816100381610038161003816119901


119890120574119904

E |119909 (119904)|119901

]

(23)



2 120582 119901 be all


119901

(R119899

timesR+timesSR


that

1198881|119909|

119901

le 119881 (119909 119905 119896) le 1198882|119909|

119901

(24)

for all (119909 119905 119896) isin R119899

times R+


E1003816100381610038161003816119863 (120593 119896)

1003816100381610038161003816119901

le 120581119901 supminus120591le120579le0

119890]1205791003817100381710038171003817120593

1003817100381710038171003817119901

120581 = max 1205811 120581

119896 120593 isin 119871

119901

F119905

(25)


E [max119896isin

S

L119881 (120593 119905 119896)]


S

119881 (120593 (0) minus 119863 (120593 119896) 119905 119896)]

(26)


F119905([minus120591 0]R119899

)satisfying

E [min119896isin

S

119881 (120593 (120579) 119905 + 120579 119896)]


S

119881 (120593 (0) minus 119863 (120593 119896) 119905 119896)]

(27)


F0([minus120591 0]R119899

) 119905 ge 0

lim sup120576rarr0

E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

119890minus]119905

(28)

where

] = min1205741

120591log

119902

(11988821198881) (1 minus 120581)

119901 (29)


1198882

1198881

(1 minus 120581)119901

119890120574120591

= 120582 (30)


Proof Let

(120593 119905 119895) =

119897

sum

119896=1

119881 (120593 119905 119896) 119868119895isinS119896 = 119881 (120593 119905 119896) if 119895 isin S

119896

(31)


(120593120576

119905 119903120576

(119905)) = 119881 (120593120576

119905 119903120576

(119905))

119872

sum

119894=1

120574119897119894 (120593 119905 119894) =

119872

sum

119894=1

120574119897119894

119897

sum

119896=1

119881 (120593 119905 119896) 119868119894isinS119896 = 0

(32)




119880 (119905) = supminus120591le120579le0

[119890120574(119905+120579)E119881 (119909

120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ]


[119890120574(119905+120579)

E (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ]

(33)


119863+

119880 (119905) = lim supℎrarr0+

119880 (119905 + ℎ) minus 119880 (119905)

ℎle 0 forall119905 ge 0 (34)


120576

(119905) minus

119863(119909120576

119905 119903

120576

(119905)) 119905 119903120576

(119905)) or 119880(119905) = 119890120574119905E119881(119909

120576

(119905) minus 119863(119909120576

119905 119903

120576

(119905))

119905 119903120576

(119905))If 119880(119905) gt 119890

120574119905E119881(119909120576

(119905) minus 119863(119909120576

119905 119903

120576

(119905)) 119905 119903120576


119890120574(119905+ℎ)E119881(119909

120576

(119905 + ℎ) minus119863(119909120576

119905+ℎ 119903

120576

(119905 + ℎ)) 119905 + ℎ 119903120576

(119905 + ℎ)) hence119880(119905 + ℎ) le 119880(119905) and 119863

+

119880(119905) le 0If 119880(119905) = 119890

120574119905E119881(119909120576

(119905) minus 119863(119909120576

119905 119903

120576

(119905)) 119905 119903120576

(119905)) we have

119890120574(119905+120579)

E119881 (119909120576

(119905 + 120579) minus 119863 (119909119905+120579

119903120576

(119905 + 120579)) 119905 + 120579 119903120576

(119905 + 120579))

le 119890120574119905

E119881 (119909120576

(119905) minus 119863 (119909119905 119903

120576

(119905)) 119905 119903120576

(119905))

(35)


E119881 (119909120576

(119905 + 120579) minus 119863 (119909119905+120579

119903120576

(119905 + 120579)) 119905 + 120579 119903120576

(119905 + 120579))

le 119890minus120574120579

E119881 (119909120576

(119905) minus 119863 (119909119905 119903

120576

(119905)) 119905 119903120576

(119905))

le 119890120574120591

E119881 (119909120576

(119905) minus 119863 (119909119905 119903

120576

(119905)) 119905 119903120576

(119905))

(36)


119890120574(119905+120579)

E119881 (119909120576

(119905 + 120579) 119905 + 120579 119903120576

(119905 + 120579))

le 1198882119890120574(119905+120579)

E1003816100381610038161003816119909

120576

(119905 + 120579)1003816100381610038161003816119901

le 1198882(1 minus 120581)

119901 sup0le119904le119905

119890120574119904

E1003816100381610038161003816119909

120576

(119904) minus 119863 (119909120576

119904 119903 (119904))

1003816100381610038161003816119901

le1198882

1198881

(1 minus 120581)119901 sup0le119904le119905

119890120574119904

E119881 (119909120576

(119904) minus 119863 (119909120576

119904 119903 (119904)) 119904 119903 (119904))

le1198882

1198881

(1 minus 120581)119901

119890120574119905

E119881 (119909120576

(119905) minus 119863 (119909120576

119905 119903 (119905)) 119905 119903 (119905))

(37)

Then

E119881 (119909120576

(119905 + 120579) 119905 + 120579 119903120576

(119905 + 120579))

lt 119902E119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

(38)

where 119902 gt (11988821198881)(1 minus 120581)

119901

119890120574120591 that is 120574 lt (1120591)(log(119902(119888

21198881)

(1 minus 120581)119901


0gt 0 such

that for any 120576 isin (0 1205760)

E [min119896isin

S

119881 (120593120576

(120579) 119905 + 120579 119896)]


S

119881 (120593120576

(0) minus 119863 (120593120576

119896) 119905 119896)]

(39)


E [max119896isin

S

L119881 (120593120576

119905 119903120576

(119905))]


S

119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(40)

which implies that

E [L119881 (120593120576

119905 119903120576

(119905))]

le minus120582E [119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(41)


E [L119881 (120593120576

119905 119903120576

(119905))]

le minus120574E [119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(42)

Next we consider

119880 (119905 + ℎ) minus 119880 (119905)


[119890120574(119905+120579+ℎ)

E [119881 (119909120576

(119905 + 120579 + ℎ)

minus 119863 (119909120576

119905+120579+ℎ 119903

120576

(119905 + 120579 + ℎ))

119905 + 120579 + ℎ 119903120576

(119905 + 120579 + ℎ)) ]

minus 119890120574(119905+120579)

E [119881 (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579))]]


Eint

119905+120579+ℎ

119905+120579

119890120574119904

[L119881 (120593120576

119904 119903120576

(119904))

+ 120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904))

119904 119903120576

(119904))] 119889119904


Eint

119905+120579+ℎ

119905+120579

119890120574119904

[L (120593120576

119904 119903120576

(119904))

+ 120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904))

119904 119903120576

(119904)) ] 119889119904

(43)



L (120593120576

119905 119903120576

(119905))

= 119905(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

+ 119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119891 (120593120576

119905 119903120576

(119905))

+1

2trace [119892

119879

(120593120576

119905 119903120576

(119905))

times 119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) ]

+

119872

sum

119894=1

120574120576

119903120576(119905)119894

(120593120576

(0) minus 119863 (119909120576

119905 119894) 119905 119894)

= 119905(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

+ 119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119891 (120593120576

119905 119903120576

(119905))

+1

2trace [119892

119879

(120593120576

119905 119903120576

(119905))

times 119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) ]

+

119872

sum

119894=1

120574119903120576(119905)119894

(120593120576

(0) minus 119863 (119909120576

119905 119894) 119905 119894)

= 119881119905(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

+ 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119891 (120593120576

119905 119903120576

(119905))

+1

2trace [119892

119879

(120593120576

119905 119903120576

(119905))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) ]

+

119897

sum

119896=1

120574120576

119903120576(119905)119896

119881 (120593120576

(0) minus 119863 (119909120576

119905 119896) 119905 119896)

+ 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times [119891 (120593120576

119905 119903120576

(119905)) minus 119891 (120593120576

119905 119903120576

(119905))]

+1

2trace [119892

119879

(120593120576

119905 119903120576

(119905))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) minus 119892119879

(120593120576

119905 119903120576

(119905))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) ]

+

119872

sum

119894=1

120574119903120576(119905)119894

(120593120576

(0) minus 119863 (119909120576

119905 119894) 119905 119894)

minus

119897

sum

119896=1

120574119903120576(119905)119896

119881 (120593120576

(0) minus 119863 (119909120576

119905 119896) 119905 119896)

= L119881 (120593120576

119905 119903120576

(119905))

+ 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times [119891 (120593120576

119905 119903120576

(119905)) minus 119891 (120593120576

119905 119903120576

(119905))]

+1

2trace [119892

119879

(120593120576

119905 119903120576

(119905))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) minus 119892119879

(120593120576

119905 119903120576

(119905))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) ]

+

119872

sum

119894=1

120574119903120576(119905)119894

(120593120576

(0) minus 119863 (119909120576

119905 119894) 119905 119894)

minus

119897

sum

119896=1

120574119903120576(119905)119896

119881 (120593120576

(0) minus 119863 (119909120576

119905 119896) 119905 119896)

(44)

Therefore

119880 (119905 + ℎ) minus 119880 (119905)


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [L119881 (120593120576

119904 119903120576

(119904))

+120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119905

119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times [119891 (120593120576

119904 119903120576

(119904)) minus 119891 (120593120576

119904 119903120576

(119904))] 119889119904

+1

2lim sup120576rarr0

E

times int

119905+120579+ℎ

119905+120579

119890120574119904



(120593120576

119904 119903120576

(119904))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times 119892 (120593120576

119904 119903120576

(119904)) minus 119892119879

(120593120576

119904 119903120576

(119904))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times 119892 (120593120576

119904 119903120576

(119904)) ] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times (

119872

sum

119894=1

120574119903120576(119904)119894

(120593120576

(0) minus 119863 (119909120576

119904 119894) 119904 119894)

minus

119897

sum

119896=1

120574119903120576(119904)119896

119881 (120593120576

(0) minus 119863 (119909120576

119904 119896) 119904 119896))119889119904

= 1198681+ 119868

2+ 119868

3+ 119868

4

(45)


119891 (120593120576

119905 119903120576

(119905)) minus 119891 (120593120576

119905 119903120576

(119905))

=

119897

sum

119896=1

119872119896

sum

119895=1

119891 (120593120576

119905 119904119896119895) times [119868

119903120576(119905)=119904119896119895

minus 120583119896

119895119868119903120576(119905)=119896

]

(46)

This implies that

lim120576rarr0

Eint

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times [119891 (120593120576

119904 119903120576

(119904)) minus 119891 (120593120576

119904 119903120576

(119904))] 119889119904

le lim120576rarr0

[E

100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times [119891 (120593120576

119904 119903120576

(119904)) minus 119891 (120593120576

119904 119903120576

(119904))] 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

]

12

= lim120576rarr0

[

[

E

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times

119897

sum

119896=1

119872119896

sum

119895=1

119891 (120593120576

119904 119904119896119895)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

] 119889119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

]

]

12

le lim120576rarr0

[

[

E

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119897

sum

119896=1

119872119896

sum

119895=1

119890120574119904

119871 (1 +1003817100381710038171003817120593

1003817100381710038171003817119901

)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

] 119889119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

]

]

12

(47)


2= 0 Similarly

we can show that

1198683=

1

2lim sup120576rarr0

E

times int

119905+120579+ℎ

119905+120579

119890120574119904


(120593120576

119904 119903120576

(119904))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times 119892 (120593120576

119904 119903120576

(119904))

minus 119892119879

(120593120576

119904 119903120576

(119904))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times 119892 (120593120576

119904 119903120576

(119904)) ] 119889119904 = 0

(48)


119872

sum

119894=1

120574119903120576(119905)119894

(120593120576

(0) minus 119863 (119909120576

119905 119894) 119905 119894)

= Γ (120593120576

(0) minus 119863 (119909120576

119905 119894) 119905 sdot) (119903

120576

(119905))

119897

sum

119896=1

120574119903120576(119905)119896

119881 (120593 (0) minus 119863 (119909120576

119905 119896) 119905 119896)

= Γ119881 (120593 (0) minus 119863 (119909120576

119905 119896) 119905 sdot) (119903

120576

(119905))

(49)

Hence

1198684= lim sup

120576rarr0

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times (

119872

sum

119894=1

120574119903120576(119904)119894

(120593120576

(0) minus 119863 (119909120576

119904 119894) 119904 119894)

minus

119897

sum

119896=1

120574119903120576(119904)119896

119881 (120593120576

(0) minus 119863 (119909120576

119904 119896) 119904 119896))119889119904



E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times (Γ (120593120576

(0) minus 119863 (119909120576

119904 119894) 119904 sdot) (119903

120576

(119904))

minus Γ119881 (120593120576

(0) minus 119863 (119909120576

119904 119896) 119904 sdot) (119904

119896119895)) 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

119897

sum

119896=1

119872119896

sum

119895=1

Γ (120593120576

(0) minus 119863 (119909120576

119904 119894) 119904 sdot) (119904

119896119895)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

] 119889119904


[

[

E

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

times

119897

sum

119896=1

119872119896

sum

119895=1

Γ (120593120576

(0) minus 119863 (119909120576

119904 119894) 119904 sdot) (119904

119896119895)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

]

]

12

(50)


4= 0 Therefore

119880 (119905 + ℎ) minus 119880 (119905)

= lim120576rarr0

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [L119881 (120593120576

119904 119903120576

(119904))

+120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))] 119889119904

le 0

(51)

That is

119880 (119905 + ℎ) le 119880 (119905) (52)


+



lim sup120576rarr0

119890120574119905

E1003816100381610038161003816119909

120576

(119905) minus 119863 (119909120576

119905 119903

120576

(119905))1003816100381610038161003816119901


sup120591le120579le0

119890120574120579

E1003816100381610038161003816119909

120576

(120579) minus 119863 (119909120576

120579 119903

120576

(120579))1003816100381610038161003816

119901



(1 + 120581)119901minus1

times [E1003816100381610038161003816119909

120576

(120579)1003816100381610038161003816119901

+ 1205811minus119901

E1003816100381610038161003816119863 (119909

120576

120579 119903

120576

(120579))1003816100381610038161003816119901

]

le 1198882(1 + 120581)

11990110038171003817100381710038171205851003817100381710038171003817119901

119905 ge 0

(53)


lim sup120576rarr0

119890120574119905

E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

(54)

That is

lim sup120576rarr0

E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

119890minus120574119905

forall119905 ge 0 (55)



119889 [120593120576

(0) minus 119863 (120593120576

119903120576

(119905))]

= 119891 (120593120576

119905 119903120576

(119905)) 119889119905 + intR119898

119887 (119909120576

119905minus 119905 119903

120576

(119905) 119911) (119889119905 119889119911)

1199090= 120585 isin 119862 ([minus120591 0] R

119899

) 119903 (0) isin S

(56)

where 119909120576

119905minus= lim

119904uarr119905119909120576

119904 119863 119862([minus120591 0]R119899

) times S rarr R119899 119887

119862([minus120591 0]R119899





119887(120575)

(120593 119905 119894 119911) = 119887(120575)

(120593 119905 119894 119911120575)

119911 = (1199111 119911

119898) isin R

119898

119894 isin S

(57)


(119889119905 119889119911) = (1(119889119905 119889119911

1)

119898(119889119905 119889119911

119898))

= (1198731(119889119905 119889119911

1) minus 120582

1(119889119911

1) 119889119905 119873

119898(119889119905 119889119911

119898)

minus 120582119898

(119889119911119898) 119889119905)

(58)






119889 [120593120576

(0) minus 119863 (119909120576

119905 119903

120576

(119905))]

= 119891 (119909120576

119905 119905 119903

120576

(119905)) 119889119905

+ intR119898

(119909120576

119905minus 119905 119903

120576

(119905) 119911) (119889119905 119889119911)

1199090= 120585 isin 119862 ([minus120591 0] R

119899

) 119903 (0) isin S

(59)

where 119909120576

119905minus= lim

119904uarr119905119909120576

119904and 119862([minus120591 0]R119899


rarr


119863(120593 119896) =

119873119898

sum

119895=1

120583119896

119895119863(120593 119904

119896119895)

(120593 119905 119896 119911) =

119873119898

sum

119895=1

120583119896

119895119887 (120593 119905 119904

119896119895 119911)

(60)

for each 119904119896119895




gt 0such that

1003816100381610038161003816119891 (120593 119905 119894) minus 119891 (120601 119905 119894)1003816100381610038161003816

or

119898

sum

120575=1

intR

10038161003816100381610038161003816119887(120575)

(120593 119905 119894 119911120575) minus 119887

(120575)

(120601 119905 119894 119911120575)10038161003816100381610038161003816120582120575(119889119911

120575)

le 119871120577

1003817100381710038171003817120593 minus 12060110038171003817100381710038172

(61)




119862([minus120591 0]R119899

) 119894 isin S

1003816100381610038161003816119891 (120593 119905 119894)1003816100381610038161003816 or

119898

sum

120575=1

intR

10038161003816100381610038161003816119887(120575)

(120593 119905 119894 119911120575)10038161003816100381610038161003816120582120575(119889119911

120575)

le 119871 (1 +1003817100381710038171003817120593

10038171003817100381710038172

)

(62)



1003816100381610038161003816119863 (120593 119894) minus 119863 (120601 119894)1003816100381610038161003816 le 120581

1003817100381710038171003817120593 minus 12060110038171003817100381710038172

119863 (0 119894) equiv 0

(63)

Given that 119881 isin 119862119901

(R119899

times R+times SR


L119881 by

L119881 (120593 119905 119894)

= 119881119905(120593 (0) minus 119863 (120593 119894) 119905 119894)

+ 119881119909(120593 (0) minus 119863 (120593 119894) 119905 119894) 119891 (120593 119905 119894)

+

119873

sum

119895=1

120574119894119895119881 (120593 (0) minus 119863 (120593 119894) 119905 119895)

+ intR

119898

sum

120575=1

119881 (120593 (0) minus 119863 (120593 119894) + 119887(120575)

(120593 119905 120580 119911120575) 119905 120580)

minus 119881 (120593 (0) minus 119863 (120593 119894) 119905 119894)

minus 119881119909(120593 (0) minus 119863 (120593 119894) 119905 119894)

times 119887(120575)

(120593 119905 120580 119911120575) 120582

120575(119889119911

120575)

(64)

where

119881119909(120593 (0) minus 119863 (120593 119894) 119905 119894)

= (120597119881 (120593 (0) minus 119863 (120593 119894) 119905 119894)

1205971199091

120597119881 (120593 (0) minus 119863 (120593 119894) 119905 119894)

120597119909119898

)

(65)


(sdot) 119903120576






2


119901

(R119899

times R+times SR

+) satisfying


1198881|119909|

119901

le 119881 (119909 119905 119896) le 1198882|119909|

119901

119896 isin S (66)

for all (119909 119905 119896) isin R119899


following

E1003816100381610038161003816119863 (120593 119896)

1003816100381610038161003816119901

le 120581119901 supminus120591le120579le0

119890]120579 1003817100381710038171003817120593

1003817100381710038171003817

119901

120581 = max 1205811 120581

119896 120593 isin 119871

119901

F119905

(67)


E [max119896isin

S


S

119881 (120593 (0) minus 119863 (120593 119896))]

(68)



F119905([minus120591 0]R119899

)satisfying

E [min119896isin

S

119881 (119909 (119905 + 120579) 119905 + 120579 119896)]


S

119881 (120593 (0) minus 119863 (120593 119896) 119905 119894)] minus120591 le 120579 le 0

(69)


) 119905 ge 0

lim sup120576rarr0

E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

119890minus]119905

(70)

where

] = min1205741

120591log

119902

(11988821198881) (1 minus 120581)

119901 (71)


1198882

1198881

(1 minus 120581)119901

119890120574120591

= 120582 (72)

Proof Define

(120593 119905 120588) =

119897

sum

119896=1

119881 (120593 119905 119896) 119868120588isinS119896 = 119881 (120593 119905 119896) if 120588 isin S

119896

(73)


119880 (119905)


[119890120574(119905+120579)

E119881 (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ]


[119890120574(119905+120579)

E (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ]

(74)


119863+

119880 (119905) = lim supℎrarr0+

119880 (119905 + ℎ) minus 119880 (119905)

ℎle 0 forall119905 ge 0 (75)


E119881 (119909120576

(119905 + 120579) 119905 + 120579 119903120576

(119905 + 120579))

lt 119902E119881 (120593 (0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

(76)


119901

119890120574120591 that is 120574 lt

(1120591)(log(119902(11988821198881)(1 minus 120581)

119901

))

Thus

E [max119896isin

S

L119881 (120593120576

119905 119903120576

(119905))]


S

119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(77)

which implies that

E [L119881 (120593120576

119905 119903120576

(119905))]

le minus120582E [119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(78)


E [L119881 (120593120576

119905 119903120576

(119905))]

le minus120574E [119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(79)

We now consider

119880 (119905 + ℎ) minus 119880 (119905)


[119890120574(119905+120579+ℎ)

E

times [119881 (119909120576

(119905 + 120579 + ℎ)

minus 119863 (119909120576

119905+120579+ℎ 119903

120576

(119905 + 120579 + ℎ))

119905 + 120579 + ℎ 119903120576

(119905 + 120579 + ℎ)) ]

minus 119890120574(119905+120579)

E

times [119881 (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ] ]


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

[L119881 (120593120576

119904 119903120576

(119904))

+120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119905))] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [L (120593120576

119904 119903120576

(119904))

+ 120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119904 119903120576

(119904)) ] 119889119905

(80)



L (120593120576

119905 119903120576

(119905))

= 119905(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

+ 119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119891 (120593120576

119905 119903120576

(119905))

+

119898

sum

120575=1

intR

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905))

+ 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 119905 119903

120576

(119905))

minus (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

minus 119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 120582

120575(119889119911

120575)

+

119873

sum

119895=1

120574120576

119903120576(119905)119895

(120593120576

(0) minus 119863 (120593120576

119895) 119905 119895)

= L119881 (120593120576

119905 119903120576

(119905))

+ 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times [119891 (120593120576

119905 119903120576

(119905)) minus 119891 (120593120576

119905 119903120576

(119905))]

+

119898

sum

120575=1

intR

119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905))

+ 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 119905 119903

120576

(119905))

minus 119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905))

+ (120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 119905 119903

120576

(119905))

times 120582120575(119889119911

120575)

minus

119898

sum

120575=1

intR

119881119909(120593

120576

(0)

minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times (119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)

minus(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)) 120582

120575(119889119911

120575)

+

119873

sum

119895=1

120574119903120576(119905)119895

(120593120576

(0) minus 119863 (120593120576

119895) 119905 119895)

minus

119897

sum

119896=1

120574119903120576(119905)119896

(120593120576

(0) minus 119863 (120593120576

119896) 119905 119896)

(81)

This implies that

119880 (119905 + ℎ) minus 119880 (119905)


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [L119881 (120593120576

119904 119903120576

(119904))

+ 120574119881 (120593120576

(0)

minus119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times [119891 (120593120576

119904 119903120576

(119904)) minus 119891 (120593120576

119904 119903120576

(119904))] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [

119898

sum

120575=1

intR

[119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904))

+119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575) 119904 119903

120576

(119904))

minus 119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904))

+ (120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

119904 119903120576

(119904) )]

times120582120575(119889119911

120575) ] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times

119898

sum

120575=1

intR

[119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times (119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

minus(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575))]

times120582120575(119889119911

120575) 119889119904



E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times (

119873

sum

119895=1

120574119903120576(119904)119895

(120593120576

(0) minus 119863 (120593120576

119895) 119904 119895)

minus

119897

sum

119896=1

120574119903120576(119904)119896

(120593120576

(0) minus 119863 (120593120576

119896) 119904 119896))119889119904

= 1198691+ 119869

2+ 119869

3+ 119869

4+ 119869

5

(82)


119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) minus

(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)

=

119897

sum

119894=1

119873119896

sum

119895=1

119887(120575)

(119909120576

119905minus 119905 119904

119896119895 119911

120575)

times [119868119903120576(119905)=119904119896119895

minus 120583119896

119895119868119903120576(119905)=119896

]

(83)


1198694= lim sup

120576rarr0

119898

sum

120575=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times intR

[119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

minus (120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)]

times 120582120575(119889119911

120575) 119889119904


119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904))

119904 119903120576

(119904))

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)

times [119868119903120576(119904)=119904119896119895

minus120583119896

119895119868119903120576(119904)=119896

]

times 120582120575(119889119911

120575) 119889119904


119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

[E

100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904))

119904 119903120576

(119904))

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

times120582120575(119889119911

120575) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

]

12

(84)


4= 0 Similarly


119905


(0) minus 119863(120593120576

119903120576

(119905)) + 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)

and 120593120576

(0) minus 119863(120593120576

119903120576

(119905)) + (120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) such that

1198693= lim

120576rarr0

119898

sum

120575=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

intR

119881119909(120578

119904)

times [119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

minus(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)]

times 120582120575(119889119911

120575) 119889119904

= lim120576rarr0

119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120578

119904)

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

times 120582120575(119889119911

120575) 119889119904

le lim120576rarr0

119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

[E

100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120578

119905)

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)


times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

times 120582120575(119889119911

120575)119889119904

100381610038161003816100381610038161003816100381610038161003816

2

]

12

(85)



5= 0Therefore


= lim120576rarr0

Eint

119905+120579+ℎ

119905+120579

119890120574119904

times [L119881 (120593120576

119904 119903120576

(119904))

+120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))] 119889119904

le 0

(86)

Then

119880 (119905 + ℎ) le 119880 (119905) (87)


E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

119890minus]119905

(88)


5 Examples



in (14) with

Γ = (

minus1 0 1 0 0

1 minus2 1 0 0

2 1 minus3 0 0

0 0 0 minus1 1

0 0 0 1 minus1

) (89)

Γ = (

minus1 0 1 0 0

0 minus1 0 1 0

0 0 minus1 0 1

0 1 0 minus1 0

1 0 0 0 minus1

) (90)


(1205871

1205872

1205873)(

minus1 0 1

1 minus2 1

2 1 minus3

) = 0 (91)

and 1205871+ 120587

2+ 120587

3= 1 we get 1205831


(1205874

1205875) (

minus1 1

1 minus1) = 0 (92)

and 1205874+ 120587

5= 1 we have 120583

2

= (12 12) So

Γ = 120583Γ1 = (

5

8

1

8

1

40 0

0 0 01

2

1

2

)

times (

minus1 0 1 0 0

0 minus1 0 1 0

0 0 minus1 0 1

0 1 0 minus1 0

1 0 0 0 minus1

)(

1 0

1 0

1 0

0 1

0 1

)

= (minus3

8

3

81 minus1

)

(93)


119889 [120593120576

(0) minus 119863 (120593120576

119903120576

(119905))]

= 119891 (120593120576

119903120576

(119905)) 119889119905 + 119892 (120593120576

119903120576

(119905)) 119889119908 (119905)

(94)

with

119863(120593 11990411) = minus06 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990412) = minus02 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990413) = minus04 int

0

minus1

120593 (120579) 119889120579

119891 (120593 11990411) = minus16120593 (0) minus 8 cos [120593 (0)]

119891 (120593 11990412) = 8120593 (0) + 4 cos [120593 (0)]

119891 (120593 11990413) = 16120593 (0)

119892 (120593 11990411) =

radic10

10int

0

minus1

120593 (120579) 119889120579 cos [int0

minus1

120593 (120579) 119889120579]

119892 (120593 11990412) = minus

radic2

2int

0

minus1

120593 (120579) 119889120579 sin [int

0

minus1

120593 (120579) 119889120579]

119892 (120593 11990413) =

radic3

2int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990421) = 05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990422) = 05 int

0

minus1

120593 (120579) 119889120579

119891 (120593 11990421) = minus2120593 (0) 119891 (120593 119904

22) = minus2120593 (0)


119892 (120593 11990421) =

int

0

minus1

120593 (120579) 119889120579 sin [int

0

minus1

120593 (120579) 119889120579]

4radic2

119892 (120593 11990422) =

int

0

minus1

120593 (120579) 119889120579 cos [int0

minus1

120593 (120579) 119889120579]

4radic2

(95)

For any 120593 isin 1198712

F119905([minus1 0]R) and 120581 = max06 02 04 = 06


E1003816100381610038161003816119863 (120593 119894)

10038161003816100381610038162


119890]120579E

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

le 0361003817100381710038171003817120593

10038171003817100381710038172

(96)


119889 [120593 (0) minus 119863 (120593 119903 (119905))] = 119891 (120593 119903 (119905)) 119889119905 + 119892 (120593 119903 (119905)) 119889119908 (119905)

(97)


119863(120593 1) = minus05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 2) = 05 int

0

minus1

120593 (120579) 119889120579

119891 (120593 1) = minus5120593 (0) 119891 (120593 2) = minus2120593 (0)

119892 (120593 1) =1

2int

0

minus1

120593 (120579) 119889120579 119892 (120593 2) =1

4int

0

minus1

120593 (120579) 119889120579

(98)

We define 119881(119909 1) = 21199092 119881(119909 2) = 119909


L119881 (120593 1) le minus203

81205932

(0) +13

32

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

L119881 (120593 2) le minus5

21205932

(0) +13

16

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

(99)

Consequently

max119894=12

L119881 (120593 119894) le minus5

21205932

(0) +13

16

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

= minus5

4[max119894=12

119881 (119909 119894)] +13

16[min119894=12

119881 (119909 119894)]

(100)


2


119894isinS120601(120579)] le


E [max119894isinS

L119881 (120593 119894)] le minus (5

4minus

13119902

16)E [max

119894=12

119881 (119909 119894)] (101)




Γ120576

=1

120576Γ + Γ =

1

120576(

minus2 0 2 0

1 minus2 0 1

0 2 minus2 0

0 1 1 minus2

) (102)



119889 [120593120576

(0) minus 119863 (120593120576

119903120576

(119905))]

= 119891 (120593120576

119903120576

(119905)) 119889119905 + int

infin

0

120590 (119903120576

(119905) 119911) 119909120576

119905minus (119889119905 119889119911)

(103)

with

119863(120593 1) = minus09 int

0

minus1

120593 (120579) 119889120579

119863 (120593 2) = minus04 int

0

minus1

120593 (120579) 119889120579

119863 (120593 3) = minus05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 4) = minus03 int

0

minus1

120593 (120579) 119889120579

119891 (120593 1) = 2 sin [120593 (0)] 119891 (120593 2) = minus11

2120593 (0)

119891 (120593 3) = minus11

3120593 (0) 119891 (120593 4) = minus2 sin [120593 (0)]

(104)

Let

120572 (119911) =2

11120590 (1 119911) +

4

11120590 (2 119911) +

3

11120590 (3 119911) +

2

11120590 (4 119911)

int

infin

0

1205722

(119911) 120582 (119889119911) lt 2

(105)

For any 120593 isin 1198712

F119905([minus1 0]R) and 120581 = max09 04 05 03 =


E1003816100381610038161003816119863 (120593 119894)

10038161003816100381610038162


119890]120579E

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

le 0811003817100381710038171003817120593

10038171003817100381710038172

(106)


119889 [120593 (0) + 05 int

0

minus1

120593 (120579) 119889120579]

= minus3120593 (0) 119889119905 + int

infin

0

120572 (119911) 119909119905minus (119889119905 119889119911)

(107)


Let 119881(119909) = 1199092 then

L119881 (120593 119894) le minus61205932

(0) + int

infin

0

1205722

(119911) 120582 (119889119911)

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

(108)


1198712


119894isinS120601(120579)] le 119902E[max

119894isinS120601(0)]


E [max119894isinS


119881 (119909 119894)] (109)





Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119889 [119909120576

minus 119863 (119909120576

119905 119903

120576

(119905))]

= 119891 (119909120576

119905 119905 119903

120576

(119905)) 119889119905 + 119892 (119909120576

119905 119905 119903

120576

(119905)) 119889119908 (119905)

119909120576

0= 120585 isin 119862 ([minus120591 0] R

119899

) 119903120576

= 1199030

(10)




1003816100381610038161003816119891 (120593 119905 119894) minus 119891 (120601 119905 119894)1003816100381610038161003816 or

1003816100381610038161003816119892 (120593 119905 119894) minus 119892 (120601 119905 119894)1003816100381610038161003816

le 119871120572

1003817100381710038171003817120593 minus 12060110038171003817100381710038172

(11)


) with 120593 or




1003816100381610038161003816119891 (120593 119905 119894)10038161003816100381610038162

or1003816100381610038161003816119892 (120593 119905 119894)

10038161003816100381610038162

le 119871 (1 +1003817100381710038171003817120593

10038171003817100381710038172

) (12)



1003816100381610038161003816119863 (120593 119894) minus 119863 (120601 119894)1003816100381610038161003816 le 120581

1003817100381710038171003817120593 minus 12060110038171003817100381710038172

119863 (0 119894) equiv 0

(13)


120576120585119894




sup(120585119894)isin119866timesS

119864[ supminus120591le119904le119905

10038161003816100381610038161003816119909120576120585119894

(119904)10038161003816100381610038161003816

119901

] le 119867 119905 ge 0 (14)



(sdot) as

119903120576

(119905) = 119896 if 119903120576

(119905) isin S119896

(15)


119896 by 120583119896

= (120583119896

1 120583

119896

119872119896

) isin R1times119872119896 and 120583 =

diag(1205831

120583119897


Γ = (120574119894119895)119897times119897

= 120583Γ1 (16)

with 1 = diag(11198721

1119872119897

) and 1119872119896

= (1 1)119879


119863(120593 119896) =

119872119896

sum

119895=1

120583119896

119895119863(120593 119904

119896119895)

119891 (120593 119905 119896) =

119872119896

sum

119895=1

120583119896

119895119891 (120593 119905 119904

119896119895)

119892 (120593 119905 119896) 119892119879

(120593 119905 119896) =

119872119896

sum

119895=1

120583119896

119895119892 (120593 119905 119904

119896119895) 119892

119879

(120593 119905 119904119896119895)

(17)

for each 119904119896119895



119896119895)119892

119879

(120593 119905 119904119896119895)


119896119895)119892

119879



119889 [120593 (0) minus 119863 (120593 119903 (119905))]

= 119891 (120593 119905 119903 (119905)) 119889119905 + 119892 (120593 119905 119903 (119905)) 119889119908 (119905)

1199090= 120585 119903 = 119903

0

(18)



119901

(R119899

times

R+times SR


defined on R119899

times R+


119862119901

(R119899

times R+times SR

+) for some 119901 ge 4


infin Moreover 120597119901119881(119909 119905 119894) = 119874(1) 120597120580119881(119909 119905 119894)(|119909|120580

+ |119910|120580

) le

119870(|119909|119901

+|119910|119901


119910|119874(119910)|119910 lt infin


E1003816100381610038161003816119863 (120593 119896)

1003816100381610038161003816119901

le 120581119901 supminus120591le120579le0

1198901205741205791003817100381710038171003817120593

1003817100381710038171003817119901

(120593 119894) isin 119871119901

F119905([minus120591 0] R

119899

) times S

(19)


Then for any 120585 isin 119871119901

F0([minus120591 0]R119899



119890120574119904

E |119909 (119904)|119901

le

10038171003817100381710038171205851003817100381710038171003817119901

1 minus 120581orsup

0le119904le119905119890120574119904E

1003816100381610038161003816119909 (119904) minus 119863 (119909119904 119903 (119904))

1003816100381610038161003816119901

(1 minus 120581)119901

119905 ge 0

(20)


(119909 + 119910)119901

= (1 minus 1205811)1minus119901

(119909119901

+ 1205811

1minus119901

119910119901

)

forall119909 119910 ge 0 1205811gt 0

(21)


119890120574119905

E|119909 (119905)|119901

le 119890120574119905

[(1 minus 120581)1minus119901

E1003816100381610038161003816119909 (119905) minus 119863 (119909

119905 119903 (119905))

1003816100381610038161003816119901

+ 1205811minus119901

E1003816100381610038161003816119863 (119909

119905 119903 (119905))

1003816100381610038161003816119901

]

le (1 minus 120581)1minus119901

119890120574119905

E1003816100381610038161003816119909 (119905) minus 119863 (119909

119905 119903 (119905))

1003816100381610038161003816119901

+ 120581119890120574119905 supminus120591le120579le0

119890120574120579

E |119909 (119905 + 120579)|119901


0le119904le119905

119890120574119904

E1003816100381610038161003816119909 (119904) minus 119863 (119909

119904 119903 (119904))

1003816100381610038161003816119901

+ 120581 supminus120591le120579le0

119890120574(119904+120579)

E |119909 (119904 + 120579)|119901


0le119904le119905

119890120574119904

E1003816100381610038161003816119909 (119904) minus 119863 (119909

119904 119903 (119904))

1003816100381610038161003816119901


119890120574119904

E |119909 (119904)|119901

(22)

Then


119890120574119904

E |119909 (119904)|119901


E |119909 (120579)|2

]


0le119904le119905

119890120574119904

E1003816100381610038161003816119909 (119904) minus 119863 (119909

119904 119903 (119904))

1003816100381610038161003816119901


119890120574119904

E |119909 (119904)|119901

]

(23)



2 120582 119901 be all


119901

(R119899

timesR+timesSR


that

1198881|119909|

119901

le 119881 (119909 119905 119896) le 1198882|119909|

119901

(24)

for all (119909 119905 119896) isin R119899

times R+


E1003816100381610038161003816119863 (120593 119896)

1003816100381610038161003816119901

le 120581119901 supminus120591le120579le0

119890]1205791003817100381710038171003817120593

1003817100381710038171003817119901

120581 = max 1205811 120581

119896 120593 isin 119871

119901

F119905

(25)


E [max119896isin

S

L119881 (120593 119905 119896)]


S

119881 (120593 (0) minus 119863 (120593 119896) 119905 119896)]

(26)


F119905([minus120591 0]R119899

)satisfying

E [min119896isin

S

119881 (120593 (120579) 119905 + 120579 119896)]


S

119881 (120593 (0) minus 119863 (120593 119896) 119905 119896)]

(27)


F0([minus120591 0]R119899

) 119905 ge 0

lim sup120576rarr0

E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

119890minus]119905

(28)

where

] = min1205741

120591log

119902

(11988821198881) (1 minus 120581)

119901 (29)


1198882

1198881

(1 minus 120581)119901

119890120574120591

= 120582 (30)


Proof Let

(120593 119905 119895) =

119897

sum

119896=1

119881 (120593 119905 119896) 119868119895isinS119896 = 119881 (120593 119905 119896) if 119895 isin S

119896

(31)


(120593120576

119905 119903120576

(119905)) = 119881 (120593120576

119905 119903120576

(119905))

119872

sum

119894=1

120574119897119894 (120593 119905 119894) =

119872

sum

119894=1

120574119897119894

119897

sum

119896=1

119881 (120593 119905 119896) 119868119894isinS119896 = 0

(32)




119880 (119905) = supminus120591le120579le0

[119890120574(119905+120579)E119881 (119909

120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ]


[119890120574(119905+120579)

E (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ]

(33)


119863+

119880 (119905) = lim supℎrarr0+

119880 (119905 + ℎ) minus 119880 (119905)

ℎle 0 forall119905 ge 0 (34)


120576

(119905) minus

119863(119909120576

119905 119903

120576

(119905)) 119905 119903120576

(119905)) or 119880(119905) = 119890120574119905E119881(119909

120576

(119905) minus 119863(119909120576

119905 119903

120576

(119905))

119905 119903120576

(119905))If 119880(119905) gt 119890

120574119905E119881(119909120576

(119905) minus 119863(119909120576

119905 119903

120576

(119905)) 119905 119903120576


119890120574(119905+ℎ)E119881(119909

120576

(119905 + ℎ) minus119863(119909120576

119905+ℎ 119903

120576

(119905 + ℎ)) 119905 + ℎ 119903120576

(119905 + ℎ)) hence119880(119905 + ℎ) le 119880(119905) and 119863

+

119880(119905) le 0If 119880(119905) = 119890

120574119905E119881(119909120576

(119905) minus 119863(119909120576

119905 119903

120576

(119905)) 119905 119903120576

(119905)) we have

119890120574(119905+120579)

E119881 (119909120576

(119905 + 120579) minus 119863 (119909119905+120579

119903120576

(119905 + 120579)) 119905 + 120579 119903120576

(119905 + 120579))

le 119890120574119905

E119881 (119909120576

(119905) minus 119863 (119909119905 119903

120576

(119905)) 119905 119903120576

(119905))

(35)


E119881 (119909120576

(119905 + 120579) minus 119863 (119909119905+120579

119903120576

(119905 + 120579)) 119905 + 120579 119903120576

(119905 + 120579))

le 119890minus120574120579

E119881 (119909120576

(119905) minus 119863 (119909119905 119903

120576

(119905)) 119905 119903120576

(119905))

le 119890120574120591

E119881 (119909120576

(119905) minus 119863 (119909119905 119903

120576

(119905)) 119905 119903120576

(119905))

(36)


119890120574(119905+120579)

E119881 (119909120576

(119905 + 120579) 119905 + 120579 119903120576

(119905 + 120579))

le 1198882119890120574(119905+120579)

E1003816100381610038161003816119909

120576

(119905 + 120579)1003816100381610038161003816119901

le 1198882(1 minus 120581)

119901 sup0le119904le119905

119890120574119904

E1003816100381610038161003816119909

120576

(119904) minus 119863 (119909120576

119904 119903 (119904))

1003816100381610038161003816119901

le1198882

1198881

(1 minus 120581)119901 sup0le119904le119905

119890120574119904

E119881 (119909120576

(119904) minus 119863 (119909120576

119904 119903 (119904)) 119904 119903 (119904))

le1198882

1198881

(1 minus 120581)119901

119890120574119905

E119881 (119909120576

(119905) minus 119863 (119909120576

119905 119903 (119905)) 119905 119903 (119905))

(37)

Then

E119881 (119909120576

(119905 + 120579) 119905 + 120579 119903120576

(119905 + 120579))

lt 119902E119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

(38)

where 119902 gt (11988821198881)(1 minus 120581)

119901

119890120574120591 that is 120574 lt (1120591)(log(119902(119888

21198881)

(1 minus 120581)119901


0gt 0 such

that for any 120576 isin (0 1205760)

E [min119896isin

S

119881 (120593120576

(120579) 119905 + 120579 119896)]


S

119881 (120593120576

(0) minus 119863 (120593120576

119896) 119905 119896)]

(39)


E [max119896isin

S

L119881 (120593120576

119905 119903120576

(119905))]


S

119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(40)

which implies that

E [L119881 (120593120576

119905 119903120576

(119905))]

le minus120582E [119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(41)


E [L119881 (120593120576

119905 119903120576

(119905))]

le minus120574E [119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(42)

Next we consider

119880 (119905 + ℎ) minus 119880 (119905)


[119890120574(119905+120579+ℎ)

E [119881 (119909120576

(119905 + 120579 + ℎ)

minus 119863 (119909120576

119905+120579+ℎ 119903

120576

(119905 + 120579 + ℎ))

119905 + 120579 + ℎ 119903120576

(119905 + 120579 + ℎ)) ]

minus 119890120574(119905+120579)

E [119881 (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579))]]


Eint

119905+120579+ℎ

119905+120579

119890120574119904

[L119881 (120593120576

119904 119903120576

(119904))

+ 120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904))

119904 119903120576

(119904))] 119889119904


Eint

119905+120579+ℎ

119905+120579

119890120574119904

[L (120593120576

119904 119903120576

(119904))

+ 120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904))

119904 119903120576

(119904)) ] 119889119904

(43)



L (120593120576

119905 119903120576

(119905))

= 119905(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

+ 119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119891 (120593120576

119905 119903120576

(119905))

+1

2trace [119892

119879

(120593120576

119905 119903120576

(119905))

times 119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) ]

+

119872

sum

119894=1

120574120576

119903120576(119905)119894

(120593120576

(0) minus 119863 (119909120576

119905 119894) 119905 119894)

= 119905(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

+ 119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119891 (120593120576

119905 119903120576

(119905))

+1

2trace [119892

119879

(120593120576

119905 119903120576

(119905))

times 119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) ]

+

119872

sum

119894=1

120574119903120576(119905)119894

(120593120576

(0) minus 119863 (119909120576

119905 119894) 119905 119894)

= 119881119905(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

+ 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119891 (120593120576

119905 119903120576

(119905))

+1

2trace [119892

119879

(120593120576

119905 119903120576

(119905))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) ]

+

119897

sum

119896=1

120574120576

119903120576(119905)119896

119881 (120593120576

(0) minus 119863 (119909120576

119905 119896) 119905 119896)

+ 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times [119891 (120593120576

119905 119903120576

(119905)) minus 119891 (120593120576

119905 119903120576

(119905))]

+1

2trace [119892

119879

(120593120576

119905 119903120576

(119905))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) minus 119892119879

(120593120576

119905 119903120576

(119905))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) ]

+

119872

sum

119894=1

120574119903120576(119905)119894

(120593120576

(0) minus 119863 (119909120576

119905 119894) 119905 119894)

minus

119897

sum

119896=1

120574119903120576(119905)119896

119881 (120593120576

(0) minus 119863 (119909120576

119905 119896) 119905 119896)

= L119881 (120593120576

119905 119903120576

(119905))

+ 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times [119891 (120593120576

119905 119903120576

(119905)) minus 119891 (120593120576

119905 119903120576

(119905))]

+1

2trace [119892

119879

(120593120576

119905 119903120576

(119905))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) minus 119892119879

(120593120576

119905 119903120576

(119905))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) ]

+

119872

sum

119894=1

120574119903120576(119905)119894

(120593120576

(0) minus 119863 (119909120576

119905 119894) 119905 119894)

minus

119897

sum

119896=1

120574119903120576(119905)119896

119881 (120593120576

(0) minus 119863 (119909120576

119905 119896) 119905 119896)

(44)

Therefore

119880 (119905 + ℎ) minus 119880 (119905)


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [L119881 (120593120576

119904 119903120576

(119904))

+120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119905

119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times [119891 (120593120576

119904 119903120576

(119904)) minus 119891 (120593120576

119904 119903120576

(119904))] 119889119904

+1

2lim sup120576rarr0

E

times int

119905+120579+ℎ

119905+120579

119890120574119904



(120593120576

119904 119903120576

(119904))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times 119892 (120593120576

119904 119903120576

(119904)) minus 119892119879

(120593120576

119904 119903120576

(119904))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times 119892 (120593120576

119904 119903120576

(119904)) ] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times (

119872

sum

119894=1

120574119903120576(119904)119894

(120593120576

(0) minus 119863 (119909120576

119904 119894) 119904 119894)

minus

119897

sum

119896=1

120574119903120576(119904)119896

119881 (120593120576

(0) minus 119863 (119909120576

119904 119896) 119904 119896))119889119904

= 1198681+ 119868

2+ 119868

3+ 119868

4

(45)


119891 (120593120576

119905 119903120576

(119905)) minus 119891 (120593120576

119905 119903120576

(119905))

=

119897

sum

119896=1

119872119896

sum

119895=1

119891 (120593120576

119905 119904119896119895) times [119868

119903120576(119905)=119904119896119895

minus 120583119896

119895119868119903120576(119905)=119896

]

(46)

This implies that

lim120576rarr0

Eint

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times [119891 (120593120576

119904 119903120576

(119904)) minus 119891 (120593120576

119904 119903120576

(119904))] 119889119904

le lim120576rarr0

[E

100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times [119891 (120593120576

119904 119903120576

(119904)) minus 119891 (120593120576

119904 119903120576

(119904))] 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

]

12

= lim120576rarr0

[

[

E

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times

119897

sum

119896=1

119872119896

sum

119895=1

119891 (120593120576

119904 119904119896119895)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

] 119889119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

]

]

12

le lim120576rarr0

[

[

E

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119897

sum

119896=1

119872119896

sum

119895=1

119890120574119904

119871 (1 +1003817100381710038171003817120593

1003817100381710038171003817119901

)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

] 119889119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

]

]

12

(47)


2= 0 Similarly

we can show that

1198683=

1

2lim sup120576rarr0

E

times int

119905+120579+ℎ

119905+120579

119890120574119904


(120593120576

119904 119903120576

(119904))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times 119892 (120593120576

119904 119903120576

(119904))

minus 119892119879

(120593120576

119904 119903120576

(119904))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times 119892 (120593120576

119904 119903120576

(119904)) ] 119889119904 = 0

(48)


119872

sum

119894=1

120574119903120576(119905)119894

(120593120576

(0) minus 119863 (119909120576

119905 119894) 119905 119894)

= Γ (120593120576

(0) minus 119863 (119909120576

119905 119894) 119905 sdot) (119903

120576

(119905))

119897

sum

119896=1

120574119903120576(119905)119896

119881 (120593 (0) minus 119863 (119909120576

119905 119896) 119905 119896)

= Γ119881 (120593 (0) minus 119863 (119909120576

119905 119896) 119905 sdot) (119903

120576

(119905))

(49)

Hence

1198684= lim sup

120576rarr0

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times (

119872

sum

119894=1

120574119903120576(119904)119894

(120593120576

(0) minus 119863 (119909120576

119904 119894) 119904 119894)

minus

119897

sum

119896=1

120574119903120576(119904)119896

119881 (120593120576

(0) minus 119863 (119909120576

119904 119896) 119904 119896))119889119904



E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times (Γ (120593120576

(0) minus 119863 (119909120576

119904 119894) 119904 sdot) (119903

120576

(119904))

minus Γ119881 (120593120576

(0) minus 119863 (119909120576

119904 119896) 119904 sdot) (119904

119896119895)) 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

119897

sum

119896=1

119872119896

sum

119895=1

Γ (120593120576

(0) minus 119863 (119909120576

119904 119894) 119904 sdot) (119904

119896119895)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

] 119889119904


[

[

E

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

times

119897

sum

119896=1

119872119896

sum

119895=1

Γ (120593120576

(0) minus 119863 (119909120576

119904 119894) 119904 sdot) (119904

119896119895)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

]

]

12

(50)


4= 0 Therefore

119880 (119905 + ℎ) minus 119880 (119905)

= lim120576rarr0

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [L119881 (120593120576

119904 119903120576

(119904))

+120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))] 119889119904

le 0

(51)

That is

119880 (119905 + ℎ) le 119880 (119905) (52)


+



lim sup120576rarr0

119890120574119905

E1003816100381610038161003816119909

120576

(119905) minus 119863 (119909120576

119905 119903

120576

(119905))1003816100381610038161003816119901


sup120591le120579le0

119890120574120579

E1003816100381610038161003816119909

120576

(120579) minus 119863 (119909120576

120579 119903

120576

(120579))1003816100381610038161003816

119901



(1 + 120581)119901minus1

times [E1003816100381610038161003816119909

120576

(120579)1003816100381610038161003816119901

+ 1205811minus119901

E1003816100381610038161003816119863 (119909

120576

120579 119903

120576

(120579))1003816100381610038161003816119901

]

le 1198882(1 + 120581)

11990110038171003817100381710038171205851003817100381710038171003817119901

119905 ge 0

(53)


lim sup120576rarr0

119890120574119905

E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

(54)

That is

lim sup120576rarr0

E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

119890minus120574119905

forall119905 ge 0 (55)



119889 [120593120576

(0) minus 119863 (120593120576

119903120576

(119905))]

= 119891 (120593120576

119905 119903120576

(119905)) 119889119905 + intR119898

119887 (119909120576

119905minus 119905 119903

120576

(119905) 119911) (119889119905 119889119911)

1199090= 120585 isin 119862 ([minus120591 0] R

119899

) 119903 (0) isin S

(56)

where 119909120576

119905minus= lim

119904uarr119905119909120576

119904 119863 119862([minus120591 0]R119899

) times S rarr R119899 119887

119862([minus120591 0]R119899





119887(120575)

(120593 119905 119894 119911) = 119887(120575)

(120593 119905 119894 119911120575)

119911 = (1199111 119911

119898) isin R

119898

119894 isin S

(57)


(119889119905 119889119911) = (1(119889119905 119889119911

1)

119898(119889119905 119889119911

119898))

= (1198731(119889119905 119889119911

1) minus 120582

1(119889119911

1) 119889119905 119873

119898(119889119905 119889119911

119898)

minus 120582119898

(119889119911119898) 119889119905)

(58)






119889 [120593120576

(0) minus 119863 (119909120576

119905 119903

120576

(119905))]

= 119891 (119909120576

119905 119905 119903

120576

(119905)) 119889119905

+ intR119898

(119909120576

119905minus 119905 119903

120576

(119905) 119911) (119889119905 119889119911)

1199090= 120585 isin 119862 ([minus120591 0] R

119899

) 119903 (0) isin S

(59)

where 119909120576

119905minus= lim

119904uarr119905119909120576

119904and 119862([minus120591 0]R119899


rarr


119863(120593 119896) =

119873119898

sum

119895=1

120583119896

119895119863(120593 119904

119896119895)

(120593 119905 119896 119911) =

119873119898

sum

119895=1

120583119896

119895119887 (120593 119905 119904

119896119895 119911)

(60)

for each 119904119896119895




gt 0such that

1003816100381610038161003816119891 (120593 119905 119894) minus 119891 (120601 119905 119894)1003816100381610038161003816

or

119898

sum

120575=1

intR

10038161003816100381610038161003816119887(120575)

(120593 119905 119894 119911120575) minus 119887

(120575)

(120601 119905 119894 119911120575)10038161003816100381610038161003816120582120575(119889119911

120575)

le 119871120577

1003817100381710038171003817120593 minus 12060110038171003817100381710038172

(61)




119862([minus120591 0]R119899

) 119894 isin S

1003816100381610038161003816119891 (120593 119905 119894)1003816100381610038161003816 or

119898

sum

120575=1

intR

10038161003816100381610038161003816119887(120575)

(120593 119905 119894 119911120575)10038161003816100381610038161003816120582120575(119889119911

120575)

le 119871 (1 +1003817100381710038171003817120593

10038171003817100381710038172

)

(62)



1003816100381610038161003816119863 (120593 119894) minus 119863 (120601 119894)1003816100381610038161003816 le 120581

1003817100381710038171003817120593 minus 12060110038171003817100381710038172

119863 (0 119894) equiv 0

(63)

Given that 119881 isin 119862119901

(R119899

times R+times SR


L119881 by

L119881 (120593 119905 119894)

= 119881119905(120593 (0) minus 119863 (120593 119894) 119905 119894)

+ 119881119909(120593 (0) minus 119863 (120593 119894) 119905 119894) 119891 (120593 119905 119894)

+

119873

sum

119895=1

120574119894119895119881 (120593 (0) minus 119863 (120593 119894) 119905 119895)

+ intR

119898

sum

120575=1

119881 (120593 (0) minus 119863 (120593 119894) + 119887(120575)

(120593 119905 120580 119911120575) 119905 120580)

minus 119881 (120593 (0) minus 119863 (120593 119894) 119905 119894)

minus 119881119909(120593 (0) minus 119863 (120593 119894) 119905 119894)

times 119887(120575)

(120593 119905 120580 119911120575) 120582

120575(119889119911

120575)

(64)

where

119881119909(120593 (0) minus 119863 (120593 119894) 119905 119894)

= (120597119881 (120593 (0) minus 119863 (120593 119894) 119905 119894)

1205971199091

120597119881 (120593 (0) minus 119863 (120593 119894) 119905 119894)

120597119909119898

)

(65)


(sdot) 119903120576






2


119901

(R119899

times R+times SR

+) satisfying


1198881|119909|

119901

le 119881 (119909 119905 119896) le 1198882|119909|

119901

119896 isin S (66)

for all (119909 119905 119896) isin R119899


following

E1003816100381610038161003816119863 (120593 119896)

1003816100381610038161003816119901

le 120581119901 supminus120591le120579le0

119890]120579 1003817100381710038171003817120593

1003817100381710038171003817

119901

120581 = max 1205811 120581

119896 120593 isin 119871

119901

F119905

(67)


E [max119896isin

S


S

119881 (120593 (0) minus 119863 (120593 119896))]

(68)



F119905([minus120591 0]R119899

)satisfying

E [min119896isin

S

119881 (119909 (119905 + 120579) 119905 + 120579 119896)]


S

119881 (120593 (0) minus 119863 (120593 119896) 119905 119894)] minus120591 le 120579 le 0

(69)


) 119905 ge 0

lim sup120576rarr0

E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

119890minus]119905

(70)

where

] = min1205741

120591log

119902

(11988821198881) (1 minus 120581)

119901 (71)


1198882

1198881

(1 minus 120581)119901

119890120574120591

= 120582 (72)

Proof Define

(120593 119905 120588) =

119897

sum

119896=1

119881 (120593 119905 119896) 119868120588isinS119896 = 119881 (120593 119905 119896) if 120588 isin S

119896

(73)


119880 (119905)


[119890120574(119905+120579)

E119881 (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ]


[119890120574(119905+120579)

E (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ]

(74)


119863+

119880 (119905) = lim supℎrarr0+

119880 (119905 + ℎ) minus 119880 (119905)

ℎle 0 forall119905 ge 0 (75)


E119881 (119909120576

(119905 + 120579) 119905 + 120579 119903120576

(119905 + 120579))

lt 119902E119881 (120593 (0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

(76)


119901

119890120574120591 that is 120574 lt

(1120591)(log(119902(11988821198881)(1 minus 120581)

119901

))

Thus

E [max119896isin

S

L119881 (120593120576

119905 119903120576

(119905))]


S

119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(77)

which implies that

E [L119881 (120593120576

119905 119903120576

(119905))]

le minus120582E [119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(78)


E [L119881 (120593120576

119905 119903120576

(119905))]

le minus120574E [119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(79)

We now consider

119880 (119905 + ℎ) minus 119880 (119905)


[119890120574(119905+120579+ℎ)

E

times [119881 (119909120576

(119905 + 120579 + ℎ)

minus 119863 (119909120576

119905+120579+ℎ 119903

120576

(119905 + 120579 + ℎ))

119905 + 120579 + ℎ 119903120576

(119905 + 120579 + ℎ)) ]

minus 119890120574(119905+120579)

E

times [119881 (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ] ]


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

[L119881 (120593120576

119904 119903120576

(119904))

+120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119905))] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [L (120593120576

119904 119903120576

(119904))

+ 120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119904 119903120576

(119904)) ] 119889119905

(80)



L (120593120576

119905 119903120576

(119905))

= 119905(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

+ 119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119891 (120593120576

119905 119903120576

(119905))

+

119898

sum

120575=1

intR

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905))

+ 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 119905 119903

120576

(119905))

minus (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

minus 119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 120582

120575(119889119911

120575)

+

119873

sum

119895=1

120574120576

119903120576(119905)119895

(120593120576

(0) minus 119863 (120593120576

119895) 119905 119895)

= L119881 (120593120576

119905 119903120576

(119905))

+ 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times [119891 (120593120576

119905 119903120576

(119905)) minus 119891 (120593120576

119905 119903120576

(119905))]

+

119898

sum

120575=1

intR

119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905))

+ 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 119905 119903

120576

(119905))

minus 119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905))

+ (120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 119905 119903

120576

(119905))

times 120582120575(119889119911

120575)

minus

119898

sum

120575=1

intR

119881119909(120593

120576

(0)

minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times (119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)

minus(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)) 120582

120575(119889119911

120575)

+

119873

sum

119895=1

120574119903120576(119905)119895

(120593120576

(0) minus 119863 (120593120576

119895) 119905 119895)

minus

119897

sum

119896=1

120574119903120576(119905)119896

(120593120576

(0) minus 119863 (120593120576

119896) 119905 119896)

(81)

This implies that

119880 (119905 + ℎ) minus 119880 (119905)


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [L119881 (120593120576

119904 119903120576

(119904))

+ 120574119881 (120593120576

(0)

minus119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times [119891 (120593120576

119904 119903120576

(119904)) minus 119891 (120593120576

119904 119903120576

(119904))] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [

119898

sum

120575=1

intR

[119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904))

+119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575) 119904 119903

120576

(119904))

minus 119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904))

+ (120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

119904 119903120576

(119904) )]

times120582120575(119889119911

120575) ] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times

119898

sum

120575=1

intR

[119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times (119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

minus(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575))]

times120582120575(119889119911

120575) 119889119904



E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times (

119873

sum

119895=1

120574119903120576(119904)119895

(120593120576

(0) minus 119863 (120593120576

119895) 119904 119895)

minus

119897

sum

119896=1

120574119903120576(119904)119896

(120593120576

(0) minus 119863 (120593120576

119896) 119904 119896))119889119904

= 1198691+ 119869

2+ 119869

3+ 119869

4+ 119869

5

(82)


119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) minus

(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)

=

119897

sum

119894=1

119873119896

sum

119895=1

119887(120575)

(119909120576

119905minus 119905 119904

119896119895 119911

120575)

times [119868119903120576(119905)=119904119896119895

minus 120583119896

119895119868119903120576(119905)=119896

]

(83)


1198694= lim sup

120576rarr0

119898

sum

120575=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times intR

[119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

minus (120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)]

times 120582120575(119889119911

120575) 119889119904


119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904))

119904 119903120576

(119904))

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)

times [119868119903120576(119904)=119904119896119895

minus120583119896

119895119868119903120576(119904)=119896

]

times 120582120575(119889119911

120575) 119889119904


119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

[E

100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904))

119904 119903120576

(119904))

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

times120582120575(119889119911

120575) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

]

12

(84)


4= 0 Similarly


119905


(0) minus 119863(120593120576

119903120576

(119905)) + 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)

and 120593120576

(0) minus 119863(120593120576

119903120576

(119905)) + (120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) such that

1198693= lim

120576rarr0

119898

sum

120575=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

intR

119881119909(120578

119904)

times [119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

minus(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)]

times 120582120575(119889119911

120575) 119889119904

= lim120576rarr0

119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120578

119904)

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

times 120582120575(119889119911

120575) 119889119904

le lim120576rarr0

119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

[E

100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120578

119905)

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)


times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

times 120582120575(119889119911

120575)119889119904

100381610038161003816100381610038161003816100381610038161003816

2

]

12

(85)



5= 0Therefore


= lim120576rarr0

Eint

119905+120579+ℎ

119905+120579

119890120574119904

times [L119881 (120593120576

119904 119903120576

(119904))

+120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))] 119889119904

le 0

(86)

Then

119880 (119905 + ℎ) le 119880 (119905) (87)


E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

119890minus]119905

(88)


5 Examples



in (14) with

Γ = (

minus1 0 1 0 0

1 minus2 1 0 0

2 1 minus3 0 0

0 0 0 minus1 1

0 0 0 1 minus1

) (89)

Γ = (

minus1 0 1 0 0

0 minus1 0 1 0

0 0 minus1 0 1

0 1 0 minus1 0

1 0 0 0 minus1

) (90)


(1205871

1205872

1205873)(

minus1 0 1

1 minus2 1

2 1 minus3

) = 0 (91)

and 1205871+ 120587

2+ 120587

3= 1 we get 1205831


(1205874

1205875) (

minus1 1

1 minus1) = 0 (92)

and 1205874+ 120587

5= 1 we have 120583

2

= (12 12) So

Γ = 120583Γ1 = (

5

8

1

8

1

40 0

0 0 01

2

1

2

)

times (

minus1 0 1 0 0

0 minus1 0 1 0

0 0 minus1 0 1

0 1 0 minus1 0

1 0 0 0 minus1

)(

1 0

1 0

1 0

0 1

0 1

)

= (minus3

8

3

81 minus1

)

(93)


119889 [120593120576

(0) minus 119863 (120593120576

119903120576

(119905))]

= 119891 (120593120576

119903120576

(119905)) 119889119905 + 119892 (120593120576

119903120576

(119905)) 119889119908 (119905)

(94)

with

119863(120593 11990411) = minus06 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990412) = minus02 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990413) = minus04 int

0

minus1

120593 (120579) 119889120579

119891 (120593 11990411) = minus16120593 (0) minus 8 cos [120593 (0)]

119891 (120593 11990412) = 8120593 (0) + 4 cos [120593 (0)]

119891 (120593 11990413) = 16120593 (0)

119892 (120593 11990411) =

radic10

10int

0

minus1

120593 (120579) 119889120579 cos [int0

minus1

120593 (120579) 119889120579]

119892 (120593 11990412) = minus

radic2

2int

0

minus1

120593 (120579) 119889120579 sin [int

0

minus1

120593 (120579) 119889120579]

119892 (120593 11990413) =

radic3

2int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990421) = 05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990422) = 05 int

0

minus1

120593 (120579) 119889120579

119891 (120593 11990421) = minus2120593 (0) 119891 (120593 119904

22) = minus2120593 (0)


119892 (120593 11990421) =

int

0

minus1

120593 (120579) 119889120579 sin [int

0

minus1

120593 (120579) 119889120579]

4radic2

119892 (120593 11990422) =

int

0

minus1

120593 (120579) 119889120579 cos [int0

minus1

120593 (120579) 119889120579]

4radic2

(95)

For any 120593 isin 1198712

F119905([minus1 0]R) and 120581 = max06 02 04 = 06


E1003816100381610038161003816119863 (120593 119894)

10038161003816100381610038162


119890]120579E

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

le 0361003817100381710038171003817120593

10038171003817100381710038172

(96)


119889 [120593 (0) minus 119863 (120593 119903 (119905))] = 119891 (120593 119903 (119905)) 119889119905 + 119892 (120593 119903 (119905)) 119889119908 (119905)

(97)


119863(120593 1) = minus05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 2) = 05 int

0

minus1

120593 (120579) 119889120579

119891 (120593 1) = minus5120593 (0) 119891 (120593 2) = minus2120593 (0)

119892 (120593 1) =1

2int

0

minus1

120593 (120579) 119889120579 119892 (120593 2) =1

4int

0

minus1

120593 (120579) 119889120579

(98)

We define 119881(119909 1) = 21199092 119881(119909 2) = 119909


L119881 (120593 1) le minus203

81205932

(0) +13

32

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

L119881 (120593 2) le minus5

21205932

(0) +13

16

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

(99)

Consequently

max119894=12

L119881 (120593 119894) le minus5

21205932

(0) +13

16

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

= minus5

4[max119894=12

119881 (119909 119894)] +13

16[min119894=12

119881 (119909 119894)]

(100)


2


119894isinS120601(120579)] le


E [max119894isinS

L119881 (120593 119894)] le minus (5

4minus

13119902

16)E [max

119894=12

119881 (119909 119894)] (101)




Γ120576

=1

120576Γ + Γ =

1

120576(

minus2 0 2 0

1 minus2 0 1

0 2 minus2 0

0 1 1 minus2

) (102)



119889 [120593120576

(0) minus 119863 (120593120576

119903120576

(119905))]

= 119891 (120593120576

119903120576

(119905)) 119889119905 + int

infin

0

120590 (119903120576

(119905) 119911) 119909120576

119905minus (119889119905 119889119911)

(103)

with

119863(120593 1) = minus09 int

0

minus1

120593 (120579) 119889120579

119863 (120593 2) = minus04 int

0

minus1

120593 (120579) 119889120579

119863 (120593 3) = minus05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 4) = minus03 int

0

minus1

120593 (120579) 119889120579

119891 (120593 1) = 2 sin [120593 (0)] 119891 (120593 2) = minus11

2120593 (0)

119891 (120593 3) = minus11

3120593 (0) 119891 (120593 4) = minus2 sin [120593 (0)]

(104)

Let

120572 (119911) =2

11120590 (1 119911) +

4

11120590 (2 119911) +

3

11120590 (3 119911) +

2

11120590 (4 119911)

int

infin

0

1205722

(119911) 120582 (119889119911) lt 2

(105)

For any 120593 isin 1198712

F119905([minus1 0]R) and 120581 = max09 04 05 03 =


E1003816100381610038161003816119863 (120593 119894)

10038161003816100381610038162


119890]120579E

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

le 0811003817100381710038171003817120593

10038171003817100381710038172

(106)


119889 [120593 (0) + 05 int

0

minus1

120593 (120579) 119889120579]

= minus3120593 (0) 119889119905 + int

infin

0

120572 (119911) 119909119905minus (119889119905 119889119911)

(107)


Let 119881(119909) = 1199092 then

L119881 (120593 119894) le minus61205932

(0) + int

infin

0

1205722

(119911) 120582 (119889119911)

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

(108)


1198712


119894isinS120601(120579)] le 119902E[max

119894isinS120601(0)]


E [max119894isinS


119881 (119909 119894)] (109)





Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Then for any 120585 isin 119871119901

F0([minus120591 0]R119899



119890120574119904

E |119909 (119904)|119901

le

10038171003817100381710038171205851003817100381710038171003817119901

1 minus 120581orsup

0le119904le119905119890120574119904E

1003816100381610038161003816119909 (119904) minus 119863 (119909119904 119903 (119904))

1003816100381610038161003816119901

(1 minus 120581)119901

119905 ge 0

(20)


(119909 + 119910)119901

= (1 minus 1205811)1minus119901

(119909119901

+ 1205811

1minus119901

119910119901

)

forall119909 119910 ge 0 1205811gt 0

(21)


119890120574119905

E|119909 (119905)|119901

le 119890120574119905

[(1 minus 120581)1minus119901

E1003816100381610038161003816119909 (119905) minus 119863 (119909

119905 119903 (119905))

1003816100381610038161003816119901

+ 1205811minus119901

E1003816100381610038161003816119863 (119909

119905 119903 (119905))

1003816100381610038161003816119901

]

le (1 minus 120581)1minus119901

119890120574119905

E1003816100381610038161003816119909 (119905) minus 119863 (119909

119905 119903 (119905))

1003816100381610038161003816119901

+ 120581119890120574119905 supminus120591le120579le0

119890120574120579

E |119909 (119905 + 120579)|119901


0le119904le119905

119890120574119904

E1003816100381610038161003816119909 (119904) minus 119863 (119909

119904 119903 (119904))

1003816100381610038161003816119901

+ 120581 supminus120591le120579le0

119890120574(119904+120579)

E |119909 (119904 + 120579)|119901


0le119904le119905

119890120574119904

E1003816100381610038161003816119909 (119904) minus 119863 (119909

119904 119903 (119904))

1003816100381610038161003816119901


119890120574119904

E |119909 (119904)|119901

(22)

Then


119890120574119904

E |119909 (119904)|119901


E |119909 (120579)|2

]


0le119904le119905

119890120574119904

E1003816100381610038161003816119909 (119904) minus 119863 (119909

119904 119903 (119904))

1003816100381610038161003816119901


119890120574119904

E |119909 (119904)|119901

]

(23)



2 120582 119901 be all


119901

(R119899

timesR+timesSR


that

1198881|119909|

119901

le 119881 (119909 119905 119896) le 1198882|119909|

119901

(24)

for all (119909 119905 119896) isin R119899

times R+


E1003816100381610038161003816119863 (120593 119896)

1003816100381610038161003816119901

le 120581119901 supminus120591le120579le0

119890]1205791003817100381710038171003817120593

1003817100381710038171003817119901

120581 = max 1205811 120581

119896 120593 isin 119871

119901

F119905

(25)


E [max119896isin

S

L119881 (120593 119905 119896)]


S

119881 (120593 (0) minus 119863 (120593 119896) 119905 119896)]

(26)


F119905([minus120591 0]R119899

)satisfying

E [min119896isin

S

119881 (120593 (120579) 119905 + 120579 119896)]


S

119881 (120593 (0) minus 119863 (120593 119896) 119905 119896)]

(27)


F0([minus120591 0]R119899

) 119905 ge 0

lim sup120576rarr0

E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

119890minus]119905

(28)

where

] = min1205741

120591log

119902

(11988821198881) (1 minus 120581)

119901 (29)


1198882

1198881

(1 minus 120581)119901

119890120574120591

= 120582 (30)


Proof Let

(120593 119905 119895) =

119897

sum

119896=1

119881 (120593 119905 119896) 119868119895isinS119896 = 119881 (120593 119905 119896) if 119895 isin S

119896

(31)


(120593120576

119905 119903120576

(119905)) = 119881 (120593120576

119905 119903120576

(119905))

119872

sum

119894=1

120574119897119894 (120593 119905 119894) =

119872

sum

119894=1

120574119897119894

119897

sum

119896=1

119881 (120593 119905 119896) 119868119894isinS119896 = 0

(32)




119880 (119905) = supminus120591le120579le0

[119890120574(119905+120579)E119881 (119909

120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ]


[119890120574(119905+120579)

E (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ]

(33)


119863+

119880 (119905) = lim supℎrarr0+

119880 (119905 + ℎ) minus 119880 (119905)

ℎle 0 forall119905 ge 0 (34)


120576

(119905) minus

119863(119909120576

119905 119903

120576

(119905)) 119905 119903120576

(119905)) or 119880(119905) = 119890120574119905E119881(119909

120576

(119905) minus 119863(119909120576

119905 119903

120576

(119905))

119905 119903120576

(119905))If 119880(119905) gt 119890

120574119905E119881(119909120576

(119905) minus 119863(119909120576

119905 119903

120576

(119905)) 119905 119903120576


119890120574(119905+ℎ)E119881(119909

120576

(119905 + ℎ) minus119863(119909120576

119905+ℎ 119903

120576

(119905 + ℎ)) 119905 + ℎ 119903120576

(119905 + ℎ)) hence119880(119905 + ℎ) le 119880(119905) and 119863

+

119880(119905) le 0If 119880(119905) = 119890

120574119905E119881(119909120576

(119905) minus 119863(119909120576

119905 119903

120576

(119905)) 119905 119903120576

(119905)) we have

119890120574(119905+120579)

E119881 (119909120576

(119905 + 120579) minus 119863 (119909119905+120579

119903120576

(119905 + 120579)) 119905 + 120579 119903120576

(119905 + 120579))

le 119890120574119905

E119881 (119909120576

(119905) minus 119863 (119909119905 119903

120576

(119905)) 119905 119903120576

(119905))

(35)


E119881 (119909120576

(119905 + 120579) minus 119863 (119909119905+120579

119903120576

(119905 + 120579)) 119905 + 120579 119903120576

(119905 + 120579))

le 119890minus120574120579

E119881 (119909120576

(119905) minus 119863 (119909119905 119903

120576

(119905)) 119905 119903120576

(119905))

le 119890120574120591

E119881 (119909120576

(119905) minus 119863 (119909119905 119903

120576

(119905)) 119905 119903120576

(119905))

(36)


119890120574(119905+120579)

E119881 (119909120576

(119905 + 120579) 119905 + 120579 119903120576

(119905 + 120579))

le 1198882119890120574(119905+120579)

E1003816100381610038161003816119909

120576

(119905 + 120579)1003816100381610038161003816119901

le 1198882(1 minus 120581)

119901 sup0le119904le119905

119890120574119904

E1003816100381610038161003816119909

120576

(119904) minus 119863 (119909120576

119904 119903 (119904))

1003816100381610038161003816119901

le1198882

1198881

(1 minus 120581)119901 sup0le119904le119905

119890120574119904

E119881 (119909120576

(119904) minus 119863 (119909120576

119904 119903 (119904)) 119904 119903 (119904))

le1198882

1198881

(1 minus 120581)119901

119890120574119905

E119881 (119909120576

(119905) minus 119863 (119909120576

119905 119903 (119905)) 119905 119903 (119905))

(37)

Then

E119881 (119909120576

(119905 + 120579) 119905 + 120579 119903120576

(119905 + 120579))

lt 119902E119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

(38)

where 119902 gt (11988821198881)(1 minus 120581)

119901

119890120574120591 that is 120574 lt (1120591)(log(119902(119888

21198881)

(1 minus 120581)119901


0gt 0 such

that for any 120576 isin (0 1205760)

E [min119896isin

S

119881 (120593120576

(120579) 119905 + 120579 119896)]


S

119881 (120593120576

(0) minus 119863 (120593120576

119896) 119905 119896)]

(39)


E [max119896isin

S

L119881 (120593120576

119905 119903120576

(119905))]


S

119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(40)

which implies that

E [L119881 (120593120576

119905 119903120576

(119905))]

le minus120582E [119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(41)


E [L119881 (120593120576

119905 119903120576

(119905))]

le minus120574E [119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(42)

Next we consider

119880 (119905 + ℎ) minus 119880 (119905)


[119890120574(119905+120579+ℎ)

E [119881 (119909120576

(119905 + 120579 + ℎ)

minus 119863 (119909120576

119905+120579+ℎ 119903

120576

(119905 + 120579 + ℎ))

119905 + 120579 + ℎ 119903120576

(119905 + 120579 + ℎ)) ]

minus 119890120574(119905+120579)

E [119881 (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579))]]


Eint

119905+120579+ℎ

119905+120579

119890120574119904

[L119881 (120593120576

119904 119903120576

(119904))

+ 120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904))

119904 119903120576

(119904))] 119889119904


Eint

119905+120579+ℎ

119905+120579

119890120574119904

[L (120593120576

119904 119903120576

(119904))

+ 120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904))

119904 119903120576

(119904)) ] 119889119904

(43)



L (120593120576

119905 119903120576

(119905))

= 119905(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

+ 119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119891 (120593120576

119905 119903120576

(119905))

+1

2trace [119892

119879

(120593120576

119905 119903120576

(119905))

times 119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) ]

+

119872

sum

119894=1

120574120576

119903120576(119905)119894

(120593120576

(0) minus 119863 (119909120576

119905 119894) 119905 119894)

= 119905(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

+ 119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119891 (120593120576

119905 119903120576

(119905))

+1

2trace [119892

119879

(120593120576

119905 119903120576

(119905))

times 119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) ]

+

119872

sum

119894=1

120574119903120576(119905)119894

(120593120576

(0) minus 119863 (119909120576

119905 119894) 119905 119894)

= 119881119905(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

+ 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119891 (120593120576

119905 119903120576

(119905))

+1

2trace [119892

119879

(120593120576

119905 119903120576

(119905))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) ]

+

119897

sum

119896=1

120574120576

119903120576(119905)119896

119881 (120593120576

(0) minus 119863 (119909120576

119905 119896) 119905 119896)

+ 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times [119891 (120593120576

119905 119903120576

(119905)) minus 119891 (120593120576

119905 119903120576

(119905))]

+1

2trace [119892

119879

(120593120576

119905 119903120576

(119905))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) minus 119892119879

(120593120576

119905 119903120576

(119905))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) ]

+

119872

sum

119894=1

120574119903120576(119905)119894

(120593120576

(0) minus 119863 (119909120576

119905 119894) 119905 119894)

minus

119897

sum

119896=1

120574119903120576(119905)119896

119881 (120593120576

(0) minus 119863 (119909120576

119905 119896) 119905 119896)

= L119881 (120593120576

119905 119903120576

(119905))

+ 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times [119891 (120593120576

119905 119903120576

(119905)) minus 119891 (120593120576

119905 119903120576

(119905))]

+1

2trace [119892

119879

(120593120576

119905 119903120576

(119905))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) minus 119892119879

(120593120576

119905 119903120576

(119905))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) ]

+

119872

sum

119894=1

120574119903120576(119905)119894

(120593120576

(0) minus 119863 (119909120576

119905 119894) 119905 119894)

minus

119897

sum

119896=1

120574119903120576(119905)119896

119881 (120593120576

(0) minus 119863 (119909120576

119905 119896) 119905 119896)

(44)

Therefore

119880 (119905 + ℎ) minus 119880 (119905)


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [L119881 (120593120576

119904 119903120576

(119904))

+120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119905

119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times [119891 (120593120576

119904 119903120576

(119904)) minus 119891 (120593120576

119904 119903120576

(119904))] 119889119904

+1

2lim sup120576rarr0

E

times int

119905+120579+ℎ

119905+120579

119890120574119904



(120593120576

119904 119903120576

(119904))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times 119892 (120593120576

119904 119903120576

(119904)) minus 119892119879

(120593120576

119904 119903120576

(119904))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times 119892 (120593120576

119904 119903120576

(119904)) ] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times (

119872

sum

119894=1

120574119903120576(119904)119894

(120593120576

(0) minus 119863 (119909120576

119904 119894) 119904 119894)

minus

119897

sum

119896=1

120574119903120576(119904)119896

119881 (120593120576

(0) minus 119863 (119909120576

119904 119896) 119904 119896))119889119904

= 1198681+ 119868

2+ 119868

3+ 119868

4

(45)


119891 (120593120576

119905 119903120576

(119905)) minus 119891 (120593120576

119905 119903120576

(119905))

=

119897

sum

119896=1

119872119896

sum

119895=1

119891 (120593120576

119905 119904119896119895) times [119868

119903120576(119905)=119904119896119895

minus 120583119896

119895119868119903120576(119905)=119896

]

(46)

This implies that

lim120576rarr0

Eint

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times [119891 (120593120576

119904 119903120576

(119904)) minus 119891 (120593120576

119904 119903120576

(119904))] 119889119904

le lim120576rarr0

[E

100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times [119891 (120593120576

119904 119903120576

(119904)) minus 119891 (120593120576

119904 119903120576

(119904))] 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

]

12

= lim120576rarr0

[

[

E

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times

119897

sum

119896=1

119872119896

sum

119895=1

119891 (120593120576

119904 119904119896119895)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

] 119889119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

]

]

12

le lim120576rarr0

[

[

E

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119897

sum

119896=1

119872119896

sum

119895=1

119890120574119904

119871 (1 +1003817100381710038171003817120593

1003817100381710038171003817119901

)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

] 119889119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

]

]

12

(47)


2= 0 Similarly

we can show that

1198683=

1

2lim sup120576rarr0

E

times int

119905+120579+ℎ

119905+120579

119890120574119904


(120593120576

119904 119903120576

(119904))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times 119892 (120593120576

119904 119903120576

(119904))

minus 119892119879

(120593120576

119904 119903120576

(119904))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times 119892 (120593120576

119904 119903120576

(119904)) ] 119889119904 = 0

(48)


119872

sum

119894=1

120574119903120576(119905)119894

(120593120576

(0) minus 119863 (119909120576

119905 119894) 119905 119894)

= Γ (120593120576

(0) minus 119863 (119909120576

119905 119894) 119905 sdot) (119903

120576

(119905))

119897

sum

119896=1

120574119903120576(119905)119896

119881 (120593 (0) minus 119863 (119909120576

119905 119896) 119905 119896)

= Γ119881 (120593 (0) minus 119863 (119909120576

119905 119896) 119905 sdot) (119903

120576

(119905))

(49)

Hence

1198684= lim sup

120576rarr0

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times (

119872

sum

119894=1

120574119903120576(119904)119894

(120593120576

(0) minus 119863 (119909120576

119904 119894) 119904 119894)

minus

119897

sum

119896=1

120574119903120576(119904)119896

119881 (120593120576

(0) minus 119863 (119909120576

119904 119896) 119904 119896))119889119904



E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times (Γ (120593120576

(0) minus 119863 (119909120576

119904 119894) 119904 sdot) (119903

120576

(119904))

minus Γ119881 (120593120576

(0) minus 119863 (119909120576

119904 119896) 119904 sdot) (119904

119896119895)) 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

119897

sum

119896=1

119872119896

sum

119895=1

Γ (120593120576

(0) minus 119863 (119909120576

119904 119894) 119904 sdot) (119904

119896119895)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

] 119889119904


[

[

E

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

times

119897

sum

119896=1

119872119896

sum

119895=1

Γ (120593120576

(0) minus 119863 (119909120576

119904 119894) 119904 sdot) (119904

119896119895)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

]

]

12

(50)


4= 0 Therefore

119880 (119905 + ℎ) minus 119880 (119905)

= lim120576rarr0

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [L119881 (120593120576

119904 119903120576

(119904))

+120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))] 119889119904

le 0

(51)

That is

119880 (119905 + ℎ) le 119880 (119905) (52)


+



lim sup120576rarr0

119890120574119905

E1003816100381610038161003816119909

120576

(119905) minus 119863 (119909120576

119905 119903

120576

(119905))1003816100381610038161003816119901


sup120591le120579le0

119890120574120579

E1003816100381610038161003816119909

120576

(120579) minus 119863 (119909120576

120579 119903

120576

(120579))1003816100381610038161003816

119901



(1 + 120581)119901minus1

times [E1003816100381610038161003816119909

120576

(120579)1003816100381610038161003816119901

+ 1205811minus119901

E1003816100381610038161003816119863 (119909

120576

120579 119903

120576

(120579))1003816100381610038161003816119901

]

le 1198882(1 + 120581)

11990110038171003817100381710038171205851003817100381710038171003817119901

119905 ge 0

(53)


lim sup120576rarr0

119890120574119905

E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

(54)

That is

lim sup120576rarr0

E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

119890minus120574119905

forall119905 ge 0 (55)



119889 [120593120576

(0) minus 119863 (120593120576

119903120576

(119905))]

= 119891 (120593120576

119905 119903120576

(119905)) 119889119905 + intR119898

119887 (119909120576

119905minus 119905 119903

120576

(119905) 119911) (119889119905 119889119911)

1199090= 120585 isin 119862 ([minus120591 0] R

119899

) 119903 (0) isin S

(56)

where 119909120576

119905minus= lim

119904uarr119905119909120576

119904 119863 119862([minus120591 0]R119899

) times S rarr R119899 119887

119862([minus120591 0]R119899





119887(120575)

(120593 119905 119894 119911) = 119887(120575)

(120593 119905 119894 119911120575)

119911 = (1199111 119911

119898) isin R

119898

119894 isin S

(57)


(119889119905 119889119911) = (1(119889119905 119889119911

1)

119898(119889119905 119889119911

119898))

= (1198731(119889119905 119889119911

1) minus 120582

1(119889119911

1) 119889119905 119873

119898(119889119905 119889119911

119898)

minus 120582119898

(119889119911119898) 119889119905)

(58)






119889 [120593120576

(0) minus 119863 (119909120576

119905 119903

120576

(119905))]

= 119891 (119909120576

119905 119905 119903

120576

(119905)) 119889119905

+ intR119898

(119909120576

119905minus 119905 119903

120576

(119905) 119911) (119889119905 119889119911)

1199090= 120585 isin 119862 ([minus120591 0] R

119899

) 119903 (0) isin S

(59)

where 119909120576

119905minus= lim

119904uarr119905119909120576

119904and 119862([minus120591 0]R119899


rarr


119863(120593 119896) =

119873119898

sum

119895=1

120583119896

119895119863(120593 119904

119896119895)

(120593 119905 119896 119911) =

119873119898

sum

119895=1

120583119896

119895119887 (120593 119905 119904

119896119895 119911)

(60)

for each 119904119896119895




gt 0such that

1003816100381610038161003816119891 (120593 119905 119894) minus 119891 (120601 119905 119894)1003816100381610038161003816

or

119898

sum

120575=1

intR

10038161003816100381610038161003816119887(120575)

(120593 119905 119894 119911120575) minus 119887

(120575)

(120601 119905 119894 119911120575)10038161003816100381610038161003816120582120575(119889119911

120575)

le 119871120577

1003817100381710038171003817120593 minus 12060110038171003817100381710038172

(61)




119862([minus120591 0]R119899

) 119894 isin S

1003816100381610038161003816119891 (120593 119905 119894)1003816100381610038161003816 or

119898

sum

120575=1

intR

10038161003816100381610038161003816119887(120575)

(120593 119905 119894 119911120575)10038161003816100381610038161003816120582120575(119889119911

120575)

le 119871 (1 +1003817100381710038171003817120593

10038171003817100381710038172

)

(62)



1003816100381610038161003816119863 (120593 119894) minus 119863 (120601 119894)1003816100381610038161003816 le 120581

1003817100381710038171003817120593 minus 12060110038171003817100381710038172

119863 (0 119894) equiv 0

(63)

Given that 119881 isin 119862119901

(R119899

times R+times SR


L119881 by

L119881 (120593 119905 119894)

= 119881119905(120593 (0) minus 119863 (120593 119894) 119905 119894)

+ 119881119909(120593 (0) minus 119863 (120593 119894) 119905 119894) 119891 (120593 119905 119894)

+

119873

sum

119895=1

120574119894119895119881 (120593 (0) minus 119863 (120593 119894) 119905 119895)

+ intR

119898

sum

120575=1

119881 (120593 (0) minus 119863 (120593 119894) + 119887(120575)

(120593 119905 120580 119911120575) 119905 120580)

minus 119881 (120593 (0) minus 119863 (120593 119894) 119905 119894)

minus 119881119909(120593 (0) minus 119863 (120593 119894) 119905 119894)

times 119887(120575)

(120593 119905 120580 119911120575) 120582

120575(119889119911

120575)

(64)

where

119881119909(120593 (0) minus 119863 (120593 119894) 119905 119894)

= (120597119881 (120593 (0) minus 119863 (120593 119894) 119905 119894)

1205971199091

120597119881 (120593 (0) minus 119863 (120593 119894) 119905 119894)

120597119909119898

)

(65)


(sdot) 119903120576






2


119901

(R119899

times R+times SR

+) satisfying


1198881|119909|

119901

le 119881 (119909 119905 119896) le 1198882|119909|

119901

119896 isin S (66)

for all (119909 119905 119896) isin R119899


following

E1003816100381610038161003816119863 (120593 119896)

1003816100381610038161003816119901

le 120581119901 supminus120591le120579le0

119890]120579 1003817100381710038171003817120593

1003817100381710038171003817

119901

120581 = max 1205811 120581

119896 120593 isin 119871

119901

F119905

(67)


E [max119896isin

S


S

119881 (120593 (0) minus 119863 (120593 119896))]

(68)



F119905([minus120591 0]R119899

)satisfying

E [min119896isin

S

119881 (119909 (119905 + 120579) 119905 + 120579 119896)]


S

119881 (120593 (0) minus 119863 (120593 119896) 119905 119894)] minus120591 le 120579 le 0

(69)


) 119905 ge 0

lim sup120576rarr0

E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

119890minus]119905

(70)

where

] = min1205741

120591log

119902

(11988821198881) (1 minus 120581)

119901 (71)


1198882

1198881

(1 minus 120581)119901

119890120574120591

= 120582 (72)

Proof Define

(120593 119905 120588) =

119897

sum

119896=1

119881 (120593 119905 119896) 119868120588isinS119896 = 119881 (120593 119905 119896) if 120588 isin S

119896

(73)


119880 (119905)


[119890120574(119905+120579)

E119881 (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ]


[119890120574(119905+120579)

E (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ]

(74)


119863+

119880 (119905) = lim supℎrarr0+

119880 (119905 + ℎ) minus 119880 (119905)

ℎle 0 forall119905 ge 0 (75)


E119881 (119909120576

(119905 + 120579) 119905 + 120579 119903120576

(119905 + 120579))

lt 119902E119881 (120593 (0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

(76)


119901

119890120574120591 that is 120574 lt

(1120591)(log(119902(11988821198881)(1 minus 120581)

119901

))

Thus

E [max119896isin

S

L119881 (120593120576

119905 119903120576

(119905))]


S

119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(77)

which implies that

E [L119881 (120593120576

119905 119903120576

(119905))]

le minus120582E [119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(78)


E [L119881 (120593120576

119905 119903120576

(119905))]

le minus120574E [119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(79)

We now consider

119880 (119905 + ℎ) minus 119880 (119905)


[119890120574(119905+120579+ℎ)

E

times [119881 (119909120576

(119905 + 120579 + ℎ)

minus 119863 (119909120576

119905+120579+ℎ 119903

120576

(119905 + 120579 + ℎ))

119905 + 120579 + ℎ 119903120576

(119905 + 120579 + ℎ)) ]

minus 119890120574(119905+120579)

E

times [119881 (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ] ]


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

[L119881 (120593120576

119904 119903120576

(119904))

+120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119905))] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [L (120593120576

119904 119903120576

(119904))

+ 120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119904 119903120576

(119904)) ] 119889119905

(80)



L (120593120576

119905 119903120576

(119905))

= 119905(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

+ 119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119891 (120593120576

119905 119903120576

(119905))

+

119898

sum

120575=1

intR

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905))

+ 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 119905 119903

120576

(119905))

minus (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

minus 119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 120582

120575(119889119911

120575)

+

119873

sum

119895=1

120574120576

119903120576(119905)119895

(120593120576

(0) minus 119863 (120593120576

119895) 119905 119895)

= L119881 (120593120576

119905 119903120576

(119905))

+ 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times [119891 (120593120576

119905 119903120576

(119905)) minus 119891 (120593120576

119905 119903120576

(119905))]

+

119898

sum

120575=1

intR

119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905))

+ 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 119905 119903

120576

(119905))

minus 119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905))

+ (120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 119905 119903

120576

(119905))

times 120582120575(119889119911

120575)

minus

119898

sum

120575=1

intR

119881119909(120593

120576

(0)

minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times (119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)

minus(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)) 120582

120575(119889119911

120575)

+

119873

sum

119895=1

120574119903120576(119905)119895

(120593120576

(0) minus 119863 (120593120576

119895) 119905 119895)

minus

119897

sum

119896=1

120574119903120576(119905)119896

(120593120576

(0) minus 119863 (120593120576

119896) 119905 119896)

(81)

This implies that

119880 (119905 + ℎ) minus 119880 (119905)


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [L119881 (120593120576

119904 119903120576

(119904))

+ 120574119881 (120593120576

(0)

minus119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times [119891 (120593120576

119904 119903120576

(119904)) minus 119891 (120593120576

119904 119903120576

(119904))] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [

119898

sum

120575=1

intR

[119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904))

+119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575) 119904 119903

120576

(119904))

minus 119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904))

+ (120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

119904 119903120576

(119904) )]

times120582120575(119889119911

120575) ] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times

119898

sum

120575=1

intR

[119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times (119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

minus(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575))]

times120582120575(119889119911

120575) 119889119904



E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times (

119873

sum

119895=1

120574119903120576(119904)119895

(120593120576

(0) minus 119863 (120593120576

119895) 119904 119895)

minus

119897

sum

119896=1

120574119903120576(119904)119896

(120593120576

(0) minus 119863 (120593120576

119896) 119904 119896))119889119904

= 1198691+ 119869

2+ 119869

3+ 119869

4+ 119869

5

(82)


119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) minus

(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)

=

119897

sum

119894=1

119873119896

sum

119895=1

119887(120575)

(119909120576

119905minus 119905 119904

119896119895 119911

120575)

times [119868119903120576(119905)=119904119896119895

minus 120583119896

119895119868119903120576(119905)=119896

]

(83)


1198694= lim sup

120576rarr0

119898

sum

120575=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times intR

[119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

minus (120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)]

times 120582120575(119889119911

120575) 119889119904


119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904))

119904 119903120576

(119904))

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)

times [119868119903120576(119904)=119904119896119895

minus120583119896

119895119868119903120576(119904)=119896

]

times 120582120575(119889119911

120575) 119889119904


119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

[E

100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904))

119904 119903120576

(119904))

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

times120582120575(119889119911

120575) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

]

12

(84)


4= 0 Similarly


119905


(0) minus 119863(120593120576

119903120576

(119905)) + 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)

and 120593120576

(0) minus 119863(120593120576

119903120576

(119905)) + (120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) such that

1198693= lim

120576rarr0

119898

sum

120575=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

intR

119881119909(120578

119904)

times [119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

minus(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)]

times 120582120575(119889119911

120575) 119889119904

= lim120576rarr0

119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120578

119904)

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

times 120582120575(119889119911

120575) 119889119904

le lim120576rarr0

119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

[E

100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120578

119905)

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)


times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

times 120582120575(119889119911

120575)119889119904

100381610038161003816100381610038161003816100381610038161003816

2

]

12

(85)



5= 0Therefore


= lim120576rarr0

Eint

119905+120579+ℎ

119905+120579

119890120574119904

times [L119881 (120593120576

119904 119903120576

(119904))

+120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))] 119889119904

le 0

(86)

Then

119880 (119905 + ℎ) le 119880 (119905) (87)


E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

119890minus]119905

(88)


5 Examples



in (14) with

Γ = (

minus1 0 1 0 0

1 minus2 1 0 0

2 1 minus3 0 0

0 0 0 minus1 1

0 0 0 1 minus1

) (89)

Γ = (

minus1 0 1 0 0

0 minus1 0 1 0

0 0 minus1 0 1

0 1 0 minus1 0

1 0 0 0 minus1

) (90)


(1205871

1205872

1205873)(

minus1 0 1

1 minus2 1

2 1 minus3

) = 0 (91)

and 1205871+ 120587

2+ 120587

3= 1 we get 1205831


(1205874

1205875) (

minus1 1

1 minus1) = 0 (92)

and 1205874+ 120587

5= 1 we have 120583

2

= (12 12) So

Γ = 120583Γ1 = (

5

8

1

8

1

40 0

0 0 01

2

1

2

)

times (

minus1 0 1 0 0

0 minus1 0 1 0

0 0 minus1 0 1

0 1 0 minus1 0

1 0 0 0 minus1

)(

1 0

1 0

1 0

0 1

0 1

)

= (minus3

8

3

81 minus1

)

(93)


119889 [120593120576

(0) minus 119863 (120593120576

119903120576

(119905))]

= 119891 (120593120576

119903120576

(119905)) 119889119905 + 119892 (120593120576

119903120576

(119905)) 119889119908 (119905)

(94)

with

119863(120593 11990411) = minus06 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990412) = minus02 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990413) = minus04 int

0

minus1

120593 (120579) 119889120579

119891 (120593 11990411) = minus16120593 (0) minus 8 cos [120593 (0)]

119891 (120593 11990412) = 8120593 (0) + 4 cos [120593 (0)]

119891 (120593 11990413) = 16120593 (0)

119892 (120593 11990411) =

radic10

10int

0

minus1

120593 (120579) 119889120579 cos [int0

minus1

120593 (120579) 119889120579]

119892 (120593 11990412) = minus

radic2

2int

0

minus1

120593 (120579) 119889120579 sin [int

0

minus1

120593 (120579) 119889120579]

119892 (120593 11990413) =

radic3

2int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990421) = 05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990422) = 05 int

0

minus1

120593 (120579) 119889120579

119891 (120593 11990421) = minus2120593 (0) 119891 (120593 119904

22) = minus2120593 (0)


119892 (120593 11990421) =

int

0

minus1

120593 (120579) 119889120579 sin [int

0

minus1

120593 (120579) 119889120579]

4radic2

119892 (120593 11990422) =

int

0

minus1

120593 (120579) 119889120579 cos [int0

minus1

120593 (120579) 119889120579]

4radic2

(95)

For any 120593 isin 1198712

F119905([minus1 0]R) and 120581 = max06 02 04 = 06


E1003816100381610038161003816119863 (120593 119894)

10038161003816100381610038162


119890]120579E

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

le 0361003817100381710038171003817120593

10038171003817100381710038172

(96)


119889 [120593 (0) minus 119863 (120593 119903 (119905))] = 119891 (120593 119903 (119905)) 119889119905 + 119892 (120593 119903 (119905)) 119889119908 (119905)

(97)


119863(120593 1) = minus05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 2) = 05 int

0

minus1

120593 (120579) 119889120579

119891 (120593 1) = minus5120593 (0) 119891 (120593 2) = minus2120593 (0)

119892 (120593 1) =1

2int

0

minus1

120593 (120579) 119889120579 119892 (120593 2) =1

4int

0

minus1

120593 (120579) 119889120579

(98)

We define 119881(119909 1) = 21199092 119881(119909 2) = 119909


L119881 (120593 1) le minus203

81205932

(0) +13

32

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

L119881 (120593 2) le minus5

21205932

(0) +13

16

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

(99)

Consequently

max119894=12

L119881 (120593 119894) le minus5

21205932

(0) +13

16

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

= minus5

4[max119894=12

119881 (119909 119894)] +13

16[min119894=12

119881 (119909 119894)]

(100)


2


119894isinS120601(120579)] le


E [max119894isinS

L119881 (120593 119894)] le minus (5

4minus

13119902

16)E [max

119894=12

119881 (119909 119894)] (101)




Γ120576

=1

120576Γ + Γ =

1

120576(

minus2 0 2 0

1 minus2 0 1

0 2 minus2 0

0 1 1 minus2

) (102)



119889 [120593120576

(0) minus 119863 (120593120576

119903120576

(119905))]

= 119891 (120593120576

119903120576

(119905)) 119889119905 + int

infin

0

120590 (119903120576

(119905) 119911) 119909120576

119905minus (119889119905 119889119911)

(103)

with

119863(120593 1) = minus09 int

0

minus1

120593 (120579) 119889120579

119863 (120593 2) = minus04 int

0

minus1

120593 (120579) 119889120579

119863 (120593 3) = minus05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 4) = minus03 int

0

minus1

120593 (120579) 119889120579

119891 (120593 1) = 2 sin [120593 (0)] 119891 (120593 2) = minus11

2120593 (0)

119891 (120593 3) = minus11

3120593 (0) 119891 (120593 4) = minus2 sin [120593 (0)]

(104)

Let

120572 (119911) =2

11120590 (1 119911) +

4

11120590 (2 119911) +

3

11120590 (3 119911) +

2

11120590 (4 119911)

int

infin

0

1205722

(119911) 120582 (119889119911) lt 2

(105)

For any 120593 isin 1198712

F119905([minus1 0]R) and 120581 = max09 04 05 03 =


E1003816100381610038161003816119863 (120593 119894)

10038161003816100381610038162


119890]120579E

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

le 0811003817100381710038171003817120593

10038171003817100381710038172

(106)


119889 [120593 (0) + 05 int

0

minus1

120593 (120579) 119889120579]

= minus3120593 (0) 119889119905 + int

infin

0

120572 (119911) 119909119905minus (119889119905 119889119911)

(107)


Let 119881(119909) = 1199092 then

L119881 (120593 119894) le minus61205932

(0) + int

infin

0

1205722

(119911) 120582 (119889119911)

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

(108)


1198712


119894isinS120601(120579)] le 119902E[max

119894isinS120601(0)]


E [max119894isinS


119881 (119909 119894)] (109)





Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119880 (119905) = supminus120591le120579le0

[119890120574(119905+120579)E119881 (119909

120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ]


[119890120574(119905+120579)

E (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ]

(33)


119863+

119880 (119905) = lim supℎrarr0+

119880 (119905 + ℎ) minus 119880 (119905)

ℎle 0 forall119905 ge 0 (34)


120576

(119905) minus

119863(119909120576

119905 119903

120576

(119905)) 119905 119903120576

(119905)) or 119880(119905) = 119890120574119905E119881(119909

120576

(119905) minus 119863(119909120576

119905 119903

120576

(119905))

119905 119903120576

(119905))If 119880(119905) gt 119890

120574119905E119881(119909120576

(119905) minus 119863(119909120576

119905 119903

120576

(119905)) 119905 119903120576


119890120574(119905+ℎ)E119881(119909

120576

(119905 + ℎ) minus119863(119909120576

119905+ℎ 119903

120576

(119905 + ℎ)) 119905 + ℎ 119903120576

(119905 + ℎ)) hence119880(119905 + ℎ) le 119880(119905) and 119863

+

119880(119905) le 0If 119880(119905) = 119890

120574119905E119881(119909120576

(119905) minus 119863(119909120576

119905 119903

120576

(119905)) 119905 119903120576

(119905)) we have

119890120574(119905+120579)

E119881 (119909120576

(119905 + 120579) minus 119863 (119909119905+120579

119903120576

(119905 + 120579)) 119905 + 120579 119903120576

(119905 + 120579))

le 119890120574119905

E119881 (119909120576

(119905) minus 119863 (119909119905 119903

120576

(119905)) 119905 119903120576

(119905))

(35)


E119881 (119909120576

(119905 + 120579) minus 119863 (119909119905+120579

119903120576

(119905 + 120579)) 119905 + 120579 119903120576

(119905 + 120579))

le 119890minus120574120579

E119881 (119909120576

(119905) minus 119863 (119909119905 119903

120576

(119905)) 119905 119903120576

(119905))

le 119890120574120591

E119881 (119909120576

(119905) minus 119863 (119909119905 119903

120576

(119905)) 119905 119903120576

(119905))

(36)


119890120574(119905+120579)

E119881 (119909120576

(119905 + 120579) 119905 + 120579 119903120576

(119905 + 120579))

le 1198882119890120574(119905+120579)

E1003816100381610038161003816119909

120576

(119905 + 120579)1003816100381610038161003816119901

le 1198882(1 minus 120581)

119901 sup0le119904le119905

119890120574119904

E1003816100381610038161003816119909

120576

(119904) minus 119863 (119909120576

119904 119903 (119904))

1003816100381610038161003816119901

le1198882

1198881

(1 minus 120581)119901 sup0le119904le119905

119890120574119904

E119881 (119909120576

(119904) minus 119863 (119909120576

119904 119903 (119904)) 119904 119903 (119904))

le1198882

1198881

(1 minus 120581)119901

119890120574119905

E119881 (119909120576

(119905) minus 119863 (119909120576

119905 119903 (119905)) 119905 119903 (119905))

(37)

Then

E119881 (119909120576

(119905 + 120579) 119905 + 120579 119903120576

(119905 + 120579))

lt 119902E119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

(38)

where 119902 gt (11988821198881)(1 minus 120581)

119901

119890120574120591 that is 120574 lt (1120591)(log(119902(119888

21198881)

(1 minus 120581)119901


0gt 0 such

that for any 120576 isin (0 1205760)

E [min119896isin

S

119881 (120593120576

(120579) 119905 + 120579 119896)]


S

119881 (120593120576

(0) minus 119863 (120593120576

119896) 119905 119896)]

(39)


E [max119896isin

S

L119881 (120593120576

119905 119903120576

(119905))]


S

119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(40)

which implies that

E [L119881 (120593120576

119905 119903120576

(119905))]

le minus120582E [119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(41)


E [L119881 (120593120576

119905 119903120576

(119905))]

le minus120574E [119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(42)

Next we consider

119880 (119905 + ℎ) minus 119880 (119905)


[119890120574(119905+120579+ℎ)

E [119881 (119909120576

(119905 + 120579 + ℎ)

minus 119863 (119909120576

119905+120579+ℎ 119903

120576

(119905 + 120579 + ℎ))

119905 + 120579 + ℎ 119903120576

(119905 + 120579 + ℎ)) ]

minus 119890120574(119905+120579)

E [119881 (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579))]]


Eint

119905+120579+ℎ

119905+120579

119890120574119904

[L119881 (120593120576

119904 119903120576

(119904))

+ 120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904))

119904 119903120576

(119904))] 119889119904


Eint

119905+120579+ℎ

119905+120579

119890120574119904

[L (120593120576

119904 119903120576

(119904))

+ 120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904))

119904 119903120576

(119904)) ] 119889119904

(43)



L (120593120576

119905 119903120576

(119905))

= 119905(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

+ 119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119891 (120593120576

119905 119903120576

(119905))

+1

2trace [119892

119879

(120593120576

119905 119903120576

(119905))

times 119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) ]

+

119872

sum

119894=1

120574120576

119903120576(119905)119894

(120593120576

(0) minus 119863 (119909120576

119905 119894) 119905 119894)

= 119905(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

+ 119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119891 (120593120576

119905 119903120576

(119905))

+1

2trace [119892

119879

(120593120576

119905 119903120576

(119905))

times 119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) ]

+

119872

sum

119894=1

120574119903120576(119905)119894

(120593120576

(0) minus 119863 (119909120576

119905 119894) 119905 119894)

= 119881119905(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

+ 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119891 (120593120576

119905 119903120576

(119905))

+1

2trace [119892

119879

(120593120576

119905 119903120576

(119905))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) ]

+

119897

sum

119896=1

120574120576

119903120576(119905)119896

119881 (120593120576

(0) minus 119863 (119909120576

119905 119896) 119905 119896)

+ 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times [119891 (120593120576

119905 119903120576

(119905)) minus 119891 (120593120576

119905 119903120576

(119905))]

+1

2trace [119892

119879

(120593120576

119905 119903120576

(119905))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) minus 119892119879

(120593120576

119905 119903120576

(119905))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) ]

+

119872

sum

119894=1

120574119903120576(119905)119894

(120593120576

(0) minus 119863 (119909120576

119905 119894) 119905 119894)

minus

119897

sum

119896=1

120574119903120576(119905)119896

119881 (120593120576

(0) minus 119863 (119909120576

119905 119896) 119905 119896)

= L119881 (120593120576

119905 119903120576

(119905))

+ 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times [119891 (120593120576

119905 119903120576

(119905)) minus 119891 (120593120576

119905 119903120576

(119905))]

+1

2trace [119892

119879

(120593120576

119905 119903120576

(119905))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) minus 119892119879

(120593120576

119905 119903120576

(119905))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) ]

+

119872

sum

119894=1

120574119903120576(119905)119894

(120593120576

(0) minus 119863 (119909120576

119905 119894) 119905 119894)

minus

119897

sum

119896=1

120574119903120576(119905)119896

119881 (120593120576

(0) minus 119863 (119909120576

119905 119896) 119905 119896)

(44)

Therefore

119880 (119905 + ℎ) minus 119880 (119905)


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [L119881 (120593120576

119904 119903120576

(119904))

+120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119905

119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times [119891 (120593120576

119904 119903120576

(119904)) minus 119891 (120593120576

119904 119903120576

(119904))] 119889119904

+1

2lim sup120576rarr0

E

times int

119905+120579+ℎ

119905+120579

119890120574119904



(120593120576

119904 119903120576

(119904))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times 119892 (120593120576

119904 119903120576

(119904)) minus 119892119879

(120593120576

119904 119903120576

(119904))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times 119892 (120593120576

119904 119903120576

(119904)) ] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times (

119872

sum

119894=1

120574119903120576(119904)119894

(120593120576

(0) minus 119863 (119909120576

119904 119894) 119904 119894)

minus

119897

sum

119896=1

120574119903120576(119904)119896

119881 (120593120576

(0) minus 119863 (119909120576

119904 119896) 119904 119896))119889119904

= 1198681+ 119868

2+ 119868

3+ 119868

4

(45)


119891 (120593120576

119905 119903120576

(119905)) minus 119891 (120593120576

119905 119903120576

(119905))

=

119897

sum

119896=1

119872119896

sum

119895=1

119891 (120593120576

119905 119904119896119895) times [119868

119903120576(119905)=119904119896119895

minus 120583119896

119895119868119903120576(119905)=119896

]

(46)

This implies that

lim120576rarr0

Eint

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times [119891 (120593120576

119904 119903120576

(119904)) minus 119891 (120593120576

119904 119903120576

(119904))] 119889119904

le lim120576rarr0

[E

100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times [119891 (120593120576

119904 119903120576

(119904)) minus 119891 (120593120576

119904 119903120576

(119904))] 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

]

12

= lim120576rarr0

[

[

E

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times

119897

sum

119896=1

119872119896

sum

119895=1

119891 (120593120576

119904 119904119896119895)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

] 119889119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

]

]

12

le lim120576rarr0

[

[

E

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119897

sum

119896=1

119872119896

sum

119895=1

119890120574119904

119871 (1 +1003817100381710038171003817120593

1003817100381710038171003817119901

)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

] 119889119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

]

]

12

(47)


2= 0 Similarly

we can show that

1198683=

1

2lim sup120576rarr0

E

times int

119905+120579+ℎ

119905+120579

119890120574119904


(120593120576

119904 119903120576

(119904))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times 119892 (120593120576

119904 119903120576

(119904))

minus 119892119879

(120593120576

119904 119903120576

(119904))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times 119892 (120593120576

119904 119903120576

(119904)) ] 119889119904 = 0

(48)


119872

sum

119894=1

120574119903120576(119905)119894

(120593120576

(0) minus 119863 (119909120576

119905 119894) 119905 119894)

= Γ (120593120576

(0) minus 119863 (119909120576

119905 119894) 119905 sdot) (119903

120576

(119905))

119897

sum

119896=1

120574119903120576(119905)119896

119881 (120593 (0) minus 119863 (119909120576

119905 119896) 119905 119896)

= Γ119881 (120593 (0) minus 119863 (119909120576

119905 119896) 119905 sdot) (119903

120576

(119905))

(49)

Hence

1198684= lim sup

120576rarr0

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times (

119872

sum

119894=1

120574119903120576(119904)119894

(120593120576

(0) minus 119863 (119909120576

119904 119894) 119904 119894)

minus

119897

sum

119896=1

120574119903120576(119904)119896

119881 (120593120576

(0) minus 119863 (119909120576

119904 119896) 119904 119896))119889119904



E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times (Γ (120593120576

(0) minus 119863 (119909120576

119904 119894) 119904 sdot) (119903

120576

(119904))

minus Γ119881 (120593120576

(0) minus 119863 (119909120576

119904 119896) 119904 sdot) (119904

119896119895)) 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

119897

sum

119896=1

119872119896

sum

119895=1

Γ (120593120576

(0) minus 119863 (119909120576

119904 119894) 119904 sdot) (119904

119896119895)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

] 119889119904


[

[

E

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

times

119897

sum

119896=1

119872119896

sum

119895=1

Γ (120593120576

(0) minus 119863 (119909120576

119904 119894) 119904 sdot) (119904

119896119895)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

]

]

12

(50)


4= 0 Therefore

119880 (119905 + ℎ) minus 119880 (119905)

= lim120576rarr0

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [L119881 (120593120576

119904 119903120576

(119904))

+120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))] 119889119904

le 0

(51)

That is

119880 (119905 + ℎ) le 119880 (119905) (52)


+



lim sup120576rarr0

119890120574119905

E1003816100381610038161003816119909

120576

(119905) minus 119863 (119909120576

119905 119903

120576

(119905))1003816100381610038161003816119901


sup120591le120579le0

119890120574120579

E1003816100381610038161003816119909

120576

(120579) minus 119863 (119909120576

120579 119903

120576

(120579))1003816100381610038161003816

119901



(1 + 120581)119901minus1

times [E1003816100381610038161003816119909

120576

(120579)1003816100381610038161003816119901

+ 1205811minus119901

E1003816100381610038161003816119863 (119909

120576

120579 119903

120576

(120579))1003816100381610038161003816119901

]

le 1198882(1 + 120581)

11990110038171003817100381710038171205851003817100381710038171003817119901

119905 ge 0

(53)


lim sup120576rarr0

119890120574119905

E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

(54)

That is

lim sup120576rarr0

E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

119890minus120574119905

forall119905 ge 0 (55)



119889 [120593120576

(0) minus 119863 (120593120576

119903120576

(119905))]

= 119891 (120593120576

119905 119903120576

(119905)) 119889119905 + intR119898

119887 (119909120576

119905minus 119905 119903

120576

(119905) 119911) (119889119905 119889119911)

1199090= 120585 isin 119862 ([minus120591 0] R

119899

) 119903 (0) isin S

(56)

where 119909120576

119905minus= lim

119904uarr119905119909120576

119904 119863 119862([minus120591 0]R119899

) times S rarr R119899 119887

119862([minus120591 0]R119899





119887(120575)

(120593 119905 119894 119911) = 119887(120575)

(120593 119905 119894 119911120575)

119911 = (1199111 119911

119898) isin R

119898

119894 isin S

(57)


(119889119905 119889119911) = (1(119889119905 119889119911

1)

119898(119889119905 119889119911

119898))

= (1198731(119889119905 119889119911

1) minus 120582

1(119889119911

1) 119889119905 119873

119898(119889119905 119889119911

119898)

minus 120582119898

(119889119911119898) 119889119905)

(58)






119889 [120593120576

(0) minus 119863 (119909120576

119905 119903

120576

(119905))]

= 119891 (119909120576

119905 119905 119903

120576

(119905)) 119889119905

+ intR119898

(119909120576

119905minus 119905 119903

120576

(119905) 119911) (119889119905 119889119911)

1199090= 120585 isin 119862 ([minus120591 0] R

119899

) 119903 (0) isin S

(59)

where 119909120576

119905minus= lim

119904uarr119905119909120576

119904and 119862([minus120591 0]R119899


rarr


119863(120593 119896) =

119873119898

sum

119895=1

120583119896

119895119863(120593 119904

119896119895)

(120593 119905 119896 119911) =

119873119898

sum

119895=1

120583119896

119895119887 (120593 119905 119904

119896119895 119911)

(60)

for each 119904119896119895




gt 0such that

1003816100381610038161003816119891 (120593 119905 119894) minus 119891 (120601 119905 119894)1003816100381610038161003816

or

119898

sum

120575=1

intR

10038161003816100381610038161003816119887(120575)

(120593 119905 119894 119911120575) minus 119887

(120575)

(120601 119905 119894 119911120575)10038161003816100381610038161003816120582120575(119889119911

120575)

le 119871120577

1003817100381710038171003817120593 minus 12060110038171003817100381710038172

(61)




119862([minus120591 0]R119899

) 119894 isin S

1003816100381610038161003816119891 (120593 119905 119894)1003816100381610038161003816 or

119898

sum

120575=1

intR

10038161003816100381610038161003816119887(120575)

(120593 119905 119894 119911120575)10038161003816100381610038161003816120582120575(119889119911

120575)

le 119871 (1 +1003817100381710038171003817120593

10038171003817100381710038172

)

(62)



1003816100381610038161003816119863 (120593 119894) minus 119863 (120601 119894)1003816100381610038161003816 le 120581

1003817100381710038171003817120593 minus 12060110038171003817100381710038172

119863 (0 119894) equiv 0

(63)

Given that 119881 isin 119862119901

(R119899

times R+times SR


L119881 by

L119881 (120593 119905 119894)

= 119881119905(120593 (0) minus 119863 (120593 119894) 119905 119894)

+ 119881119909(120593 (0) minus 119863 (120593 119894) 119905 119894) 119891 (120593 119905 119894)

+

119873

sum

119895=1

120574119894119895119881 (120593 (0) minus 119863 (120593 119894) 119905 119895)

+ intR

119898

sum

120575=1

119881 (120593 (0) minus 119863 (120593 119894) + 119887(120575)

(120593 119905 120580 119911120575) 119905 120580)

minus 119881 (120593 (0) minus 119863 (120593 119894) 119905 119894)

minus 119881119909(120593 (0) minus 119863 (120593 119894) 119905 119894)

times 119887(120575)

(120593 119905 120580 119911120575) 120582

120575(119889119911

120575)

(64)

where

119881119909(120593 (0) minus 119863 (120593 119894) 119905 119894)

= (120597119881 (120593 (0) minus 119863 (120593 119894) 119905 119894)

1205971199091

120597119881 (120593 (0) minus 119863 (120593 119894) 119905 119894)

120597119909119898

)

(65)


(sdot) 119903120576






2


119901

(R119899

times R+times SR

+) satisfying


1198881|119909|

119901

le 119881 (119909 119905 119896) le 1198882|119909|

119901

119896 isin S (66)

for all (119909 119905 119896) isin R119899


following

E1003816100381610038161003816119863 (120593 119896)

1003816100381610038161003816119901

le 120581119901 supminus120591le120579le0

119890]120579 1003817100381710038171003817120593

1003817100381710038171003817

119901

120581 = max 1205811 120581

119896 120593 isin 119871

119901

F119905

(67)


E [max119896isin

S


S

119881 (120593 (0) minus 119863 (120593 119896))]

(68)



F119905([minus120591 0]R119899

)satisfying

E [min119896isin

S

119881 (119909 (119905 + 120579) 119905 + 120579 119896)]


S

119881 (120593 (0) minus 119863 (120593 119896) 119905 119894)] minus120591 le 120579 le 0

(69)


) 119905 ge 0

lim sup120576rarr0

E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

119890minus]119905

(70)

where

] = min1205741

120591log

119902

(11988821198881) (1 minus 120581)

119901 (71)


1198882

1198881

(1 minus 120581)119901

119890120574120591

= 120582 (72)

Proof Define

(120593 119905 120588) =

119897

sum

119896=1

119881 (120593 119905 119896) 119868120588isinS119896 = 119881 (120593 119905 119896) if 120588 isin S

119896

(73)


119880 (119905)


[119890120574(119905+120579)

E119881 (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ]


[119890120574(119905+120579)

E (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ]

(74)


119863+

119880 (119905) = lim supℎrarr0+

119880 (119905 + ℎ) minus 119880 (119905)

ℎle 0 forall119905 ge 0 (75)


E119881 (119909120576

(119905 + 120579) 119905 + 120579 119903120576

(119905 + 120579))

lt 119902E119881 (120593 (0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

(76)


119901

119890120574120591 that is 120574 lt

(1120591)(log(119902(11988821198881)(1 minus 120581)

119901

))

Thus

E [max119896isin

S

L119881 (120593120576

119905 119903120576

(119905))]


S

119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(77)

which implies that

E [L119881 (120593120576

119905 119903120576

(119905))]

le minus120582E [119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(78)


E [L119881 (120593120576

119905 119903120576

(119905))]

le minus120574E [119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(79)

We now consider

119880 (119905 + ℎ) minus 119880 (119905)


[119890120574(119905+120579+ℎ)

E

times [119881 (119909120576

(119905 + 120579 + ℎ)

minus 119863 (119909120576

119905+120579+ℎ 119903

120576

(119905 + 120579 + ℎ))

119905 + 120579 + ℎ 119903120576

(119905 + 120579 + ℎ)) ]

minus 119890120574(119905+120579)

E

times [119881 (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ] ]


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

[L119881 (120593120576

119904 119903120576

(119904))

+120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119905))] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [L (120593120576

119904 119903120576

(119904))

+ 120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119904 119903120576

(119904)) ] 119889119905

(80)



L (120593120576

119905 119903120576

(119905))

= 119905(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

+ 119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119891 (120593120576

119905 119903120576

(119905))

+

119898

sum

120575=1

intR

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905))

+ 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 119905 119903

120576

(119905))

minus (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

minus 119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 120582

120575(119889119911

120575)

+

119873

sum

119895=1

120574120576

119903120576(119905)119895

(120593120576

(0) minus 119863 (120593120576

119895) 119905 119895)

= L119881 (120593120576

119905 119903120576

(119905))

+ 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times [119891 (120593120576

119905 119903120576

(119905)) minus 119891 (120593120576

119905 119903120576

(119905))]

+

119898

sum

120575=1

intR

119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905))

+ 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 119905 119903

120576

(119905))

minus 119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905))

+ (120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 119905 119903

120576

(119905))

times 120582120575(119889119911

120575)

minus

119898

sum

120575=1

intR

119881119909(120593

120576

(0)

minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times (119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)

minus(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)) 120582

120575(119889119911

120575)

+

119873

sum

119895=1

120574119903120576(119905)119895

(120593120576

(0) minus 119863 (120593120576

119895) 119905 119895)

minus

119897

sum

119896=1

120574119903120576(119905)119896

(120593120576

(0) minus 119863 (120593120576

119896) 119905 119896)

(81)

This implies that

119880 (119905 + ℎ) minus 119880 (119905)


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [L119881 (120593120576

119904 119903120576

(119904))

+ 120574119881 (120593120576

(0)

minus119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times [119891 (120593120576

119904 119903120576

(119904)) minus 119891 (120593120576

119904 119903120576

(119904))] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [

119898

sum

120575=1

intR

[119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904))

+119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575) 119904 119903

120576

(119904))

minus 119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904))

+ (120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

119904 119903120576

(119904) )]

times120582120575(119889119911

120575) ] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times

119898

sum

120575=1

intR

[119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times (119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

minus(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575))]

times120582120575(119889119911

120575) 119889119904



E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times (

119873

sum

119895=1

120574119903120576(119904)119895

(120593120576

(0) minus 119863 (120593120576

119895) 119904 119895)

minus

119897

sum

119896=1

120574119903120576(119904)119896

(120593120576

(0) minus 119863 (120593120576

119896) 119904 119896))119889119904

= 1198691+ 119869

2+ 119869

3+ 119869

4+ 119869

5

(82)


119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) minus

(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)

=

119897

sum

119894=1

119873119896

sum

119895=1

119887(120575)

(119909120576

119905minus 119905 119904

119896119895 119911

120575)

times [119868119903120576(119905)=119904119896119895

minus 120583119896

119895119868119903120576(119905)=119896

]

(83)


1198694= lim sup

120576rarr0

119898

sum

120575=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times intR

[119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

minus (120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)]

times 120582120575(119889119911

120575) 119889119904


119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904))

119904 119903120576

(119904))

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)

times [119868119903120576(119904)=119904119896119895

minus120583119896

119895119868119903120576(119904)=119896

]

times 120582120575(119889119911

120575) 119889119904


119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

[E

100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904))

119904 119903120576

(119904))

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

times120582120575(119889119911

120575) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

]

12

(84)


4= 0 Similarly


119905


(0) minus 119863(120593120576

119903120576

(119905)) + 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)

and 120593120576

(0) minus 119863(120593120576

119903120576

(119905)) + (120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) such that

1198693= lim

120576rarr0

119898

sum

120575=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

intR

119881119909(120578

119904)

times [119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

minus(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)]

times 120582120575(119889119911

120575) 119889119904

= lim120576rarr0

119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120578

119904)

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

times 120582120575(119889119911

120575) 119889119904

le lim120576rarr0

119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

[E

100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120578

119905)

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)


times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

times 120582120575(119889119911

120575)119889119904

100381610038161003816100381610038161003816100381610038161003816

2

]

12

(85)



5= 0Therefore


= lim120576rarr0

Eint

119905+120579+ℎ

119905+120579

119890120574119904

times [L119881 (120593120576

119904 119903120576

(119904))

+120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))] 119889119904

le 0

(86)

Then

119880 (119905 + ℎ) le 119880 (119905) (87)


E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

119890minus]119905

(88)


5 Examples



in (14) with

Γ = (

minus1 0 1 0 0

1 minus2 1 0 0

2 1 minus3 0 0

0 0 0 minus1 1

0 0 0 1 minus1

) (89)

Γ = (

minus1 0 1 0 0

0 minus1 0 1 0

0 0 minus1 0 1

0 1 0 minus1 0

1 0 0 0 minus1

) (90)


(1205871

1205872

1205873)(

minus1 0 1

1 minus2 1

2 1 minus3

) = 0 (91)

and 1205871+ 120587

2+ 120587

3= 1 we get 1205831


(1205874

1205875) (

minus1 1

1 minus1) = 0 (92)

and 1205874+ 120587

5= 1 we have 120583

2

= (12 12) So

Γ = 120583Γ1 = (

5

8

1

8

1

40 0

0 0 01

2

1

2

)

times (

minus1 0 1 0 0

0 minus1 0 1 0

0 0 minus1 0 1

0 1 0 minus1 0

1 0 0 0 minus1

)(

1 0

1 0

1 0

0 1

0 1

)

= (minus3

8

3

81 minus1

)

(93)


119889 [120593120576

(0) minus 119863 (120593120576

119903120576

(119905))]

= 119891 (120593120576

119903120576

(119905)) 119889119905 + 119892 (120593120576

119903120576

(119905)) 119889119908 (119905)

(94)

with

119863(120593 11990411) = minus06 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990412) = minus02 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990413) = minus04 int

0

minus1

120593 (120579) 119889120579

119891 (120593 11990411) = minus16120593 (0) minus 8 cos [120593 (0)]

119891 (120593 11990412) = 8120593 (0) + 4 cos [120593 (0)]

119891 (120593 11990413) = 16120593 (0)

119892 (120593 11990411) =

radic10

10int

0

minus1

120593 (120579) 119889120579 cos [int0

minus1

120593 (120579) 119889120579]

119892 (120593 11990412) = minus

radic2

2int

0

minus1

120593 (120579) 119889120579 sin [int

0

minus1

120593 (120579) 119889120579]

119892 (120593 11990413) =

radic3

2int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990421) = 05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990422) = 05 int

0

minus1

120593 (120579) 119889120579

119891 (120593 11990421) = minus2120593 (0) 119891 (120593 119904

22) = minus2120593 (0)


119892 (120593 11990421) =

int

0

minus1

120593 (120579) 119889120579 sin [int

0

minus1

120593 (120579) 119889120579]

4radic2

119892 (120593 11990422) =

int

0

minus1

120593 (120579) 119889120579 cos [int0

minus1

120593 (120579) 119889120579]

4radic2

(95)

For any 120593 isin 1198712

F119905([minus1 0]R) and 120581 = max06 02 04 = 06


E1003816100381610038161003816119863 (120593 119894)

10038161003816100381610038162


119890]120579E

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

le 0361003817100381710038171003817120593

10038171003817100381710038172

(96)


119889 [120593 (0) minus 119863 (120593 119903 (119905))] = 119891 (120593 119903 (119905)) 119889119905 + 119892 (120593 119903 (119905)) 119889119908 (119905)

(97)


119863(120593 1) = minus05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 2) = 05 int

0

minus1

120593 (120579) 119889120579

119891 (120593 1) = minus5120593 (0) 119891 (120593 2) = minus2120593 (0)

119892 (120593 1) =1

2int

0

minus1

120593 (120579) 119889120579 119892 (120593 2) =1

4int

0

minus1

120593 (120579) 119889120579

(98)

We define 119881(119909 1) = 21199092 119881(119909 2) = 119909


L119881 (120593 1) le minus203

81205932

(0) +13

32

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

L119881 (120593 2) le minus5

21205932

(0) +13

16

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

(99)

Consequently

max119894=12

L119881 (120593 119894) le minus5

21205932

(0) +13

16

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

= minus5

4[max119894=12

119881 (119909 119894)] +13

16[min119894=12

119881 (119909 119894)]

(100)


2


119894isinS120601(120579)] le


E [max119894isinS

L119881 (120593 119894)] le minus (5

4minus

13119902

16)E [max

119894=12

119881 (119909 119894)] (101)




Γ120576

=1

120576Γ + Γ =

1

120576(

minus2 0 2 0

1 minus2 0 1

0 2 minus2 0

0 1 1 minus2

) (102)



119889 [120593120576

(0) minus 119863 (120593120576

119903120576

(119905))]

= 119891 (120593120576

119903120576

(119905)) 119889119905 + int

infin

0

120590 (119903120576

(119905) 119911) 119909120576

119905minus (119889119905 119889119911)

(103)

with

119863(120593 1) = minus09 int

0

minus1

120593 (120579) 119889120579

119863 (120593 2) = minus04 int

0

minus1

120593 (120579) 119889120579

119863 (120593 3) = minus05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 4) = minus03 int

0

minus1

120593 (120579) 119889120579

119891 (120593 1) = 2 sin [120593 (0)] 119891 (120593 2) = minus11

2120593 (0)

119891 (120593 3) = minus11

3120593 (0) 119891 (120593 4) = minus2 sin [120593 (0)]

(104)

Let

120572 (119911) =2

11120590 (1 119911) +

4

11120590 (2 119911) +

3

11120590 (3 119911) +

2

11120590 (4 119911)

int

infin

0

1205722

(119911) 120582 (119889119911) lt 2

(105)

For any 120593 isin 1198712

F119905([minus1 0]R) and 120581 = max09 04 05 03 =


E1003816100381610038161003816119863 (120593 119894)

10038161003816100381610038162


119890]120579E

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

le 0811003817100381710038171003817120593

10038171003817100381710038172

(106)


119889 [120593 (0) + 05 int

0

minus1

120593 (120579) 119889120579]

= minus3120593 (0) 119889119905 + int

infin

0

120572 (119911) 119909119905minus (119889119905 119889119911)

(107)


Let 119881(119909) = 1199092 then

L119881 (120593 119894) le minus61205932

(0) + int

infin

0

1205722

(119911) 120582 (119889119911)

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

(108)


1198712


119894isinS120601(120579)] le 119902E[max

119894isinS120601(0)]


E [max119894isinS


119881 (119909 119894)] (109)





Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











L (120593120576

119905 119903120576

(119905))

= 119905(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

+ 119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119891 (120593120576

119905 119903120576

(119905))

+1

2trace [119892

119879

(120593120576

119905 119903120576

(119905))

times 119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) ]

+

119872

sum

119894=1

120574120576

119903120576(119905)119894

(120593120576

(0) minus 119863 (119909120576

119905 119894) 119905 119894)

= 119905(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

+ 119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119891 (120593120576

119905 119903120576

(119905))

+1

2trace [119892

119879

(120593120576

119905 119903120576

(119905))

times 119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) ]

+

119872

sum

119894=1

120574119903120576(119905)119894

(120593120576

(0) minus 119863 (119909120576

119905 119894) 119905 119894)

= 119881119905(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

+ 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119891 (120593120576

119905 119903120576

(119905))

+1

2trace [119892

119879

(120593120576

119905 119903120576

(119905))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) ]

+

119897

sum

119896=1

120574120576

119903120576(119905)119896

119881 (120593120576

(0) minus 119863 (119909120576

119905 119896) 119905 119896)

+ 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times [119891 (120593120576

119905 119903120576

(119905)) minus 119891 (120593120576

119905 119903120576

(119905))]

+1

2trace [119892

119879

(120593120576

119905 119903120576

(119905))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) minus 119892119879

(120593120576

119905 119903120576

(119905))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) ]

+

119872

sum

119894=1

120574119903120576(119905)119894

(120593120576

(0) minus 119863 (119909120576

119905 119894) 119905 119894)

minus

119897

sum

119896=1

120574119903120576(119905)119896

119881 (120593120576

(0) minus 119863 (119909120576

119905 119896) 119905 119896)

= L119881 (120593120576

119905 119903120576

(119905))

+ 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times [119891 (120593120576

119905 119903120576

(119905)) minus 119891 (120593120576

119905 119903120576

(119905))]

+1

2trace [119892

119879

(120593120576

119905 119903120576

(119905))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) minus 119892119879

(120593120576

119905 119903120576

(119905))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119892 (120593120576

119905 119903120576

(119905)) ]

+

119872

sum

119894=1

120574119903120576(119905)119894

(120593120576

(0) minus 119863 (119909120576

119905 119894) 119905 119894)

minus

119897

sum

119896=1

120574119903120576(119905)119896

119881 (120593120576

(0) minus 119863 (119909120576

119905 119896) 119905 119896)

(44)

Therefore

119880 (119905 + ℎ) minus 119880 (119905)


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [L119881 (120593120576

119904 119903120576

(119904))

+120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119905

119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times [119891 (120593120576

119904 119903120576

(119904)) minus 119891 (120593120576

119904 119903120576

(119904))] 119889119904

+1

2lim sup120576rarr0

E

times int

119905+120579+ℎ

119905+120579

119890120574119904



(120593120576

119904 119903120576

(119904))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times 119892 (120593120576

119904 119903120576

(119904)) minus 119892119879

(120593120576

119904 119903120576

(119904))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times 119892 (120593120576

119904 119903120576

(119904)) ] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times (

119872

sum

119894=1

120574119903120576(119904)119894

(120593120576

(0) minus 119863 (119909120576

119904 119894) 119904 119894)

minus

119897

sum

119896=1

120574119903120576(119904)119896

119881 (120593120576

(0) minus 119863 (119909120576

119904 119896) 119904 119896))119889119904

= 1198681+ 119868

2+ 119868

3+ 119868

4

(45)


119891 (120593120576

119905 119903120576

(119905)) minus 119891 (120593120576

119905 119903120576

(119905))

=

119897

sum

119896=1

119872119896

sum

119895=1

119891 (120593120576

119905 119904119896119895) times [119868

119903120576(119905)=119904119896119895

minus 120583119896

119895119868119903120576(119905)=119896

]

(46)

This implies that

lim120576rarr0

Eint

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times [119891 (120593120576

119904 119903120576

(119904)) minus 119891 (120593120576

119904 119903120576

(119904))] 119889119904

le lim120576rarr0

[E

100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times [119891 (120593120576

119904 119903120576

(119904)) minus 119891 (120593120576

119904 119903120576

(119904))] 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

]

12

= lim120576rarr0

[

[

E

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times

119897

sum

119896=1

119872119896

sum

119895=1

119891 (120593120576

119904 119904119896119895)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

] 119889119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

]

]

12

le lim120576rarr0

[

[

E

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119897

sum

119896=1

119872119896

sum

119895=1

119890120574119904

119871 (1 +1003817100381710038171003817120593

1003817100381710038171003817119901

)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

] 119889119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

]

]

12

(47)


2= 0 Similarly

we can show that

1198683=

1

2lim sup120576rarr0

E

times int

119905+120579+ℎ

119905+120579

119890120574119904


(120593120576

119904 119903120576

(119904))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times 119892 (120593120576

119904 119903120576

(119904))

minus 119892119879

(120593120576

119904 119903120576

(119904))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times 119892 (120593120576

119904 119903120576

(119904)) ] 119889119904 = 0

(48)


119872

sum

119894=1

120574119903120576(119905)119894

(120593120576

(0) minus 119863 (119909120576

119905 119894) 119905 119894)

= Γ (120593120576

(0) minus 119863 (119909120576

119905 119894) 119905 sdot) (119903

120576

(119905))

119897

sum

119896=1

120574119903120576(119905)119896

119881 (120593 (0) minus 119863 (119909120576

119905 119896) 119905 119896)

= Γ119881 (120593 (0) minus 119863 (119909120576

119905 119896) 119905 sdot) (119903

120576

(119905))

(49)

Hence

1198684= lim sup

120576rarr0

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times (

119872

sum

119894=1

120574119903120576(119904)119894

(120593120576

(0) minus 119863 (119909120576

119904 119894) 119904 119894)

minus

119897

sum

119896=1

120574119903120576(119904)119896

119881 (120593120576

(0) minus 119863 (119909120576

119904 119896) 119904 119896))119889119904



E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times (Γ (120593120576

(0) minus 119863 (119909120576

119904 119894) 119904 sdot) (119903

120576

(119904))

minus Γ119881 (120593120576

(0) minus 119863 (119909120576

119904 119896) 119904 sdot) (119904

119896119895)) 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

119897

sum

119896=1

119872119896

sum

119895=1

Γ (120593120576

(0) minus 119863 (119909120576

119904 119894) 119904 sdot) (119904

119896119895)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

] 119889119904


[

[

E

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

times

119897

sum

119896=1

119872119896

sum

119895=1

Γ (120593120576

(0) minus 119863 (119909120576

119904 119894) 119904 sdot) (119904

119896119895)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

]

]

12

(50)


4= 0 Therefore

119880 (119905 + ℎ) minus 119880 (119905)

= lim120576rarr0

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [L119881 (120593120576

119904 119903120576

(119904))

+120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))] 119889119904

le 0

(51)

That is

119880 (119905 + ℎ) le 119880 (119905) (52)


+



lim sup120576rarr0

119890120574119905

E1003816100381610038161003816119909

120576

(119905) minus 119863 (119909120576

119905 119903

120576

(119905))1003816100381610038161003816119901


sup120591le120579le0

119890120574120579

E1003816100381610038161003816119909

120576

(120579) minus 119863 (119909120576

120579 119903

120576

(120579))1003816100381610038161003816

119901



(1 + 120581)119901minus1

times [E1003816100381610038161003816119909

120576

(120579)1003816100381610038161003816119901

+ 1205811minus119901

E1003816100381610038161003816119863 (119909

120576

120579 119903

120576

(120579))1003816100381610038161003816119901

]

le 1198882(1 + 120581)

11990110038171003817100381710038171205851003817100381710038171003817119901

119905 ge 0

(53)


lim sup120576rarr0

119890120574119905

E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

(54)

That is

lim sup120576rarr0

E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

119890minus120574119905

forall119905 ge 0 (55)



119889 [120593120576

(0) minus 119863 (120593120576

119903120576

(119905))]

= 119891 (120593120576

119905 119903120576

(119905)) 119889119905 + intR119898

119887 (119909120576

119905minus 119905 119903

120576

(119905) 119911) (119889119905 119889119911)

1199090= 120585 isin 119862 ([minus120591 0] R

119899

) 119903 (0) isin S

(56)

where 119909120576

119905minus= lim

119904uarr119905119909120576

119904 119863 119862([minus120591 0]R119899

) times S rarr R119899 119887

119862([minus120591 0]R119899





119887(120575)

(120593 119905 119894 119911) = 119887(120575)

(120593 119905 119894 119911120575)

119911 = (1199111 119911

119898) isin R

119898

119894 isin S

(57)


(119889119905 119889119911) = (1(119889119905 119889119911

1)

119898(119889119905 119889119911

119898))

= (1198731(119889119905 119889119911

1) minus 120582

1(119889119911

1) 119889119905 119873

119898(119889119905 119889119911

119898)

minus 120582119898

(119889119911119898) 119889119905)

(58)






119889 [120593120576

(0) minus 119863 (119909120576

119905 119903

120576

(119905))]

= 119891 (119909120576

119905 119905 119903

120576

(119905)) 119889119905

+ intR119898

(119909120576

119905minus 119905 119903

120576

(119905) 119911) (119889119905 119889119911)

1199090= 120585 isin 119862 ([minus120591 0] R

119899

) 119903 (0) isin S

(59)

where 119909120576

119905minus= lim

119904uarr119905119909120576

119904and 119862([minus120591 0]R119899


rarr


119863(120593 119896) =

119873119898

sum

119895=1

120583119896

119895119863(120593 119904

119896119895)

(120593 119905 119896 119911) =

119873119898

sum

119895=1

120583119896

119895119887 (120593 119905 119904

119896119895 119911)

(60)

for each 119904119896119895




gt 0such that

1003816100381610038161003816119891 (120593 119905 119894) minus 119891 (120601 119905 119894)1003816100381610038161003816

or

119898

sum

120575=1

intR

10038161003816100381610038161003816119887(120575)

(120593 119905 119894 119911120575) minus 119887

(120575)

(120601 119905 119894 119911120575)10038161003816100381610038161003816120582120575(119889119911

120575)

le 119871120577

1003817100381710038171003817120593 minus 12060110038171003817100381710038172

(61)




119862([minus120591 0]R119899

) 119894 isin S

1003816100381610038161003816119891 (120593 119905 119894)1003816100381610038161003816 or

119898

sum

120575=1

intR

10038161003816100381610038161003816119887(120575)

(120593 119905 119894 119911120575)10038161003816100381610038161003816120582120575(119889119911

120575)

le 119871 (1 +1003817100381710038171003817120593

10038171003817100381710038172

)

(62)



1003816100381610038161003816119863 (120593 119894) minus 119863 (120601 119894)1003816100381610038161003816 le 120581

1003817100381710038171003817120593 minus 12060110038171003817100381710038172

119863 (0 119894) equiv 0

(63)

Given that 119881 isin 119862119901

(R119899

times R+times SR


L119881 by

L119881 (120593 119905 119894)

= 119881119905(120593 (0) minus 119863 (120593 119894) 119905 119894)

+ 119881119909(120593 (0) minus 119863 (120593 119894) 119905 119894) 119891 (120593 119905 119894)

+

119873

sum

119895=1

120574119894119895119881 (120593 (0) minus 119863 (120593 119894) 119905 119895)

+ intR

119898

sum

120575=1

119881 (120593 (0) minus 119863 (120593 119894) + 119887(120575)

(120593 119905 120580 119911120575) 119905 120580)

minus 119881 (120593 (0) minus 119863 (120593 119894) 119905 119894)

minus 119881119909(120593 (0) minus 119863 (120593 119894) 119905 119894)

times 119887(120575)

(120593 119905 120580 119911120575) 120582

120575(119889119911

120575)

(64)

where

119881119909(120593 (0) minus 119863 (120593 119894) 119905 119894)

= (120597119881 (120593 (0) minus 119863 (120593 119894) 119905 119894)

1205971199091

120597119881 (120593 (0) minus 119863 (120593 119894) 119905 119894)

120597119909119898

)

(65)


(sdot) 119903120576






2


119901

(R119899

times R+times SR

+) satisfying


1198881|119909|

119901

le 119881 (119909 119905 119896) le 1198882|119909|

119901

119896 isin S (66)

for all (119909 119905 119896) isin R119899


following

E1003816100381610038161003816119863 (120593 119896)

1003816100381610038161003816119901

le 120581119901 supminus120591le120579le0

119890]120579 1003817100381710038171003817120593

1003817100381710038171003817

119901

120581 = max 1205811 120581

119896 120593 isin 119871

119901

F119905

(67)


E [max119896isin

S


S

119881 (120593 (0) minus 119863 (120593 119896))]

(68)



F119905([minus120591 0]R119899

)satisfying

E [min119896isin

S

119881 (119909 (119905 + 120579) 119905 + 120579 119896)]


S

119881 (120593 (0) minus 119863 (120593 119896) 119905 119894)] minus120591 le 120579 le 0

(69)


) 119905 ge 0

lim sup120576rarr0

E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

119890minus]119905

(70)

where

] = min1205741

120591log

119902

(11988821198881) (1 minus 120581)

119901 (71)


1198882

1198881

(1 minus 120581)119901

119890120574120591

= 120582 (72)

Proof Define

(120593 119905 120588) =

119897

sum

119896=1

119881 (120593 119905 119896) 119868120588isinS119896 = 119881 (120593 119905 119896) if 120588 isin S

119896

(73)


119880 (119905)


[119890120574(119905+120579)

E119881 (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ]


[119890120574(119905+120579)

E (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ]

(74)


119863+

119880 (119905) = lim supℎrarr0+

119880 (119905 + ℎ) minus 119880 (119905)

ℎle 0 forall119905 ge 0 (75)


E119881 (119909120576

(119905 + 120579) 119905 + 120579 119903120576

(119905 + 120579))

lt 119902E119881 (120593 (0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

(76)


119901

119890120574120591 that is 120574 lt

(1120591)(log(119902(11988821198881)(1 minus 120581)

119901

))

Thus

E [max119896isin

S

L119881 (120593120576

119905 119903120576

(119905))]


S

119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(77)

which implies that

E [L119881 (120593120576

119905 119903120576

(119905))]

le minus120582E [119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(78)


E [L119881 (120593120576

119905 119903120576

(119905))]

le minus120574E [119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(79)

We now consider

119880 (119905 + ℎ) minus 119880 (119905)


[119890120574(119905+120579+ℎ)

E

times [119881 (119909120576

(119905 + 120579 + ℎ)

minus 119863 (119909120576

119905+120579+ℎ 119903

120576

(119905 + 120579 + ℎ))

119905 + 120579 + ℎ 119903120576

(119905 + 120579 + ℎ)) ]

minus 119890120574(119905+120579)

E

times [119881 (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ] ]


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

[L119881 (120593120576

119904 119903120576

(119904))

+120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119905))] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [L (120593120576

119904 119903120576

(119904))

+ 120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119904 119903120576

(119904)) ] 119889119905

(80)



L (120593120576

119905 119903120576

(119905))

= 119905(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

+ 119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119891 (120593120576

119905 119903120576

(119905))

+

119898

sum

120575=1

intR

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905))

+ 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 119905 119903

120576

(119905))

minus (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

minus 119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 120582

120575(119889119911

120575)

+

119873

sum

119895=1

120574120576

119903120576(119905)119895

(120593120576

(0) minus 119863 (120593120576

119895) 119905 119895)

= L119881 (120593120576

119905 119903120576

(119905))

+ 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times [119891 (120593120576

119905 119903120576

(119905)) minus 119891 (120593120576

119905 119903120576

(119905))]

+

119898

sum

120575=1

intR

119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905))

+ 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 119905 119903

120576

(119905))

minus 119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905))

+ (120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 119905 119903

120576

(119905))

times 120582120575(119889119911

120575)

minus

119898

sum

120575=1

intR

119881119909(120593

120576

(0)

minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times (119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)

minus(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)) 120582

120575(119889119911

120575)

+

119873

sum

119895=1

120574119903120576(119905)119895

(120593120576

(0) minus 119863 (120593120576

119895) 119905 119895)

minus

119897

sum

119896=1

120574119903120576(119905)119896

(120593120576

(0) minus 119863 (120593120576

119896) 119905 119896)

(81)

This implies that

119880 (119905 + ℎ) minus 119880 (119905)


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [L119881 (120593120576

119904 119903120576

(119904))

+ 120574119881 (120593120576

(0)

minus119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times [119891 (120593120576

119904 119903120576

(119904)) minus 119891 (120593120576

119904 119903120576

(119904))] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [

119898

sum

120575=1

intR

[119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904))

+119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575) 119904 119903

120576

(119904))

minus 119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904))

+ (120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

119904 119903120576

(119904) )]

times120582120575(119889119911

120575) ] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times

119898

sum

120575=1

intR

[119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times (119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

minus(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575))]

times120582120575(119889119911

120575) 119889119904



E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times (

119873

sum

119895=1

120574119903120576(119904)119895

(120593120576

(0) minus 119863 (120593120576

119895) 119904 119895)

minus

119897

sum

119896=1

120574119903120576(119904)119896

(120593120576

(0) minus 119863 (120593120576

119896) 119904 119896))119889119904

= 1198691+ 119869

2+ 119869

3+ 119869

4+ 119869

5

(82)


119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) minus

(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)

=

119897

sum

119894=1

119873119896

sum

119895=1

119887(120575)

(119909120576

119905minus 119905 119904

119896119895 119911

120575)

times [119868119903120576(119905)=119904119896119895

minus 120583119896

119895119868119903120576(119905)=119896

]

(83)


1198694= lim sup

120576rarr0

119898

sum

120575=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times intR

[119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

minus (120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)]

times 120582120575(119889119911

120575) 119889119904


119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904))

119904 119903120576

(119904))

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)

times [119868119903120576(119904)=119904119896119895

minus120583119896

119895119868119903120576(119904)=119896

]

times 120582120575(119889119911

120575) 119889119904


119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

[E

100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904))

119904 119903120576

(119904))

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

times120582120575(119889119911

120575) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

]

12

(84)


4= 0 Similarly


119905


(0) minus 119863(120593120576

119903120576

(119905)) + 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)

and 120593120576

(0) minus 119863(120593120576

119903120576

(119905)) + (120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) such that

1198693= lim

120576rarr0

119898

sum

120575=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

intR

119881119909(120578

119904)

times [119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

minus(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)]

times 120582120575(119889119911

120575) 119889119904

= lim120576rarr0

119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120578

119904)

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

times 120582120575(119889119911

120575) 119889119904

le lim120576rarr0

119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

[E

100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120578

119905)

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)


times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

times 120582120575(119889119911

120575)119889119904

100381610038161003816100381610038161003816100381610038161003816

2

]

12

(85)



5= 0Therefore


= lim120576rarr0

Eint

119905+120579+ℎ

119905+120579

119890120574119904

times [L119881 (120593120576

119904 119903120576

(119904))

+120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))] 119889119904

le 0

(86)

Then

119880 (119905 + ℎ) le 119880 (119905) (87)


E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

119890minus]119905

(88)


5 Examples



in (14) with

Γ = (

minus1 0 1 0 0

1 minus2 1 0 0

2 1 minus3 0 0

0 0 0 minus1 1

0 0 0 1 minus1

) (89)

Γ = (

minus1 0 1 0 0

0 minus1 0 1 0

0 0 minus1 0 1

0 1 0 minus1 0

1 0 0 0 minus1

) (90)


(1205871

1205872

1205873)(

minus1 0 1

1 minus2 1

2 1 minus3

) = 0 (91)

and 1205871+ 120587

2+ 120587

3= 1 we get 1205831


(1205874

1205875) (

minus1 1

1 minus1) = 0 (92)

and 1205874+ 120587

5= 1 we have 120583

2

= (12 12) So

Γ = 120583Γ1 = (

5

8

1

8

1

40 0

0 0 01

2

1

2

)

times (

minus1 0 1 0 0

0 minus1 0 1 0

0 0 minus1 0 1

0 1 0 minus1 0

1 0 0 0 minus1

)(

1 0

1 0

1 0

0 1

0 1

)

= (minus3

8

3

81 minus1

)

(93)


119889 [120593120576

(0) minus 119863 (120593120576

119903120576

(119905))]

= 119891 (120593120576

119903120576

(119905)) 119889119905 + 119892 (120593120576

119903120576

(119905)) 119889119908 (119905)

(94)

with

119863(120593 11990411) = minus06 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990412) = minus02 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990413) = minus04 int

0

minus1

120593 (120579) 119889120579

119891 (120593 11990411) = minus16120593 (0) minus 8 cos [120593 (0)]

119891 (120593 11990412) = 8120593 (0) + 4 cos [120593 (0)]

119891 (120593 11990413) = 16120593 (0)

119892 (120593 11990411) =

radic10

10int

0

minus1

120593 (120579) 119889120579 cos [int0

minus1

120593 (120579) 119889120579]

119892 (120593 11990412) = minus

radic2

2int

0

minus1

120593 (120579) 119889120579 sin [int

0

minus1

120593 (120579) 119889120579]

119892 (120593 11990413) =

radic3

2int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990421) = 05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990422) = 05 int

0

minus1

120593 (120579) 119889120579

119891 (120593 11990421) = minus2120593 (0) 119891 (120593 119904

22) = minus2120593 (0)


119892 (120593 11990421) =

int

0

minus1

120593 (120579) 119889120579 sin [int

0

minus1

120593 (120579) 119889120579]

4radic2

119892 (120593 11990422) =

int

0

minus1

120593 (120579) 119889120579 cos [int0

minus1

120593 (120579) 119889120579]

4radic2

(95)

For any 120593 isin 1198712

F119905([minus1 0]R) and 120581 = max06 02 04 = 06


E1003816100381610038161003816119863 (120593 119894)

10038161003816100381610038162


119890]120579E

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

le 0361003817100381710038171003817120593

10038171003817100381710038172

(96)


119889 [120593 (0) minus 119863 (120593 119903 (119905))] = 119891 (120593 119903 (119905)) 119889119905 + 119892 (120593 119903 (119905)) 119889119908 (119905)

(97)


119863(120593 1) = minus05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 2) = 05 int

0

minus1

120593 (120579) 119889120579

119891 (120593 1) = minus5120593 (0) 119891 (120593 2) = minus2120593 (0)

119892 (120593 1) =1

2int

0

minus1

120593 (120579) 119889120579 119892 (120593 2) =1

4int

0

minus1

120593 (120579) 119889120579

(98)

We define 119881(119909 1) = 21199092 119881(119909 2) = 119909


L119881 (120593 1) le minus203

81205932

(0) +13

32

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

L119881 (120593 2) le minus5

21205932

(0) +13

16

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

(99)

Consequently

max119894=12

L119881 (120593 119894) le minus5

21205932

(0) +13

16

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

= minus5

4[max119894=12

119881 (119909 119894)] +13

16[min119894=12

119881 (119909 119894)]

(100)


2


119894isinS120601(120579)] le


E [max119894isinS

L119881 (120593 119894)] le minus (5

4minus

13119902

16)E [max

119894=12

119881 (119909 119894)] (101)




Γ120576

=1

120576Γ + Γ =

1

120576(

minus2 0 2 0

1 minus2 0 1

0 2 minus2 0

0 1 1 minus2

) (102)



119889 [120593120576

(0) minus 119863 (120593120576

119903120576

(119905))]

= 119891 (120593120576

119903120576

(119905)) 119889119905 + int

infin

0

120590 (119903120576

(119905) 119911) 119909120576

119905minus (119889119905 119889119911)

(103)

with

119863(120593 1) = minus09 int

0

minus1

120593 (120579) 119889120579

119863 (120593 2) = minus04 int

0

minus1

120593 (120579) 119889120579

119863 (120593 3) = minus05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 4) = minus03 int

0

minus1

120593 (120579) 119889120579

119891 (120593 1) = 2 sin [120593 (0)] 119891 (120593 2) = minus11

2120593 (0)

119891 (120593 3) = minus11

3120593 (0) 119891 (120593 4) = minus2 sin [120593 (0)]

(104)

Let

120572 (119911) =2

11120590 (1 119911) +

4

11120590 (2 119911) +

3

11120590 (3 119911) +

2

11120590 (4 119911)

int

infin

0

1205722

(119911) 120582 (119889119911) lt 2

(105)

For any 120593 isin 1198712

F119905([minus1 0]R) and 120581 = max09 04 05 03 =


E1003816100381610038161003816119863 (120593 119894)

10038161003816100381610038162


119890]120579E

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

le 0811003817100381710038171003817120593

10038171003817100381710038172

(106)


119889 [120593 (0) + 05 int

0

minus1

120593 (120579) 119889120579]

= minus3120593 (0) 119889119905 + int

infin

0

120572 (119911) 119909119905minus (119889119905 119889119911)

(107)


Let 119881(119909) = 1199092 then

L119881 (120593 119894) le minus61205932

(0) + int

infin

0

1205722

(119911) 120582 (119889119911)

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

(108)


1198712


119894isinS120601(120579)] le 119902E[max

119894isinS120601(0)]


E [max119894isinS


119881 (119909 119894)] (109)





Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(120593120576

119904 119903120576

(119904))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times 119892 (120593120576

119904 119903120576

(119904)) minus 119892119879

(120593120576

119904 119903120576

(119904))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times 119892 (120593120576

119904 119903120576

(119904)) ] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times (

119872

sum

119894=1

120574119903120576(119904)119894

(120593120576

(0) minus 119863 (119909120576

119904 119894) 119904 119894)

minus

119897

sum

119896=1

120574119903120576(119904)119896

119881 (120593120576

(0) minus 119863 (119909120576

119904 119896) 119904 119896))119889119904

= 1198681+ 119868

2+ 119868

3+ 119868

4

(45)


119891 (120593120576

119905 119903120576

(119905)) minus 119891 (120593120576

119905 119903120576

(119905))

=

119897

sum

119896=1

119872119896

sum

119895=1

119891 (120593120576

119905 119904119896119895) times [119868

119903120576(119905)=119904119896119895

minus 120583119896

119895119868119903120576(119905)=119896

]

(46)

This implies that

lim120576rarr0

Eint

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times [119891 (120593120576

119904 119903120576

(119904)) minus 119891 (120593120576

119904 119903120576

(119904))] 119889119904

le lim120576rarr0

[E

100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times [119891 (120593120576

119904 119903120576

(119904)) minus 119891 (120593120576

119904 119903120576

(119904))] 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

]

12

= lim120576rarr0

[

[

E

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times

119897

sum

119896=1

119872119896

sum

119895=1

119891 (120593120576

119904 119904119896119895)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

] 119889119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

]

]

12

le lim120576rarr0

[

[

E

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119897

sum

119896=1

119872119896

sum

119895=1

119890120574119904

119871 (1 +1003817100381710038171003817120593

1003817100381710038171003817119901

)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

] 119889119904

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

]

]

12

(47)


2= 0 Similarly

we can show that

1198683=

1

2lim sup120576rarr0

E

times int

119905+120579+ℎ

119905+120579

119890120574119904


(120593120576

119904 119903120576

(119904))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times 119892 (120593120576

119904 119903120576

(119904))

minus 119892119879

(120593120576

119904 119903120576

(119904))

times 119881119909119909

(120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times 119892 (120593120576

119904 119903120576

(119904)) ] 119889119904 = 0

(48)


119872

sum

119894=1

120574119903120576(119905)119894

(120593120576

(0) minus 119863 (119909120576

119905 119894) 119905 119894)

= Γ (120593120576

(0) minus 119863 (119909120576

119905 119894) 119905 sdot) (119903

120576

(119905))

119897

sum

119896=1

120574119903120576(119905)119896

119881 (120593 (0) minus 119863 (119909120576

119905 119896) 119905 119896)

= Γ119881 (120593 (0) minus 119863 (119909120576

119905 119896) 119905 sdot) (119903

120576

(119905))

(49)

Hence

1198684= lim sup

120576rarr0

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times (

119872

sum

119894=1

120574119903120576(119904)119894

(120593120576

(0) minus 119863 (119909120576

119904 119894) 119904 119894)

minus

119897

sum

119896=1

120574119903120576(119904)119896

119881 (120593120576

(0) minus 119863 (119909120576

119904 119896) 119904 119896))119889119904



E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times (Γ (120593120576

(0) minus 119863 (119909120576

119904 119894) 119904 sdot) (119903

120576

(119904))

minus Γ119881 (120593120576

(0) minus 119863 (119909120576

119904 119896) 119904 sdot) (119904

119896119895)) 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

119897

sum

119896=1

119872119896

sum

119895=1

Γ (120593120576

(0) minus 119863 (119909120576

119904 119894) 119904 sdot) (119904

119896119895)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

] 119889119904


[

[

E

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

times

119897

sum

119896=1

119872119896

sum

119895=1

Γ (120593120576

(0) minus 119863 (119909120576

119904 119894) 119904 sdot) (119904

119896119895)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

]

]

12

(50)


4= 0 Therefore

119880 (119905 + ℎ) minus 119880 (119905)

= lim120576rarr0

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [L119881 (120593120576

119904 119903120576

(119904))

+120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))] 119889119904

le 0

(51)

That is

119880 (119905 + ℎ) le 119880 (119905) (52)


+



lim sup120576rarr0

119890120574119905

E1003816100381610038161003816119909

120576

(119905) minus 119863 (119909120576

119905 119903

120576

(119905))1003816100381610038161003816119901


sup120591le120579le0

119890120574120579

E1003816100381610038161003816119909

120576

(120579) minus 119863 (119909120576

120579 119903

120576

(120579))1003816100381610038161003816

119901



(1 + 120581)119901minus1

times [E1003816100381610038161003816119909

120576

(120579)1003816100381610038161003816119901

+ 1205811minus119901

E1003816100381610038161003816119863 (119909

120576

120579 119903

120576

(120579))1003816100381610038161003816119901

]

le 1198882(1 + 120581)

11990110038171003817100381710038171205851003817100381710038171003817119901

119905 ge 0

(53)


lim sup120576rarr0

119890120574119905

E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

(54)

That is

lim sup120576rarr0

E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

119890minus120574119905

forall119905 ge 0 (55)



119889 [120593120576

(0) minus 119863 (120593120576

119903120576

(119905))]

= 119891 (120593120576

119905 119903120576

(119905)) 119889119905 + intR119898

119887 (119909120576

119905minus 119905 119903

120576

(119905) 119911) (119889119905 119889119911)

1199090= 120585 isin 119862 ([minus120591 0] R

119899

) 119903 (0) isin S

(56)

where 119909120576

119905minus= lim

119904uarr119905119909120576

119904 119863 119862([minus120591 0]R119899

) times S rarr R119899 119887

119862([minus120591 0]R119899





119887(120575)

(120593 119905 119894 119911) = 119887(120575)

(120593 119905 119894 119911120575)

119911 = (1199111 119911

119898) isin R

119898

119894 isin S

(57)


(119889119905 119889119911) = (1(119889119905 119889119911

1)

119898(119889119905 119889119911

119898))

= (1198731(119889119905 119889119911

1) minus 120582

1(119889119911

1) 119889119905 119873

119898(119889119905 119889119911

119898)

minus 120582119898

(119889119911119898) 119889119905)

(58)






119889 [120593120576

(0) minus 119863 (119909120576

119905 119903

120576

(119905))]

= 119891 (119909120576

119905 119905 119903

120576

(119905)) 119889119905

+ intR119898

(119909120576

119905minus 119905 119903

120576

(119905) 119911) (119889119905 119889119911)

1199090= 120585 isin 119862 ([minus120591 0] R

119899

) 119903 (0) isin S

(59)

where 119909120576

119905minus= lim

119904uarr119905119909120576

119904and 119862([minus120591 0]R119899


rarr


119863(120593 119896) =

119873119898

sum

119895=1

120583119896

119895119863(120593 119904

119896119895)

(120593 119905 119896 119911) =

119873119898

sum

119895=1

120583119896

119895119887 (120593 119905 119904

119896119895 119911)

(60)

for each 119904119896119895




gt 0such that

1003816100381610038161003816119891 (120593 119905 119894) minus 119891 (120601 119905 119894)1003816100381610038161003816

or

119898

sum

120575=1

intR

10038161003816100381610038161003816119887(120575)

(120593 119905 119894 119911120575) minus 119887

(120575)

(120601 119905 119894 119911120575)10038161003816100381610038161003816120582120575(119889119911

120575)

le 119871120577

1003817100381710038171003817120593 minus 12060110038171003817100381710038172

(61)




119862([minus120591 0]R119899

) 119894 isin S

1003816100381610038161003816119891 (120593 119905 119894)1003816100381610038161003816 or

119898

sum

120575=1

intR

10038161003816100381610038161003816119887(120575)

(120593 119905 119894 119911120575)10038161003816100381610038161003816120582120575(119889119911

120575)

le 119871 (1 +1003817100381710038171003817120593

10038171003817100381710038172

)

(62)



1003816100381610038161003816119863 (120593 119894) minus 119863 (120601 119894)1003816100381610038161003816 le 120581

1003817100381710038171003817120593 minus 12060110038171003817100381710038172

119863 (0 119894) equiv 0

(63)

Given that 119881 isin 119862119901

(R119899

times R+times SR


L119881 by

L119881 (120593 119905 119894)

= 119881119905(120593 (0) minus 119863 (120593 119894) 119905 119894)

+ 119881119909(120593 (0) minus 119863 (120593 119894) 119905 119894) 119891 (120593 119905 119894)

+

119873

sum

119895=1

120574119894119895119881 (120593 (0) minus 119863 (120593 119894) 119905 119895)

+ intR

119898

sum

120575=1

119881 (120593 (0) minus 119863 (120593 119894) + 119887(120575)

(120593 119905 120580 119911120575) 119905 120580)

minus 119881 (120593 (0) minus 119863 (120593 119894) 119905 119894)

minus 119881119909(120593 (0) minus 119863 (120593 119894) 119905 119894)

times 119887(120575)

(120593 119905 120580 119911120575) 120582

120575(119889119911

120575)

(64)

where

119881119909(120593 (0) minus 119863 (120593 119894) 119905 119894)

= (120597119881 (120593 (0) minus 119863 (120593 119894) 119905 119894)

1205971199091

120597119881 (120593 (0) minus 119863 (120593 119894) 119905 119894)

120597119909119898

)

(65)


(sdot) 119903120576






2


119901

(R119899

times R+times SR

+) satisfying


1198881|119909|

119901

le 119881 (119909 119905 119896) le 1198882|119909|

119901

119896 isin S (66)

for all (119909 119905 119896) isin R119899


following

E1003816100381610038161003816119863 (120593 119896)

1003816100381610038161003816119901

le 120581119901 supminus120591le120579le0

119890]120579 1003817100381710038171003817120593

1003817100381710038171003817

119901

120581 = max 1205811 120581

119896 120593 isin 119871

119901

F119905

(67)


E [max119896isin

S


S

119881 (120593 (0) minus 119863 (120593 119896))]

(68)



F119905([minus120591 0]R119899

)satisfying

E [min119896isin

S

119881 (119909 (119905 + 120579) 119905 + 120579 119896)]


S

119881 (120593 (0) minus 119863 (120593 119896) 119905 119894)] minus120591 le 120579 le 0

(69)


) 119905 ge 0

lim sup120576rarr0

E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

119890minus]119905

(70)

where

] = min1205741

120591log

119902

(11988821198881) (1 minus 120581)

119901 (71)


1198882

1198881

(1 minus 120581)119901

119890120574120591

= 120582 (72)

Proof Define

(120593 119905 120588) =

119897

sum

119896=1

119881 (120593 119905 119896) 119868120588isinS119896 = 119881 (120593 119905 119896) if 120588 isin S

119896

(73)


119880 (119905)


[119890120574(119905+120579)

E119881 (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ]


[119890120574(119905+120579)

E (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ]

(74)


119863+

119880 (119905) = lim supℎrarr0+

119880 (119905 + ℎ) minus 119880 (119905)

ℎle 0 forall119905 ge 0 (75)


E119881 (119909120576

(119905 + 120579) 119905 + 120579 119903120576

(119905 + 120579))

lt 119902E119881 (120593 (0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

(76)


119901

119890120574120591 that is 120574 lt

(1120591)(log(119902(11988821198881)(1 minus 120581)

119901

))

Thus

E [max119896isin

S

L119881 (120593120576

119905 119903120576

(119905))]


S

119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(77)

which implies that

E [L119881 (120593120576

119905 119903120576

(119905))]

le minus120582E [119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(78)


E [L119881 (120593120576

119905 119903120576

(119905))]

le minus120574E [119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(79)

We now consider

119880 (119905 + ℎ) minus 119880 (119905)


[119890120574(119905+120579+ℎ)

E

times [119881 (119909120576

(119905 + 120579 + ℎ)

minus 119863 (119909120576

119905+120579+ℎ 119903

120576

(119905 + 120579 + ℎ))

119905 + 120579 + ℎ 119903120576

(119905 + 120579 + ℎ)) ]

minus 119890120574(119905+120579)

E

times [119881 (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ] ]


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

[L119881 (120593120576

119904 119903120576

(119904))

+120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119905))] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [L (120593120576

119904 119903120576

(119904))

+ 120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119904 119903120576

(119904)) ] 119889119905

(80)



L (120593120576

119905 119903120576

(119905))

= 119905(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

+ 119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119891 (120593120576

119905 119903120576

(119905))

+

119898

sum

120575=1

intR

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905))

+ 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 119905 119903

120576

(119905))

minus (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

minus 119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 120582

120575(119889119911

120575)

+

119873

sum

119895=1

120574120576

119903120576(119905)119895

(120593120576

(0) minus 119863 (120593120576

119895) 119905 119895)

= L119881 (120593120576

119905 119903120576

(119905))

+ 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times [119891 (120593120576

119905 119903120576

(119905)) minus 119891 (120593120576

119905 119903120576

(119905))]

+

119898

sum

120575=1

intR

119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905))

+ 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 119905 119903

120576

(119905))

minus 119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905))

+ (120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 119905 119903

120576

(119905))

times 120582120575(119889119911

120575)

minus

119898

sum

120575=1

intR

119881119909(120593

120576

(0)

minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times (119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)

minus(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)) 120582

120575(119889119911

120575)

+

119873

sum

119895=1

120574119903120576(119905)119895

(120593120576

(0) minus 119863 (120593120576

119895) 119905 119895)

minus

119897

sum

119896=1

120574119903120576(119905)119896

(120593120576

(0) minus 119863 (120593120576

119896) 119905 119896)

(81)

This implies that

119880 (119905 + ℎ) minus 119880 (119905)


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [L119881 (120593120576

119904 119903120576

(119904))

+ 120574119881 (120593120576

(0)

minus119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times [119891 (120593120576

119904 119903120576

(119904)) minus 119891 (120593120576

119904 119903120576

(119904))] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [

119898

sum

120575=1

intR

[119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904))

+119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575) 119904 119903

120576

(119904))

minus 119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904))

+ (120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

119904 119903120576

(119904) )]

times120582120575(119889119911

120575) ] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times

119898

sum

120575=1

intR

[119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times (119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

minus(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575))]

times120582120575(119889119911

120575) 119889119904



E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times (

119873

sum

119895=1

120574119903120576(119904)119895

(120593120576

(0) minus 119863 (120593120576

119895) 119904 119895)

minus

119897

sum

119896=1

120574119903120576(119904)119896

(120593120576

(0) minus 119863 (120593120576

119896) 119904 119896))119889119904

= 1198691+ 119869

2+ 119869

3+ 119869

4+ 119869

5

(82)


119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) minus

(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)

=

119897

sum

119894=1

119873119896

sum

119895=1

119887(120575)

(119909120576

119905minus 119905 119904

119896119895 119911

120575)

times [119868119903120576(119905)=119904119896119895

minus 120583119896

119895119868119903120576(119905)=119896

]

(83)


1198694= lim sup

120576rarr0

119898

sum

120575=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times intR

[119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

minus (120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)]

times 120582120575(119889119911

120575) 119889119904


119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904))

119904 119903120576

(119904))

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)

times [119868119903120576(119904)=119904119896119895

minus120583119896

119895119868119903120576(119904)=119896

]

times 120582120575(119889119911

120575) 119889119904


119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

[E

100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904))

119904 119903120576

(119904))

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

times120582120575(119889119911

120575) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

]

12

(84)


4= 0 Similarly


119905


(0) minus 119863(120593120576

119903120576

(119905)) + 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)

and 120593120576

(0) minus 119863(120593120576

119903120576

(119905)) + (120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) such that

1198693= lim

120576rarr0

119898

sum

120575=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

intR

119881119909(120578

119904)

times [119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

minus(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)]

times 120582120575(119889119911

120575) 119889119904

= lim120576rarr0

119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120578

119904)

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

times 120582120575(119889119911

120575) 119889119904

le lim120576rarr0

119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

[E

100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120578

119905)

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)


times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

times 120582120575(119889119911

120575)119889119904

100381610038161003816100381610038161003816100381610038161003816

2

]

12

(85)



5= 0Therefore


= lim120576rarr0

Eint

119905+120579+ℎ

119905+120579

119890120574119904

times [L119881 (120593120576

119904 119903120576

(119904))

+120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))] 119889119904

le 0

(86)

Then

119880 (119905 + ℎ) le 119880 (119905) (87)


E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

119890minus]119905

(88)


5 Examples



in (14) with

Γ = (

minus1 0 1 0 0

1 minus2 1 0 0

2 1 minus3 0 0

0 0 0 minus1 1

0 0 0 1 minus1

) (89)

Γ = (

minus1 0 1 0 0

0 minus1 0 1 0

0 0 minus1 0 1

0 1 0 minus1 0

1 0 0 0 minus1

) (90)


(1205871

1205872

1205873)(

minus1 0 1

1 minus2 1

2 1 minus3

) = 0 (91)

and 1205871+ 120587

2+ 120587

3= 1 we get 1205831


(1205874

1205875) (

minus1 1

1 minus1) = 0 (92)

and 1205874+ 120587

5= 1 we have 120583

2

= (12 12) So

Γ = 120583Γ1 = (

5

8

1

8

1

40 0

0 0 01

2

1

2

)

times (

minus1 0 1 0 0

0 minus1 0 1 0

0 0 minus1 0 1

0 1 0 minus1 0

1 0 0 0 minus1

)(

1 0

1 0

1 0

0 1

0 1

)

= (minus3

8

3

81 minus1

)

(93)


119889 [120593120576

(0) minus 119863 (120593120576

119903120576

(119905))]

= 119891 (120593120576

119903120576

(119905)) 119889119905 + 119892 (120593120576

119903120576

(119905)) 119889119908 (119905)

(94)

with

119863(120593 11990411) = minus06 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990412) = minus02 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990413) = minus04 int

0

minus1

120593 (120579) 119889120579

119891 (120593 11990411) = minus16120593 (0) minus 8 cos [120593 (0)]

119891 (120593 11990412) = 8120593 (0) + 4 cos [120593 (0)]

119891 (120593 11990413) = 16120593 (0)

119892 (120593 11990411) =

radic10

10int

0

minus1

120593 (120579) 119889120579 cos [int0

minus1

120593 (120579) 119889120579]

119892 (120593 11990412) = minus

radic2

2int

0

minus1

120593 (120579) 119889120579 sin [int

0

minus1

120593 (120579) 119889120579]

119892 (120593 11990413) =

radic3

2int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990421) = 05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990422) = 05 int

0

minus1

120593 (120579) 119889120579

119891 (120593 11990421) = minus2120593 (0) 119891 (120593 119904

22) = minus2120593 (0)


119892 (120593 11990421) =

int

0

minus1

120593 (120579) 119889120579 sin [int

0

minus1

120593 (120579) 119889120579]

4radic2

119892 (120593 11990422) =

int

0

minus1

120593 (120579) 119889120579 cos [int0

minus1

120593 (120579) 119889120579]

4radic2

(95)

For any 120593 isin 1198712

F119905([minus1 0]R) and 120581 = max06 02 04 = 06


E1003816100381610038161003816119863 (120593 119894)

10038161003816100381610038162


119890]120579E

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

le 0361003817100381710038171003817120593

10038171003817100381710038172

(96)


119889 [120593 (0) minus 119863 (120593 119903 (119905))] = 119891 (120593 119903 (119905)) 119889119905 + 119892 (120593 119903 (119905)) 119889119908 (119905)

(97)


119863(120593 1) = minus05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 2) = 05 int

0

minus1

120593 (120579) 119889120579

119891 (120593 1) = minus5120593 (0) 119891 (120593 2) = minus2120593 (0)

119892 (120593 1) =1

2int

0

minus1

120593 (120579) 119889120579 119892 (120593 2) =1

4int

0

minus1

120593 (120579) 119889120579

(98)

We define 119881(119909 1) = 21199092 119881(119909 2) = 119909


L119881 (120593 1) le minus203

81205932

(0) +13

32

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

L119881 (120593 2) le minus5

21205932

(0) +13

16

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

(99)

Consequently

max119894=12

L119881 (120593 119894) le minus5

21205932

(0) +13

16

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

= minus5

4[max119894=12

119881 (119909 119894)] +13

16[min119894=12

119881 (119909 119894)]

(100)


2


119894isinS120601(120579)] le


E [max119894isinS

L119881 (120593 119894)] le minus (5

4minus

13119902

16)E [max

119894=12

119881 (119909 119894)] (101)




Γ120576

=1

120576Γ + Γ =

1

120576(

minus2 0 2 0

1 minus2 0 1

0 2 minus2 0

0 1 1 minus2

) (102)



119889 [120593120576

(0) minus 119863 (120593120576

119903120576

(119905))]

= 119891 (120593120576

119903120576

(119905)) 119889119905 + int

infin

0

120590 (119903120576

(119905) 119911) 119909120576

119905minus (119889119905 119889119911)

(103)

with

119863(120593 1) = minus09 int

0

minus1

120593 (120579) 119889120579

119863 (120593 2) = minus04 int

0

minus1

120593 (120579) 119889120579

119863 (120593 3) = minus05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 4) = minus03 int

0

minus1

120593 (120579) 119889120579

119891 (120593 1) = 2 sin [120593 (0)] 119891 (120593 2) = minus11

2120593 (0)

119891 (120593 3) = minus11

3120593 (0) 119891 (120593 4) = minus2 sin [120593 (0)]

(104)

Let

120572 (119911) =2

11120590 (1 119911) +

4

11120590 (2 119911) +

3

11120590 (3 119911) +

2

11120590 (4 119911)

int

infin

0

1205722

(119911) 120582 (119889119911) lt 2

(105)

For any 120593 isin 1198712

F119905([minus1 0]R) and 120581 = max09 04 05 03 =


E1003816100381610038161003816119863 (120593 119894)

10038161003816100381610038162


119890]120579E

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

le 0811003817100381710038171003817120593

10038171003817100381710038172

(106)


119889 [120593 (0) + 05 int

0

minus1

120593 (120579) 119889120579]

= minus3120593 (0) 119889119905 + int

infin

0

120572 (119911) 119909119905minus (119889119905 119889119911)

(107)


Let 119881(119909) = 1199092 then

L119881 (120593 119894) le minus61205932

(0) + int

infin

0

1205722

(119911) 120582 (119889119911)

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

(108)


1198712


119894isinS120601(120579)] le 119902E[max

119894isinS120601(0)]


E [max119894isinS


119881 (119909 119894)] (109)





Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times (Γ (120593120576

(0) minus 119863 (119909120576

119904 119894) 119904 sdot) (119903

120576

(119904))

minus Γ119881 (120593120576

(0) minus 119863 (119909120576

119904 119896) 119904 sdot) (119904

119896119895)) 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

119897

sum

119896=1

119872119896

sum

119895=1

Γ (120593120576

(0) minus 119863 (119909120576

119904 119894) 119904 sdot) (119904

119896119895)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

] 119889119904


[

[

E

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

times

119897

sum

119896=1

119872119896

sum

119895=1

Γ (120593120576

(0) minus 119863 (119909120576

119904 119894) 119904 sdot) (119904

119896119895)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

]

]

12

(50)


4= 0 Therefore

119880 (119905 + ℎ) minus 119880 (119905)

= lim120576rarr0

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [L119881 (120593120576

119904 119903120576

(119904))

+120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))] 119889119904

le 0

(51)

That is

119880 (119905 + ℎ) le 119880 (119905) (52)


+



lim sup120576rarr0

119890120574119905

E1003816100381610038161003816119909

120576

(119905) minus 119863 (119909120576

119905 119903

120576

(119905))1003816100381610038161003816119901


sup120591le120579le0

119890120574120579

E1003816100381610038161003816119909

120576

(120579) minus 119863 (119909120576

120579 119903

120576

(120579))1003816100381610038161003816

119901



(1 + 120581)119901minus1

times [E1003816100381610038161003816119909

120576

(120579)1003816100381610038161003816119901

+ 1205811minus119901

E1003816100381610038161003816119863 (119909

120576

120579 119903

120576

(120579))1003816100381610038161003816119901

]

le 1198882(1 + 120581)

11990110038171003817100381710038171205851003817100381710038171003817119901

119905 ge 0

(53)


lim sup120576rarr0

119890120574119905

E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

(54)

That is

lim sup120576rarr0

E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

119890minus120574119905

forall119905 ge 0 (55)



119889 [120593120576

(0) minus 119863 (120593120576

119903120576

(119905))]

= 119891 (120593120576

119905 119903120576

(119905)) 119889119905 + intR119898

119887 (119909120576

119905minus 119905 119903

120576

(119905) 119911) (119889119905 119889119911)

1199090= 120585 isin 119862 ([minus120591 0] R

119899

) 119903 (0) isin S

(56)

where 119909120576

119905minus= lim

119904uarr119905119909120576

119904 119863 119862([minus120591 0]R119899

) times S rarr R119899 119887

119862([minus120591 0]R119899





119887(120575)

(120593 119905 119894 119911) = 119887(120575)

(120593 119905 119894 119911120575)

119911 = (1199111 119911

119898) isin R

119898

119894 isin S

(57)


(119889119905 119889119911) = (1(119889119905 119889119911

1)

119898(119889119905 119889119911

119898))

= (1198731(119889119905 119889119911

1) minus 120582

1(119889119911

1) 119889119905 119873

119898(119889119905 119889119911

119898)

minus 120582119898

(119889119911119898) 119889119905)

(58)






119889 [120593120576

(0) minus 119863 (119909120576

119905 119903

120576

(119905))]

= 119891 (119909120576

119905 119905 119903

120576

(119905)) 119889119905

+ intR119898

(119909120576

119905minus 119905 119903

120576

(119905) 119911) (119889119905 119889119911)

1199090= 120585 isin 119862 ([minus120591 0] R

119899

) 119903 (0) isin S

(59)

where 119909120576

119905minus= lim

119904uarr119905119909120576

119904and 119862([minus120591 0]R119899


rarr


119863(120593 119896) =

119873119898

sum

119895=1

120583119896

119895119863(120593 119904

119896119895)

(120593 119905 119896 119911) =

119873119898

sum

119895=1

120583119896

119895119887 (120593 119905 119904

119896119895 119911)

(60)

for each 119904119896119895




gt 0such that

1003816100381610038161003816119891 (120593 119905 119894) minus 119891 (120601 119905 119894)1003816100381610038161003816

or

119898

sum

120575=1

intR

10038161003816100381610038161003816119887(120575)

(120593 119905 119894 119911120575) minus 119887

(120575)

(120601 119905 119894 119911120575)10038161003816100381610038161003816120582120575(119889119911

120575)

le 119871120577

1003817100381710038171003817120593 minus 12060110038171003817100381710038172

(61)




119862([minus120591 0]R119899

) 119894 isin S

1003816100381610038161003816119891 (120593 119905 119894)1003816100381610038161003816 or

119898

sum

120575=1

intR

10038161003816100381610038161003816119887(120575)

(120593 119905 119894 119911120575)10038161003816100381610038161003816120582120575(119889119911

120575)

le 119871 (1 +1003817100381710038171003817120593

10038171003817100381710038172

)

(62)



1003816100381610038161003816119863 (120593 119894) minus 119863 (120601 119894)1003816100381610038161003816 le 120581

1003817100381710038171003817120593 minus 12060110038171003817100381710038172

119863 (0 119894) equiv 0

(63)

Given that 119881 isin 119862119901

(R119899

times R+times SR


L119881 by

L119881 (120593 119905 119894)

= 119881119905(120593 (0) minus 119863 (120593 119894) 119905 119894)

+ 119881119909(120593 (0) minus 119863 (120593 119894) 119905 119894) 119891 (120593 119905 119894)

+

119873

sum

119895=1

120574119894119895119881 (120593 (0) minus 119863 (120593 119894) 119905 119895)

+ intR

119898

sum

120575=1

119881 (120593 (0) minus 119863 (120593 119894) + 119887(120575)

(120593 119905 120580 119911120575) 119905 120580)

minus 119881 (120593 (0) minus 119863 (120593 119894) 119905 119894)

minus 119881119909(120593 (0) minus 119863 (120593 119894) 119905 119894)

times 119887(120575)

(120593 119905 120580 119911120575) 120582

120575(119889119911

120575)

(64)

where

119881119909(120593 (0) minus 119863 (120593 119894) 119905 119894)

= (120597119881 (120593 (0) minus 119863 (120593 119894) 119905 119894)

1205971199091

120597119881 (120593 (0) minus 119863 (120593 119894) 119905 119894)

120597119909119898

)

(65)


(sdot) 119903120576






2


119901

(R119899

times R+times SR

+) satisfying


1198881|119909|

119901

le 119881 (119909 119905 119896) le 1198882|119909|

119901

119896 isin S (66)

for all (119909 119905 119896) isin R119899


following

E1003816100381610038161003816119863 (120593 119896)

1003816100381610038161003816119901

le 120581119901 supminus120591le120579le0

119890]120579 1003817100381710038171003817120593

1003817100381710038171003817

119901

120581 = max 1205811 120581

119896 120593 isin 119871

119901

F119905

(67)


E [max119896isin

S


S

119881 (120593 (0) minus 119863 (120593 119896))]

(68)



F119905([minus120591 0]R119899

)satisfying

E [min119896isin

S

119881 (119909 (119905 + 120579) 119905 + 120579 119896)]


S

119881 (120593 (0) minus 119863 (120593 119896) 119905 119894)] minus120591 le 120579 le 0

(69)


) 119905 ge 0

lim sup120576rarr0

E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

119890minus]119905

(70)

where

] = min1205741

120591log

119902

(11988821198881) (1 minus 120581)

119901 (71)


1198882

1198881

(1 minus 120581)119901

119890120574120591

= 120582 (72)

Proof Define

(120593 119905 120588) =

119897

sum

119896=1

119881 (120593 119905 119896) 119868120588isinS119896 = 119881 (120593 119905 119896) if 120588 isin S

119896

(73)


119880 (119905)


[119890120574(119905+120579)

E119881 (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ]


[119890120574(119905+120579)

E (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ]

(74)


119863+

119880 (119905) = lim supℎrarr0+

119880 (119905 + ℎ) minus 119880 (119905)

ℎle 0 forall119905 ge 0 (75)


E119881 (119909120576

(119905 + 120579) 119905 + 120579 119903120576

(119905 + 120579))

lt 119902E119881 (120593 (0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

(76)


119901

119890120574120591 that is 120574 lt

(1120591)(log(119902(11988821198881)(1 minus 120581)

119901

))

Thus

E [max119896isin

S

L119881 (120593120576

119905 119903120576

(119905))]


S

119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(77)

which implies that

E [L119881 (120593120576

119905 119903120576

(119905))]

le minus120582E [119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(78)


E [L119881 (120593120576

119905 119903120576

(119905))]

le minus120574E [119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(79)

We now consider

119880 (119905 + ℎ) minus 119880 (119905)


[119890120574(119905+120579+ℎ)

E

times [119881 (119909120576

(119905 + 120579 + ℎ)

minus 119863 (119909120576

119905+120579+ℎ 119903

120576

(119905 + 120579 + ℎ))

119905 + 120579 + ℎ 119903120576

(119905 + 120579 + ℎ)) ]

minus 119890120574(119905+120579)

E

times [119881 (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ] ]


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

[L119881 (120593120576

119904 119903120576

(119904))

+120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119905))] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [L (120593120576

119904 119903120576

(119904))

+ 120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119904 119903120576

(119904)) ] 119889119905

(80)



L (120593120576

119905 119903120576

(119905))

= 119905(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

+ 119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119891 (120593120576

119905 119903120576

(119905))

+

119898

sum

120575=1

intR

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905))

+ 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 119905 119903

120576

(119905))

minus (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

minus 119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 120582

120575(119889119911

120575)

+

119873

sum

119895=1

120574120576

119903120576(119905)119895

(120593120576

(0) minus 119863 (120593120576

119895) 119905 119895)

= L119881 (120593120576

119905 119903120576

(119905))

+ 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times [119891 (120593120576

119905 119903120576

(119905)) minus 119891 (120593120576

119905 119903120576

(119905))]

+

119898

sum

120575=1

intR

119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905))

+ 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 119905 119903

120576

(119905))

minus 119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905))

+ (120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 119905 119903

120576

(119905))

times 120582120575(119889119911

120575)

minus

119898

sum

120575=1

intR

119881119909(120593

120576

(0)

minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times (119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)

minus(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)) 120582

120575(119889119911

120575)

+

119873

sum

119895=1

120574119903120576(119905)119895

(120593120576

(0) minus 119863 (120593120576

119895) 119905 119895)

minus

119897

sum

119896=1

120574119903120576(119905)119896

(120593120576

(0) minus 119863 (120593120576

119896) 119905 119896)

(81)

This implies that

119880 (119905 + ℎ) minus 119880 (119905)


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [L119881 (120593120576

119904 119903120576

(119904))

+ 120574119881 (120593120576

(0)

minus119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times [119891 (120593120576

119904 119903120576

(119904)) minus 119891 (120593120576

119904 119903120576

(119904))] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [

119898

sum

120575=1

intR

[119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904))

+119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575) 119904 119903

120576

(119904))

minus 119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904))

+ (120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

119904 119903120576

(119904) )]

times120582120575(119889119911

120575) ] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times

119898

sum

120575=1

intR

[119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times (119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

minus(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575))]

times120582120575(119889119911

120575) 119889119904



E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times (

119873

sum

119895=1

120574119903120576(119904)119895

(120593120576

(0) minus 119863 (120593120576

119895) 119904 119895)

minus

119897

sum

119896=1

120574119903120576(119904)119896

(120593120576

(0) minus 119863 (120593120576

119896) 119904 119896))119889119904

= 1198691+ 119869

2+ 119869

3+ 119869

4+ 119869

5

(82)


119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) minus

(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)

=

119897

sum

119894=1

119873119896

sum

119895=1

119887(120575)

(119909120576

119905minus 119905 119904

119896119895 119911

120575)

times [119868119903120576(119905)=119904119896119895

minus 120583119896

119895119868119903120576(119905)=119896

]

(83)


1198694= lim sup

120576rarr0

119898

sum

120575=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times intR

[119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

minus (120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)]

times 120582120575(119889119911

120575) 119889119904


119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904))

119904 119903120576

(119904))

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)

times [119868119903120576(119904)=119904119896119895

minus120583119896

119895119868119903120576(119904)=119896

]

times 120582120575(119889119911

120575) 119889119904


119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

[E

100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904))

119904 119903120576

(119904))

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

times120582120575(119889119911

120575) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

]

12

(84)


4= 0 Similarly


119905


(0) minus 119863(120593120576

119903120576

(119905)) + 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)

and 120593120576

(0) minus 119863(120593120576

119903120576

(119905)) + (120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) such that

1198693= lim

120576rarr0

119898

sum

120575=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

intR

119881119909(120578

119904)

times [119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

minus(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)]

times 120582120575(119889119911

120575) 119889119904

= lim120576rarr0

119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120578

119904)

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

times 120582120575(119889119911

120575) 119889119904

le lim120576rarr0

119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

[E

100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120578

119905)

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)


times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

times 120582120575(119889119911

120575)119889119904

100381610038161003816100381610038161003816100381610038161003816

2

]

12

(85)



5= 0Therefore


= lim120576rarr0

Eint

119905+120579+ℎ

119905+120579

119890120574119904

times [L119881 (120593120576

119904 119903120576

(119904))

+120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))] 119889119904

le 0

(86)

Then

119880 (119905 + ℎ) le 119880 (119905) (87)


E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

119890minus]119905

(88)


5 Examples



in (14) with

Γ = (

minus1 0 1 0 0

1 minus2 1 0 0

2 1 minus3 0 0

0 0 0 minus1 1

0 0 0 1 minus1

) (89)

Γ = (

minus1 0 1 0 0

0 minus1 0 1 0

0 0 minus1 0 1

0 1 0 minus1 0

1 0 0 0 minus1

) (90)


(1205871

1205872

1205873)(

minus1 0 1

1 minus2 1

2 1 minus3

) = 0 (91)

and 1205871+ 120587

2+ 120587

3= 1 we get 1205831


(1205874

1205875) (

minus1 1

1 minus1) = 0 (92)

and 1205874+ 120587

5= 1 we have 120583

2

= (12 12) So

Γ = 120583Γ1 = (

5

8

1

8

1

40 0

0 0 01

2

1

2

)

times (

minus1 0 1 0 0

0 minus1 0 1 0

0 0 minus1 0 1

0 1 0 minus1 0

1 0 0 0 minus1

)(

1 0

1 0

1 0

0 1

0 1

)

= (minus3

8

3

81 minus1

)

(93)


119889 [120593120576

(0) minus 119863 (120593120576

119903120576

(119905))]

= 119891 (120593120576

119903120576

(119905)) 119889119905 + 119892 (120593120576

119903120576

(119905)) 119889119908 (119905)

(94)

with

119863(120593 11990411) = minus06 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990412) = minus02 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990413) = minus04 int

0

minus1

120593 (120579) 119889120579

119891 (120593 11990411) = minus16120593 (0) minus 8 cos [120593 (0)]

119891 (120593 11990412) = 8120593 (0) + 4 cos [120593 (0)]

119891 (120593 11990413) = 16120593 (0)

119892 (120593 11990411) =

radic10

10int

0

minus1

120593 (120579) 119889120579 cos [int0

minus1

120593 (120579) 119889120579]

119892 (120593 11990412) = minus

radic2

2int

0

minus1

120593 (120579) 119889120579 sin [int

0

minus1

120593 (120579) 119889120579]

119892 (120593 11990413) =

radic3

2int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990421) = 05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990422) = 05 int

0

minus1

120593 (120579) 119889120579

119891 (120593 11990421) = minus2120593 (0) 119891 (120593 119904

22) = minus2120593 (0)


119892 (120593 11990421) =

int

0

minus1

120593 (120579) 119889120579 sin [int

0

minus1

120593 (120579) 119889120579]

4radic2

119892 (120593 11990422) =

int

0

minus1

120593 (120579) 119889120579 cos [int0

minus1

120593 (120579) 119889120579]

4radic2

(95)

For any 120593 isin 1198712

F119905([minus1 0]R) and 120581 = max06 02 04 = 06


E1003816100381610038161003816119863 (120593 119894)

10038161003816100381610038162


119890]120579E

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

le 0361003817100381710038171003817120593

10038171003817100381710038172

(96)


119889 [120593 (0) minus 119863 (120593 119903 (119905))] = 119891 (120593 119903 (119905)) 119889119905 + 119892 (120593 119903 (119905)) 119889119908 (119905)

(97)


119863(120593 1) = minus05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 2) = 05 int

0

minus1

120593 (120579) 119889120579

119891 (120593 1) = minus5120593 (0) 119891 (120593 2) = minus2120593 (0)

119892 (120593 1) =1

2int

0

minus1

120593 (120579) 119889120579 119892 (120593 2) =1

4int

0

minus1

120593 (120579) 119889120579

(98)

We define 119881(119909 1) = 21199092 119881(119909 2) = 119909


L119881 (120593 1) le minus203

81205932

(0) +13

32

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

L119881 (120593 2) le minus5

21205932

(0) +13

16

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

(99)

Consequently

max119894=12

L119881 (120593 119894) le minus5

21205932

(0) +13

16

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

= minus5

4[max119894=12

119881 (119909 119894)] +13

16[min119894=12

119881 (119909 119894)]

(100)


2


119894isinS120601(120579)] le


E [max119894isinS

L119881 (120593 119894)] le minus (5

4minus

13119902

16)E [max

119894=12

119881 (119909 119894)] (101)




Γ120576

=1

120576Γ + Γ =

1

120576(

minus2 0 2 0

1 minus2 0 1

0 2 minus2 0

0 1 1 minus2

) (102)



119889 [120593120576

(0) minus 119863 (120593120576

119903120576

(119905))]

= 119891 (120593120576

119903120576

(119905)) 119889119905 + int

infin

0

120590 (119903120576

(119905) 119911) 119909120576

119905minus (119889119905 119889119911)

(103)

with

119863(120593 1) = minus09 int

0

minus1

120593 (120579) 119889120579

119863 (120593 2) = minus04 int

0

minus1

120593 (120579) 119889120579

119863 (120593 3) = minus05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 4) = minus03 int

0

minus1

120593 (120579) 119889120579

119891 (120593 1) = 2 sin [120593 (0)] 119891 (120593 2) = minus11

2120593 (0)

119891 (120593 3) = minus11

3120593 (0) 119891 (120593 4) = minus2 sin [120593 (0)]

(104)

Let

120572 (119911) =2

11120590 (1 119911) +

4

11120590 (2 119911) +

3

11120590 (3 119911) +

2

11120590 (4 119911)

int

infin

0

1205722

(119911) 120582 (119889119911) lt 2

(105)

For any 120593 isin 1198712

F119905([minus1 0]R) and 120581 = max09 04 05 03 =


E1003816100381610038161003816119863 (120593 119894)

10038161003816100381610038162


119890]120579E

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

le 0811003817100381710038171003817120593

10038171003817100381710038172

(106)


119889 [120593 (0) + 05 int

0

minus1

120593 (120579) 119889120579]

= minus3120593 (0) 119889119905 + int

infin

0

120572 (119911) 119909119905minus (119889119905 119889119911)

(107)


Let 119881(119909) = 1199092 then

L119881 (120593 119894) le minus61205932

(0) + int

infin

0

1205722

(119911) 120582 (119889119911)

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

(108)


1198712


119894isinS120601(120579)] le 119902E[max

119894isinS120601(0)]


E [max119894isinS


119881 (119909 119894)] (109)





Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119889 [120593120576

(0) minus 119863 (119909120576

119905 119903

120576

(119905))]

= 119891 (119909120576

119905 119905 119903

120576

(119905)) 119889119905

+ intR119898

(119909120576

119905minus 119905 119903

120576

(119905) 119911) (119889119905 119889119911)

1199090= 120585 isin 119862 ([minus120591 0] R

119899

) 119903 (0) isin S

(59)

where 119909120576

119905minus= lim

119904uarr119905119909120576

119904and 119862([minus120591 0]R119899


rarr


119863(120593 119896) =

119873119898

sum

119895=1

120583119896

119895119863(120593 119904

119896119895)

(120593 119905 119896 119911) =

119873119898

sum

119895=1

120583119896

119895119887 (120593 119905 119904

119896119895 119911)

(60)

for each 119904119896119895




gt 0such that

1003816100381610038161003816119891 (120593 119905 119894) minus 119891 (120601 119905 119894)1003816100381610038161003816

or

119898

sum

120575=1

intR

10038161003816100381610038161003816119887(120575)

(120593 119905 119894 119911120575) minus 119887

(120575)

(120601 119905 119894 119911120575)10038161003816100381610038161003816120582120575(119889119911

120575)

le 119871120577

1003817100381710038171003817120593 minus 12060110038171003817100381710038172

(61)




119862([minus120591 0]R119899

) 119894 isin S

1003816100381610038161003816119891 (120593 119905 119894)1003816100381610038161003816 or

119898

sum

120575=1

intR

10038161003816100381610038161003816119887(120575)

(120593 119905 119894 119911120575)10038161003816100381610038161003816120582120575(119889119911

120575)

le 119871 (1 +1003817100381710038171003817120593

10038171003817100381710038172

)

(62)



1003816100381610038161003816119863 (120593 119894) minus 119863 (120601 119894)1003816100381610038161003816 le 120581

1003817100381710038171003817120593 minus 12060110038171003817100381710038172

119863 (0 119894) equiv 0

(63)

Given that 119881 isin 119862119901

(R119899

times R+times SR


L119881 by

L119881 (120593 119905 119894)

= 119881119905(120593 (0) minus 119863 (120593 119894) 119905 119894)

+ 119881119909(120593 (0) minus 119863 (120593 119894) 119905 119894) 119891 (120593 119905 119894)

+

119873

sum

119895=1

120574119894119895119881 (120593 (0) minus 119863 (120593 119894) 119905 119895)

+ intR

119898

sum

120575=1

119881 (120593 (0) minus 119863 (120593 119894) + 119887(120575)

(120593 119905 120580 119911120575) 119905 120580)

minus 119881 (120593 (0) minus 119863 (120593 119894) 119905 119894)

minus 119881119909(120593 (0) minus 119863 (120593 119894) 119905 119894)

times 119887(120575)

(120593 119905 120580 119911120575) 120582

120575(119889119911

120575)

(64)

where

119881119909(120593 (0) minus 119863 (120593 119894) 119905 119894)

= (120597119881 (120593 (0) minus 119863 (120593 119894) 119905 119894)

1205971199091

120597119881 (120593 (0) minus 119863 (120593 119894) 119905 119894)

120597119909119898

)

(65)


(sdot) 119903120576






2


119901

(R119899

times R+times SR

+) satisfying


1198881|119909|

119901

le 119881 (119909 119905 119896) le 1198882|119909|

119901

119896 isin S (66)

for all (119909 119905 119896) isin R119899


following

E1003816100381610038161003816119863 (120593 119896)

1003816100381610038161003816119901

le 120581119901 supminus120591le120579le0

119890]120579 1003817100381710038171003817120593

1003817100381710038171003817

119901

120581 = max 1205811 120581

119896 120593 isin 119871

119901

F119905

(67)


E [max119896isin

S


S

119881 (120593 (0) minus 119863 (120593 119896))]

(68)



F119905([minus120591 0]R119899

)satisfying

E [min119896isin

S

119881 (119909 (119905 + 120579) 119905 + 120579 119896)]


S

119881 (120593 (0) minus 119863 (120593 119896) 119905 119894)] minus120591 le 120579 le 0

(69)


) 119905 ge 0

lim sup120576rarr0

E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

119890minus]119905

(70)

where

] = min1205741

120591log

119902

(11988821198881) (1 minus 120581)

119901 (71)


1198882

1198881

(1 minus 120581)119901

119890120574120591

= 120582 (72)

Proof Define

(120593 119905 120588) =

119897

sum

119896=1

119881 (120593 119905 119896) 119868120588isinS119896 = 119881 (120593 119905 119896) if 120588 isin S

119896

(73)


119880 (119905)


[119890120574(119905+120579)

E119881 (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ]


[119890120574(119905+120579)

E (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ]

(74)


119863+

119880 (119905) = lim supℎrarr0+

119880 (119905 + ℎ) minus 119880 (119905)

ℎle 0 forall119905 ge 0 (75)


E119881 (119909120576

(119905 + 120579) 119905 + 120579 119903120576

(119905 + 120579))

lt 119902E119881 (120593 (0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

(76)


119901

119890120574120591 that is 120574 lt

(1120591)(log(119902(11988821198881)(1 minus 120581)

119901

))

Thus

E [max119896isin

S

L119881 (120593120576

119905 119903120576

(119905))]


S

119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(77)

which implies that

E [L119881 (120593120576

119905 119903120576

(119905))]

le minus120582E [119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(78)


E [L119881 (120593120576

119905 119903120576

(119905))]

le minus120574E [119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(79)

We now consider

119880 (119905 + ℎ) minus 119880 (119905)


[119890120574(119905+120579+ℎ)

E

times [119881 (119909120576

(119905 + 120579 + ℎ)

minus 119863 (119909120576

119905+120579+ℎ 119903

120576

(119905 + 120579 + ℎ))

119905 + 120579 + ℎ 119903120576

(119905 + 120579 + ℎ)) ]

minus 119890120574(119905+120579)

E

times [119881 (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ] ]


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

[L119881 (120593120576

119904 119903120576

(119904))

+120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119905))] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [L (120593120576

119904 119903120576

(119904))

+ 120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119904 119903120576

(119904)) ] 119889119905

(80)



L (120593120576

119905 119903120576

(119905))

= 119905(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

+ 119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119891 (120593120576

119905 119903120576

(119905))

+

119898

sum

120575=1

intR

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905))

+ 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 119905 119903

120576

(119905))

minus (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

minus 119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 120582

120575(119889119911

120575)

+

119873

sum

119895=1

120574120576

119903120576(119905)119895

(120593120576

(0) minus 119863 (120593120576

119895) 119905 119895)

= L119881 (120593120576

119905 119903120576

(119905))

+ 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times [119891 (120593120576

119905 119903120576

(119905)) minus 119891 (120593120576

119905 119903120576

(119905))]

+

119898

sum

120575=1

intR

119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905))

+ 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 119905 119903

120576

(119905))

minus 119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905))

+ (120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 119905 119903

120576

(119905))

times 120582120575(119889119911

120575)

minus

119898

sum

120575=1

intR

119881119909(120593

120576

(0)

minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times (119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)

minus(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)) 120582

120575(119889119911

120575)

+

119873

sum

119895=1

120574119903120576(119905)119895

(120593120576

(0) minus 119863 (120593120576

119895) 119905 119895)

minus

119897

sum

119896=1

120574119903120576(119905)119896

(120593120576

(0) minus 119863 (120593120576

119896) 119905 119896)

(81)

This implies that

119880 (119905 + ℎ) minus 119880 (119905)


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [L119881 (120593120576

119904 119903120576

(119904))

+ 120574119881 (120593120576

(0)

minus119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times [119891 (120593120576

119904 119903120576

(119904)) minus 119891 (120593120576

119904 119903120576

(119904))] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [

119898

sum

120575=1

intR

[119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904))

+119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575) 119904 119903

120576

(119904))

minus 119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904))

+ (120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

119904 119903120576

(119904) )]

times120582120575(119889119911

120575) ] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times

119898

sum

120575=1

intR

[119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times (119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

minus(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575))]

times120582120575(119889119911

120575) 119889119904



E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times (

119873

sum

119895=1

120574119903120576(119904)119895

(120593120576

(0) minus 119863 (120593120576

119895) 119904 119895)

minus

119897

sum

119896=1

120574119903120576(119904)119896

(120593120576

(0) minus 119863 (120593120576

119896) 119904 119896))119889119904

= 1198691+ 119869

2+ 119869

3+ 119869

4+ 119869

5

(82)


119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) minus

(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)

=

119897

sum

119894=1

119873119896

sum

119895=1

119887(120575)

(119909120576

119905minus 119905 119904

119896119895 119911

120575)

times [119868119903120576(119905)=119904119896119895

minus 120583119896

119895119868119903120576(119905)=119896

]

(83)


1198694= lim sup

120576rarr0

119898

sum

120575=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times intR

[119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

minus (120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)]

times 120582120575(119889119911

120575) 119889119904


119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904))

119904 119903120576

(119904))

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)

times [119868119903120576(119904)=119904119896119895

minus120583119896

119895119868119903120576(119904)=119896

]

times 120582120575(119889119911

120575) 119889119904


119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

[E

100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904))

119904 119903120576

(119904))

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

times120582120575(119889119911

120575) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

]

12

(84)


4= 0 Similarly


119905


(0) minus 119863(120593120576

119903120576

(119905)) + 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)

and 120593120576

(0) minus 119863(120593120576

119903120576

(119905)) + (120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) such that

1198693= lim

120576rarr0

119898

sum

120575=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

intR

119881119909(120578

119904)

times [119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

minus(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)]

times 120582120575(119889119911

120575) 119889119904

= lim120576rarr0

119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120578

119904)

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

times 120582120575(119889119911

120575) 119889119904

le lim120576rarr0

119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

[E

100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120578

119905)

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)


times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

times 120582120575(119889119911

120575)119889119904

100381610038161003816100381610038161003816100381610038161003816

2

]

12

(85)



5= 0Therefore


= lim120576rarr0

Eint

119905+120579+ℎ

119905+120579

119890120574119904

times [L119881 (120593120576

119904 119903120576

(119904))

+120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))] 119889119904

le 0

(86)

Then

119880 (119905 + ℎ) le 119880 (119905) (87)


E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

119890minus]119905

(88)


5 Examples



in (14) with

Γ = (

minus1 0 1 0 0

1 minus2 1 0 0

2 1 minus3 0 0

0 0 0 minus1 1

0 0 0 1 minus1

) (89)

Γ = (

minus1 0 1 0 0

0 minus1 0 1 0

0 0 minus1 0 1

0 1 0 minus1 0

1 0 0 0 minus1

) (90)


(1205871

1205872

1205873)(

minus1 0 1

1 minus2 1

2 1 minus3

) = 0 (91)

and 1205871+ 120587

2+ 120587

3= 1 we get 1205831


(1205874

1205875) (

minus1 1

1 minus1) = 0 (92)

and 1205874+ 120587

5= 1 we have 120583

2

= (12 12) So

Γ = 120583Γ1 = (

5

8

1

8

1

40 0

0 0 01

2

1

2

)

times (

minus1 0 1 0 0

0 minus1 0 1 0

0 0 minus1 0 1

0 1 0 minus1 0

1 0 0 0 minus1

)(

1 0

1 0

1 0

0 1

0 1

)

= (minus3

8

3

81 minus1

)

(93)


119889 [120593120576

(0) minus 119863 (120593120576

119903120576

(119905))]

= 119891 (120593120576

119903120576

(119905)) 119889119905 + 119892 (120593120576

119903120576

(119905)) 119889119908 (119905)

(94)

with

119863(120593 11990411) = minus06 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990412) = minus02 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990413) = minus04 int

0

minus1

120593 (120579) 119889120579

119891 (120593 11990411) = minus16120593 (0) minus 8 cos [120593 (0)]

119891 (120593 11990412) = 8120593 (0) + 4 cos [120593 (0)]

119891 (120593 11990413) = 16120593 (0)

119892 (120593 11990411) =

radic10

10int

0

minus1

120593 (120579) 119889120579 cos [int0

minus1

120593 (120579) 119889120579]

119892 (120593 11990412) = minus

radic2

2int

0

minus1

120593 (120579) 119889120579 sin [int

0

minus1

120593 (120579) 119889120579]

119892 (120593 11990413) =

radic3

2int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990421) = 05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990422) = 05 int

0

minus1

120593 (120579) 119889120579

119891 (120593 11990421) = minus2120593 (0) 119891 (120593 119904

22) = minus2120593 (0)


119892 (120593 11990421) =

int

0

minus1

120593 (120579) 119889120579 sin [int

0

minus1

120593 (120579) 119889120579]

4radic2

119892 (120593 11990422) =

int

0

minus1

120593 (120579) 119889120579 cos [int0

minus1

120593 (120579) 119889120579]

4radic2

(95)

For any 120593 isin 1198712

F119905([minus1 0]R) and 120581 = max06 02 04 = 06


E1003816100381610038161003816119863 (120593 119894)

10038161003816100381610038162


119890]120579E

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

le 0361003817100381710038171003817120593

10038171003817100381710038172

(96)


119889 [120593 (0) minus 119863 (120593 119903 (119905))] = 119891 (120593 119903 (119905)) 119889119905 + 119892 (120593 119903 (119905)) 119889119908 (119905)

(97)


119863(120593 1) = minus05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 2) = 05 int

0

minus1

120593 (120579) 119889120579

119891 (120593 1) = minus5120593 (0) 119891 (120593 2) = minus2120593 (0)

119892 (120593 1) =1

2int

0

minus1

120593 (120579) 119889120579 119892 (120593 2) =1

4int

0

minus1

120593 (120579) 119889120579

(98)

We define 119881(119909 1) = 21199092 119881(119909 2) = 119909


L119881 (120593 1) le minus203

81205932

(0) +13

32

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

L119881 (120593 2) le minus5

21205932

(0) +13

16

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

(99)

Consequently

max119894=12

L119881 (120593 119894) le minus5

21205932

(0) +13

16

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

= minus5

4[max119894=12

119881 (119909 119894)] +13

16[min119894=12

119881 (119909 119894)]

(100)


2


119894isinS120601(120579)] le


E [max119894isinS

L119881 (120593 119894)] le minus (5

4minus

13119902

16)E [max

119894=12

119881 (119909 119894)] (101)




Γ120576

=1

120576Γ + Γ =

1

120576(

minus2 0 2 0

1 minus2 0 1

0 2 minus2 0

0 1 1 minus2

) (102)



119889 [120593120576

(0) minus 119863 (120593120576

119903120576

(119905))]

= 119891 (120593120576

119903120576

(119905)) 119889119905 + int

infin

0

120590 (119903120576

(119905) 119911) 119909120576

119905minus (119889119905 119889119911)

(103)

with

119863(120593 1) = minus09 int

0

minus1

120593 (120579) 119889120579

119863 (120593 2) = minus04 int

0

minus1

120593 (120579) 119889120579

119863 (120593 3) = minus05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 4) = minus03 int

0

minus1

120593 (120579) 119889120579

119891 (120593 1) = 2 sin [120593 (0)] 119891 (120593 2) = minus11

2120593 (0)

119891 (120593 3) = minus11

3120593 (0) 119891 (120593 4) = minus2 sin [120593 (0)]

(104)

Let

120572 (119911) =2

11120590 (1 119911) +

4

11120590 (2 119911) +

3

11120590 (3 119911) +

2

11120590 (4 119911)

int

infin

0

1205722

(119911) 120582 (119889119911) lt 2

(105)

For any 120593 isin 1198712

F119905([minus1 0]R) and 120581 = max09 04 05 03 =


E1003816100381610038161003816119863 (120593 119894)

10038161003816100381610038162


119890]120579E

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

le 0811003817100381710038171003817120593

10038171003817100381710038172

(106)


119889 [120593 (0) + 05 int

0

minus1

120593 (120579) 119889120579]

= minus3120593 (0) 119889119905 + int

infin

0

120572 (119911) 119909119905minus (119889119905 119889119911)

(107)


Let 119881(119909) = 1199092 then

L119881 (120593 119894) le minus61205932

(0) + int

infin

0

1205722

(119911) 120582 (119889119911)

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

(108)


1198712


119894isinS120601(120579)] le 119902E[max

119894isinS120601(0)]


E [max119894isinS


119881 (119909 119894)] (109)





Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











F119905([minus120591 0]R119899

)satisfying

E [min119896isin

S

119881 (119909 (119905 + 120579) 119905 + 120579 119896)]


S

119881 (120593 (0) minus 119863 (120593 119896) 119905 119894)] minus120591 le 120579 le 0

(69)


) 119905 ge 0

lim sup120576rarr0

E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

119890minus]119905

(70)

where

] = min1205741

120591log

119902

(11988821198881) (1 minus 120581)

119901 (71)


1198882

1198881

(1 minus 120581)119901

119890120574120591

= 120582 (72)

Proof Define

(120593 119905 120588) =

119897

sum

119896=1

119881 (120593 119905 119896) 119868120588isinS119896 = 119881 (120593 119905 119896) if 120588 isin S

119896

(73)


119880 (119905)


[119890120574(119905+120579)

E119881 (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ]


[119890120574(119905+120579)

E (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ]

(74)


119863+

119880 (119905) = lim supℎrarr0+

119880 (119905 + ℎ) minus 119880 (119905)

ℎle 0 forall119905 ge 0 (75)


E119881 (119909120576

(119905 + 120579) 119905 + 120579 119903120576

(119905 + 120579))

lt 119902E119881 (120593 (0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

(76)


119901

119890120574120591 that is 120574 lt

(1120591)(log(119902(11988821198881)(1 minus 120581)

119901

))

Thus

E [max119896isin

S

L119881 (120593120576

119905 119903120576

(119905))]


S

119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(77)

which implies that

E [L119881 (120593120576

119905 119903120576

(119905))]

le minus120582E [119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(78)


E [L119881 (120593120576

119905 119903120576

(119905))]

le minus120574E [119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))]

(79)

We now consider

119880 (119905 + ℎ) minus 119880 (119905)


[119890120574(119905+120579+ℎ)

E

times [119881 (119909120576

(119905 + 120579 + ℎ)

minus 119863 (119909120576

119905+120579+ℎ 119903

120576

(119905 + 120579 + ℎ))

119905 + 120579 + ℎ 119903120576

(119905 + 120579 + ℎ)) ]

minus 119890120574(119905+120579)

E

times [119881 (119909120576

(119905 + 120579) minus 119863 (119909120576

119905+120579 119903

120576

(119905 + 120579))

119905 + 120579 119903120576

(119905 + 120579)) ] ]


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

[L119881 (120593120576

119904 119903120576

(119904))

+120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119905))] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [L (120593120576

119904 119903120576

(119904))

+ 120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119904 119903120576

(119904)) ] 119889119905

(80)



L (120593120576

119905 119903120576

(119905))

= 119905(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

+ 119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119891 (120593120576

119905 119903120576

(119905))

+

119898

sum

120575=1

intR

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905))

+ 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 119905 119903

120576

(119905))

minus (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

minus 119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 120582

120575(119889119911

120575)

+

119873

sum

119895=1

120574120576

119903120576(119905)119895

(120593120576

(0) minus 119863 (120593120576

119895) 119905 119895)

= L119881 (120593120576

119905 119903120576

(119905))

+ 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times [119891 (120593120576

119905 119903120576

(119905)) minus 119891 (120593120576

119905 119903120576

(119905))]

+

119898

sum

120575=1

intR

119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905))

+ 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 119905 119903

120576

(119905))

minus 119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905))

+ (120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 119905 119903

120576

(119905))

times 120582120575(119889119911

120575)

minus

119898

sum

120575=1

intR

119881119909(120593

120576

(0)

minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times (119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)

minus(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)) 120582

120575(119889119911

120575)

+

119873

sum

119895=1

120574119903120576(119905)119895

(120593120576

(0) minus 119863 (120593120576

119895) 119905 119895)

minus

119897

sum

119896=1

120574119903120576(119905)119896

(120593120576

(0) minus 119863 (120593120576

119896) 119905 119896)

(81)

This implies that

119880 (119905 + ℎ) minus 119880 (119905)


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [L119881 (120593120576

119904 119903120576

(119904))

+ 120574119881 (120593120576

(0)

minus119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times [119891 (120593120576

119904 119903120576

(119904)) minus 119891 (120593120576

119904 119903120576

(119904))] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [

119898

sum

120575=1

intR

[119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904))

+119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575) 119904 119903

120576

(119904))

minus 119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904))

+ (120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

119904 119903120576

(119904) )]

times120582120575(119889119911

120575) ] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times

119898

sum

120575=1

intR

[119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times (119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

minus(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575))]

times120582120575(119889119911

120575) 119889119904



E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times (

119873

sum

119895=1

120574119903120576(119904)119895

(120593120576

(0) minus 119863 (120593120576

119895) 119904 119895)

minus

119897

sum

119896=1

120574119903120576(119904)119896

(120593120576

(0) minus 119863 (120593120576

119896) 119904 119896))119889119904

= 1198691+ 119869

2+ 119869

3+ 119869

4+ 119869

5

(82)


119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) minus

(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)

=

119897

sum

119894=1

119873119896

sum

119895=1

119887(120575)

(119909120576

119905minus 119905 119904

119896119895 119911

120575)

times [119868119903120576(119905)=119904119896119895

minus 120583119896

119895119868119903120576(119905)=119896

]

(83)


1198694= lim sup

120576rarr0

119898

sum

120575=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times intR

[119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

minus (120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)]

times 120582120575(119889119911

120575) 119889119904


119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904))

119904 119903120576

(119904))

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)

times [119868119903120576(119904)=119904119896119895

minus120583119896

119895119868119903120576(119904)=119896

]

times 120582120575(119889119911

120575) 119889119904


119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

[E

100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904))

119904 119903120576

(119904))

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

times120582120575(119889119911

120575) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

]

12

(84)


4= 0 Similarly


119905


(0) minus 119863(120593120576

119903120576

(119905)) + 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)

and 120593120576

(0) minus 119863(120593120576

119903120576

(119905)) + (120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) such that

1198693= lim

120576rarr0

119898

sum

120575=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

intR

119881119909(120578

119904)

times [119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

minus(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)]

times 120582120575(119889119911

120575) 119889119904

= lim120576rarr0

119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120578

119904)

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

times 120582120575(119889119911

120575) 119889119904

le lim120576rarr0

119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

[E

100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120578

119905)

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)


times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

times 120582120575(119889119911

120575)119889119904

100381610038161003816100381610038161003816100381610038161003816

2

]

12

(85)



5= 0Therefore


= lim120576rarr0

Eint

119905+120579+ℎ

119905+120579

119890120574119904

times [L119881 (120593120576

119904 119903120576

(119904))

+120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))] 119889119904

le 0

(86)

Then

119880 (119905 + ℎ) le 119880 (119905) (87)


E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

119890minus]119905

(88)


5 Examples



in (14) with

Γ = (

minus1 0 1 0 0

1 minus2 1 0 0

2 1 minus3 0 0

0 0 0 minus1 1

0 0 0 1 minus1

) (89)

Γ = (

minus1 0 1 0 0

0 minus1 0 1 0

0 0 minus1 0 1

0 1 0 minus1 0

1 0 0 0 minus1

) (90)


(1205871

1205872

1205873)(

minus1 0 1

1 minus2 1

2 1 minus3

) = 0 (91)

and 1205871+ 120587

2+ 120587

3= 1 we get 1205831


(1205874

1205875) (

minus1 1

1 minus1) = 0 (92)

and 1205874+ 120587

5= 1 we have 120583

2

= (12 12) So

Γ = 120583Γ1 = (

5

8

1

8

1

40 0

0 0 01

2

1

2

)

times (

minus1 0 1 0 0

0 minus1 0 1 0

0 0 minus1 0 1

0 1 0 minus1 0

1 0 0 0 minus1

)(

1 0

1 0

1 0

0 1

0 1

)

= (minus3

8

3

81 minus1

)

(93)


119889 [120593120576

(0) minus 119863 (120593120576

119903120576

(119905))]

= 119891 (120593120576

119903120576

(119905)) 119889119905 + 119892 (120593120576

119903120576

(119905)) 119889119908 (119905)

(94)

with

119863(120593 11990411) = minus06 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990412) = minus02 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990413) = minus04 int

0

minus1

120593 (120579) 119889120579

119891 (120593 11990411) = minus16120593 (0) minus 8 cos [120593 (0)]

119891 (120593 11990412) = 8120593 (0) + 4 cos [120593 (0)]

119891 (120593 11990413) = 16120593 (0)

119892 (120593 11990411) =

radic10

10int

0

minus1

120593 (120579) 119889120579 cos [int0

minus1

120593 (120579) 119889120579]

119892 (120593 11990412) = minus

radic2

2int

0

minus1

120593 (120579) 119889120579 sin [int

0

minus1

120593 (120579) 119889120579]

119892 (120593 11990413) =

radic3

2int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990421) = 05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990422) = 05 int

0

minus1

120593 (120579) 119889120579

119891 (120593 11990421) = minus2120593 (0) 119891 (120593 119904

22) = minus2120593 (0)


119892 (120593 11990421) =

int

0

minus1

120593 (120579) 119889120579 sin [int

0

minus1

120593 (120579) 119889120579]

4radic2

119892 (120593 11990422) =

int

0

minus1

120593 (120579) 119889120579 cos [int0

minus1

120593 (120579) 119889120579]

4radic2

(95)

For any 120593 isin 1198712

F119905([minus1 0]R) and 120581 = max06 02 04 = 06


E1003816100381610038161003816119863 (120593 119894)

10038161003816100381610038162


119890]120579E

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

le 0361003817100381710038171003817120593

10038171003817100381710038172

(96)


119889 [120593 (0) minus 119863 (120593 119903 (119905))] = 119891 (120593 119903 (119905)) 119889119905 + 119892 (120593 119903 (119905)) 119889119908 (119905)

(97)


119863(120593 1) = minus05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 2) = 05 int

0

minus1

120593 (120579) 119889120579

119891 (120593 1) = minus5120593 (0) 119891 (120593 2) = minus2120593 (0)

119892 (120593 1) =1

2int

0

minus1

120593 (120579) 119889120579 119892 (120593 2) =1

4int

0

minus1

120593 (120579) 119889120579

(98)

We define 119881(119909 1) = 21199092 119881(119909 2) = 119909


L119881 (120593 1) le minus203

81205932

(0) +13

32

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

L119881 (120593 2) le minus5

21205932

(0) +13

16

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

(99)

Consequently

max119894=12

L119881 (120593 119894) le minus5

21205932

(0) +13

16

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

= minus5

4[max119894=12

119881 (119909 119894)] +13

16[min119894=12

119881 (119909 119894)]

(100)


2


119894isinS120601(120579)] le


E [max119894isinS

L119881 (120593 119894)] le minus (5

4minus

13119902

16)E [max

119894=12

119881 (119909 119894)] (101)




Γ120576

=1

120576Γ + Γ =

1

120576(

minus2 0 2 0

1 minus2 0 1

0 2 minus2 0

0 1 1 minus2

) (102)



119889 [120593120576

(0) minus 119863 (120593120576

119903120576

(119905))]

= 119891 (120593120576

119903120576

(119905)) 119889119905 + int

infin

0

120590 (119903120576

(119905) 119911) 119909120576

119905minus (119889119905 119889119911)

(103)

with

119863(120593 1) = minus09 int

0

minus1

120593 (120579) 119889120579

119863 (120593 2) = minus04 int

0

minus1

120593 (120579) 119889120579

119863 (120593 3) = minus05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 4) = minus03 int

0

minus1

120593 (120579) 119889120579

119891 (120593 1) = 2 sin [120593 (0)] 119891 (120593 2) = minus11

2120593 (0)

119891 (120593 3) = minus11

3120593 (0) 119891 (120593 4) = minus2 sin [120593 (0)]

(104)

Let

120572 (119911) =2

11120590 (1 119911) +

4

11120590 (2 119911) +

3

11120590 (3 119911) +

2

11120590 (4 119911)

int

infin

0

1205722

(119911) 120582 (119889119911) lt 2

(105)

For any 120593 isin 1198712

F119905([minus1 0]R) and 120581 = max09 04 05 03 =


E1003816100381610038161003816119863 (120593 119894)

10038161003816100381610038162


119890]120579E

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

le 0811003817100381710038171003817120593

10038171003817100381710038172

(106)


119889 [120593 (0) + 05 int

0

minus1

120593 (120579) 119889120579]

= minus3120593 (0) 119889119905 + int

infin

0

120572 (119911) 119909119905minus (119889119905 119889119911)

(107)


Let 119881(119909) = 1199092 then

L119881 (120593 119894) le minus61205932

(0) + int

infin

0

1205722

(119911) 120582 (119889119911)

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

(108)


1198712


119894isinS120601(120579)] le 119902E[max

119894isinS120601(0)]


E [max119894isinS


119881 (119909 119894)] (109)





Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











L (120593120576

119905 119903120576

(119905))

= 119905(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

+ 119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119891 (120593120576

119905 119903120576

(119905))

+

119898

sum

120575=1

intR

(120593120576

(0) minus 119863 (120593120576

119903120576

(119905))

+ 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 119905 119903

120576

(119905))

minus (120593120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

minus 119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 120582

120575(119889119911

120575)

+

119873

sum

119895=1

120574120576

119903120576(119905)119895

(120593120576

(0) minus 119863 (120593120576

119895) 119905 119895)

= L119881 (120593120576

119905 119903120576

(119905))

+ 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times [119891 (120593120576

119905 119903120576

(119905)) minus 119891 (120593120576

119905 119903120576

(119905))]

+

119898

sum

120575=1

intR

119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905))

+ 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 119905 119903

120576

(119905))

minus 119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119905))

+ (120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) 119905 119903

120576

(119905))

times 120582120575(119889119911

120575)

minus

119898

sum

120575=1

intR

119881119909(120593

120576

(0)

minus 119863 (120593120576

119903120576

(119905)) 119905 119903120576

(119905))

times (119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)

minus(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)) 120582

120575(119889119911

120575)

+

119873

sum

119895=1

120574119903120576(119905)119895

(120593120576

(0) minus 119863 (120593120576

119895) 119905 119895)

minus

119897

sum

119896=1

120574119903120576(119905)119896

(120593120576

(0) minus 119863 (120593120576

119896) 119905 119896)

(81)

This implies that

119880 (119905 + ℎ) minus 119880 (119905)


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [L119881 (120593120576

119904 119903120576

(119904))

+ 120574119881 (120593120576

(0)

minus119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times [119891 (120593120576

119904 119903120576

(119904)) minus 119891 (120593120576

119904 119903120576

(119904))] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times [

119898

sum

120575=1

intR

[119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904))

+119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575) 119904 119903

120576

(119904))

minus 119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904))

+ (120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

119904 119903120576

(119904) )]

times120582120575(119889119911

120575) ] 119889119904


E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times

119898

sum

120575=1

intR

[119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times (119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

minus(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575))]

times120582120575(119889119911

120575) 119889119904



E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times (

119873

sum

119895=1

120574119903120576(119904)119895

(120593120576

(0) minus 119863 (120593120576

119895) 119904 119895)

minus

119897

sum

119896=1

120574119903120576(119904)119896

(120593120576

(0) minus 119863 (120593120576

119896) 119904 119896))119889119904

= 1198691+ 119869

2+ 119869

3+ 119869

4+ 119869

5

(82)


119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) minus

(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)

=

119897

sum

119894=1

119873119896

sum

119895=1

119887(120575)

(119909120576

119905minus 119905 119904

119896119895 119911

120575)

times [119868119903120576(119905)=119904119896119895

minus 120583119896

119895119868119903120576(119905)=119896

]

(83)


1198694= lim sup

120576rarr0

119898

sum

120575=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times intR

[119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

minus (120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)]

times 120582120575(119889119911

120575) 119889119904


119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904))

119904 119903120576

(119904))

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)

times [119868119903120576(119904)=119904119896119895

minus120583119896

119895119868119903120576(119904)=119896

]

times 120582120575(119889119911

120575) 119889119904


119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

[E

100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904))

119904 119903120576

(119904))

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

times120582120575(119889119911

120575) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

]

12

(84)


4= 0 Similarly


119905


(0) minus 119863(120593120576

119903120576

(119905)) + 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)

and 120593120576

(0) minus 119863(120593120576

119903120576

(119905)) + (120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) such that

1198693= lim

120576rarr0

119898

sum

120575=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

intR

119881119909(120578

119904)

times [119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

minus(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)]

times 120582120575(119889119911

120575) 119889119904

= lim120576rarr0

119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120578

119904)

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

times 120582120575(119889119911

120575) 119889119904

le lim120576rarr0

119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

[E

100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120578

119905)

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)


times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

times 120582120575(119889119911

120575)119889119904

100381610038161003816100381610038161003816100381610038161003816

2

]

12

(85)



5= 0Therefore


= lim120576rarr0

Eint

119905+120579+ℎ

119905+120579

119890120574119904

times [L119881 (120593120576

119904 119903120576

(119904))

+120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))] 119889119904

le 0

(86)

Then

119880 (119905 + ℎ) le 119880 (119905) (87)


E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

119890minus]119905

(88)


5 Examples



in (14) with

Γ = (

minus1 0 1 0 0

1 minus2 1 0 0

2 1 minus3 0 0

0 0 0 minus1 1

0 0 0 1 minus1

) (89)

Γ = (

minus1 0 1 0 0

0 minus1 0 1 0

0 0 minus1 0 1

0 1 0 minus1 0

1 0 0 0 minus1

) (90)


(1205871

1205872

1205873)(

minus1 0 1

1 minus2 1

2 1 minus3

) = 0 (91)

and 1205871+ 120587

2+ 120587

3= 1 we get 1205831


(1205874

1205875) (

minus1 1

1 minus1) = 0 (92)

and 1205874+ 120587

5= 1 we have 120583

2

= (12 12) So

Γ = 120583Γ1 = (

5

8

1

8

1

40 0

0 0 01

2

1

2

)

times (

minus1 0 1 0 0

0 minus1 0 1 0

0 0 minus1 0 1

0 1 0 minus1 0

1 0 0 0 minus1

)(

1 0

1 0

1 0

0 1

0 1

)

= (minus3

8

3

81 minus1

)

(93)


119889 [120593120576

(0) minus 119863 (120593120576

119903120576

(119905))]

= 119891 (120593120576

119903120576

(119905)) 119889119905 + 119892 (120593120576

119903120576

(119905)) 119889119908 (119905)

(94)

with

119863(120593 11990411) = minus06 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990412) = minus02 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990413) = minus04 int

0

minus1

120593 (120579) 119889120579

119891 (120593 11990411) = minus16120593 (0) minus 8 cos [120593 (0)]

119891 (120593 11990412) = 8120593 (0) + 4 cos [120593 (0)]

119891 (120593 11990413) = 16120593 (0)

119892 (120593 11990411) =

radic10

10int

0

minus1

120593 (120579) 119889120579 cos [int0

minus1

120593 (120579) 119889120579]

119892 (120593 11990412) = minus

radic2

2int

0

minus1

120593 (120579) 119889120579 sin [int

0

minus1

120593 (120579) 119889120579]

119892 (120593 11990413) =

radic3

2int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990421) = 05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990422) = 05 int

0

minus1

120593 (120579) 119889120579

119891 (120593 11990421) = minus2120593 (0) 119891 (120593 119904

22) = minus2120593 (0)


119892 (120593 11990421) =

int

0

minus1

120593 (120579) 119889120579 sin [int

0

minus1

120593 (120579) 119889120579]

4radic2

119892 (120593 11990422) =

int

0

minus1

120593 (120579) 119889120579 cos [int0

minus1

120593 (120579) 119889120579]

4radic2

(95)

For any 120593 isin 1198712

F119905([minus1 0]R) and 120581 = max06 02 04 = 06


E1003816100381610038161003816119863 (120593 119894)

10038161003816100381610038162


119890]120579E

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

le 0361003817100381710038171003817120593

10038171003817100381710038172

(96)


119889 [120593 (0) minus 119863 (120593 119903 (119905))] = 119891 (120593 119903 (119905)) 119889119905 + 119892 (120593 119903 (119905)) 119889119908 (119905)

(97)


119863(120593 1) = minus05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 2) = 05 int

0

minus1

120593 (120579) 119889120579

119891 (120593 1) = minus5120593 (0) 119891 (120593 2) = minus2120593 (0)

119892 (120593 1) =1

2int

0

minus1

120593 (120579) 119889120579 119892 (120593 2) =1

4int

0

minus1

120593 (120579) 119889120579

(98)

We define 119881(119909 1) = 21199092 119881(119909 2) = 119909


L119881 (120593 1) le minus203

81205932

(0) +13

32

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

L119881 (120593 2) le minus5

21205932

(0) +13

16

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

(99)

Consequently

max119894=12

L119881 (120593 119894) le minus5

21205932

(0) +13

16

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

= minus5

4[max119894=12

119881 (119909 119894)] +13

16[min119894=12

119881 (119909 119894)]

(100)


2


119894isinS120601(120579)] le


E [max119894isinS

L119881 (120593 119894)] le minus (5

4minus

13119902

16)E [max

119894=12

119881 (119909 119894)] (101)




Γ120576

=1

120576Γ + Γ =

1

120576(

minus2 0 2 0

1 minus2 0 1

0 2 minus2 0

0 1 1 minus2

) (102)



119889 [120593120576

(0) minus 119863 (120593120576

119903120576

(119905))]

= 119891 (120593120576

119903120576

(119905)) 119889119905 + int

infin

0

120590 (119903120576

(119905) 119911) 119909120576

119905minus (119889119905 119889119911)

(103)

with

119863(120593 1) = minus09 int

0

minus1

120593 (120579) 119889120579

119863 (120593 2) = minus04 int

0

minus1

120593 (120579) 119889120579

119863 (120593 3) = minus05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 4) = minus03 int

0

minus1

120593 (120579) 119889120579

119891 (120593 1) = 2 sin [120593 (0)] 119891 (120593 2) = minus11

2120593 (0)

119891 (120593 3) = minus11

3120593 (0) 119891 (120593 4) = minus2 sin [120593 (0)]

(104)

Let

120572 (119911) =2

11120590 (1 119911) +

4

11120590 (2 119911) +

3

11120590 (3 119911) +

2

11120590 (4 119911)

int

infin

0

1205722

(119911) 120582 (119889119911) lt 2

(105)

For any 120593 isin 1198712

F119905([minus1 0]R) and 120581 = max09 04 05 03 =


E1003816100381610038161003816119863 (120593 119894)

10038161003816100381610038162


119890]120579E

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

le 0811003817100381710038171003817120593

10038171003817100381710038172

(106)


119889 [120593 (0) + 05 int

0

minus1

120593 (120579) 119889120579]

= minus3120593 (0) 119889119905 + int

infin

0

120572 (119911) 119909119905minus (119889119905 119889119911)

(107)


Let 119881(119909) = 1199092 then

L119881 (120593 119894) le minus61205932

(0) + int

infin

0

1205722

(119911) 120582 (119889119911)

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

(108)


1198712


119894isinS120601(120579)] le 119902E[max

119894isinS120601(0)]


E [max119894isinS


119881 (119909 119894)] (109)





Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times (

119873

sum

119895=1

120574119903120576(119904)119895

(120593120576

(0) minus 119863 (120593120576

119895) 119904 119895)

minus

119897

sum

119896=1

120574119903120576(119904)119896

(120593120576

(0) minus 119863 (120593120576

119896) 119904 119896))119889119904

= 1198691+ 119869

2+ 119869

3+ 119869

4+ 119869

5

(82)


119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) minus

(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)

=

119897

sum

119894=1

119873119896

sum

119895=1

119887(120575)

(119909120576

119905minus 119905 119904

119896119895 119911

120575)

times [119868119903120576(119905)=119904119896119895

minus 120583119896

119895119868119903120576(119905)=119896

]

(83)


1198694= lim sup

120576rarr0

119898

sum

120575=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))

times intR

[119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

minus (120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)]

times 120582120575(119889119911

120575) 119889119904


119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904))

119904 119903120576

(119904))

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)

times [119868119903120576(119904)=119904119896119895

minus120583119896

119895119868119903120576(119904)=119896

]

times 120582120575(119889119911

120575) 119889119904


119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

[E

100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

times 119881119909(120593

120576

(0) minus 119863 (120593120576

119903120576

(119904))

119904 119903120576

(119904))

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

times120582120575(119889119911

120575) 119889119904

100381610038161003816100381610038161003816100381610038161003816

2

]

12

(84)


4= 0 Similarly


119905


(0) minus 119863(120593120576

119903120576

(119905)) + 119887(120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575)

and 120593120576

(0) minus 119863(120593120576

119903120576

(119905)) + (120575)

(119909120576

119905minus 119905 119903

120576

(119905) 119911120575) such that

1198693= lim

120576rarr0

119898

sum

120575=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

intR

119881119909(120578

119904)

times [119887(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)

minus(120575)

(119909120576

119904minus 119904 119903

120576

(119904) 119911120575)]

times 120582120575(119889119911

120575) 119889119904

= lim120576rarr0

119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

E

times int

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120578

119904)

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)

times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

times 120582120575(119889119911

120575) 119889119904

le lim120576rarr0

119898

sum

120575=1

119897

sum

119896=1

119873119896

sum

119895=1

[E

100381610038161003816100381610038161003816100381610038161003816

int

119905+120579+ℎ

119905+120579

119890120574119904

119881119909(120578

119905)

times intR

119887(120575)

(119909120576

119904minus 119904 119904

119896119895 119911

120575)


times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

times 120582120575(119889119911

120575)119889119904

100381610038161003816100381610038161003816100381610038161003816

2

]

12

(85)



5= 0Therefore


= lim120576rarr0

Eint

119905+120579+ℎ

119905+120579

119890120574119904

times [L119881 (120593120576

119904 119903120576

(119904))

+120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))] 119889119904

le 0

(86)

Then

119880 (119905 + ℎ) le 119880 (119905) (87)


E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

119890minus]119905

(88)


5 Examples



in (14) with

Γ = (

minus1 0 1 0 0

1 minus2 1 0 0

2 1 minus3 0 0

0 0 0 minus1 1

0 0 0 1 minus1

) (89)

Γ = (

minus1 0 1 0 0

0 minus1 0 1 0

0 0 minus1 0 1

0 1 0 minus1 0

1 0 0 0 minus1

) (90)


(1205871

1205872

1205873)(

minus1 0 1

1 minus2 1

2 1 minus3

) = 0 (91)

and 1205871+ 120587

2+ 120587

3= 1 we get 1205831


(1205874

1205875) (

minus1 1

1 minus1) = 0 (92)

and 1205874+ 120587

5= 1 we have 120583

2

= (12 12) So

Γ = 120583Γ1 = (

5

8

1

8

1

40 0

0 0 01

2

1

2

)

times (

minus1 0 1 0 0

0 minus1 0 1 0

0 0 minus1 0 1

0 1 0 minus1 0

1 0 0 0 minus1

)(

1 0

1 0

1 0

0 1

0 1

)

= (minus3

8

3

81 minus1

)

(93)


119889 [120593120576

(0) minus 119863 (120593120576

119903120576

(119905))]

= 119891 (120593120576

119903120576

(119905)) 119889119905 + 119892 (120593120576

119903120576

(119905)) 119889119908 (119905)

(94)

with

119863(120593 11990411) = minus06 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990412) = minus02 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990413) = minus04 int

0

minus1

120593 (120579) 119889120579

119891 (120593 11990411) = minus16120593 (0) minus 8 cos [120593 (0)]

119891 (120593 11990412) = 8120593 (0) + 4 cos [120593 (0)]

119891 (120593 11990413) = 16120593 (0)

119892 (120593 11990411) =

radic10

10int

0

minus1

120593 (120579) 119889120579 cos [int0

minus1

120593 (120579) 119889120579]

119892 (120593 11990412) = minus

radic2

2int

0

minus1

120593 (120579) 119889120579 sin [int

0

minus1

120593 (120579) 119889120579]

119892 (120593 11990413) =

radic3

2int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990421) = 05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990422) = 05 int

0

minus1

120593 (120579) 119889120579

119891 (120593 11990421) = minus2120593 (0) 119891 (120593 119904

22) = minus2120593 (0)


119892 (120593 11990421) =

int

0

minus1

120593 (120579) 119889120579 sin [int

0

minus1

120593 (120579) 119889120579]

4radic2

119892 (120593 11990422) =

int

0

minus1

120593 (120579) 119889120579 cos [int0

minus1

120593 (120579) 119889120579]

4radic2

(95)

For any 120593 isin 1198712

F119905([minus1 0]R) and 120581 = max06 02 04 = 06


E1003816100381610038161003816119863 (120593 119894)

10038161003816100381610038162


119890]120579E

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

le 0361003817100381710038171003817120593

10038171003817100381710038172

(96)


119889 [120593 (0) minus 119863 (120593 119903 (119905))] = 119891 (120593 119903 (119905)) 119889119905 + 119892 (120593 119903 (119905)) 119889119908 (119905)

(97)


119863(120593 1) = minus05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 2) = 05 int

0

minus1

120593 (120579) 119889120579

119891 (120593 1) = minus5120593 (0) 119891 (120593 2) = minus2120593 (0)

119892 (120593 1) =1

2int

0

minus1

120593 (120579) 119889120579 119892 (120593 2) =1

4int

0

minus1

120593 (120579) 119889120579

(98)

We define 119881(119909 1) = 21199092 119881(119909 2) = 119909


L119881 (120593 1) le minus203

81205932

(0) +13

32

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

L119881 (120593 2) le minus5

21205932

(0) +13

16

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

(99)

Consequently

max119894=12

L119881 (120593 119894) le minus5

21205932

(0) +13

16

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

= minus5

4[max119894=12

119881 (119909 119894)] +13

16[min119894=12

119881 (119909 119894)]

(100)


2


119894isinS120601(120579)] le


E [max119894isinS

L119881 (120593 119894)] le minus (5

4minus

13119902

16)E [max

119894=12

119881 (119909 119894)] (101)




Γ120576

=1

120576Γ + Γ =

1

120576(

minus2 0 2 0

1 minus2 0 1

0 2 minus2 0

0 1 1 minus2

) (102)



119889 [120593120576

(0) minus 119863 (120593120576

119903120576

(119905))]

= 119891 (120593120576

119903120576

(119905)) 119889119905 + int

infin

0

120590 (119903120576

(119905) 119911) 119909120576

119905minus (119889119905 119889119911)

(103)

with

119863(120593 1) = minus09 int

0

minus1

120593 (120579) 119889120579

119863 (120593 2) = minus04 int

0

minus1

120593 (120579) 119889120579

119863 (120593 3) = minus05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 4) = minus03 int

0

minus1

120593 (120579) 119889120579

119891 (120593 1) = 2 sin [120593 (0)] 119891 (120593 2) = minus11

2120593 (0)

119891 (120593 3) = minus11

3120593 (0) 119891 (120593 4) = minus2 sin [120593 (0)]

(104)

Let

120572 (119911) =2

11120590 (1 119911) +

4

11120590 (2 119911) +

3

11120590 (3 119911) +

2

11120590 (4 119911)

int

infin

0

1205722

(119911) 120582 (119889119911) lt 2

(105)

For any 120593 isin 1198712

F119905([minus1 0]R) and 120581 = max09 04 05 03 =


E1003816100381610038161003816119863 (120593 119894)

10038161003816100381610038162


119890]120579E

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

le 0811003817100381710038171003817120593

10038171003817100381710038172

(106)


119889 [120593 (0) + 05 int

0

minus1

120593 (120579) 119889120579]

= minus3120593 (0) 119889119905 + int

infin

0

120572 (119911) 119909119905minus (119889119905 119889119911)

(107)


Let 119881(119909) = 1199092 then

L119881 (120593 119894) le minus61205932

(0) + int

infin

0

1205722

(119911) 120582 (119889119911)

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

(108)


1198712


119894isinS120601(120579)] le 119902E[max

119894isinS120601(0)]


E [max119894isinS


119881 (119909 119894)] (109)





Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










times [119868119903120576(119904)=119904119896119895

minus 120583119896

119895119868119903120576(119904)=119896

]

times 120582120575(119889119911

120575)119889119904

100381610038161003816100381610038161003816100381610038161003816

2

]

12

(85)



5= 0Therefore


= lim120576rarr0

Eint

119905+120579+ℎ

119905+120579

119890120574119904

times [L119881 (120593120576

119904 119903120576

(119904))

+120574119881 (120593120576

(0) minus 119863 (120593120576

119903120576

(119904)) 119904 119903120576

(119904))] 119889119904

le 0

(86)

Then

119880 (119905 + ℎ) le 119880 (119905) (87)


E1003816100381610038161003816119909

120576

(119905)1003816100381610038161003816119901

le1198882(1 + 120581)

119901

1198881(1 minus 120581)

119901

10038171003817100381710038171205851003817100381710038171003817119901

119890minus]119905

(88)


5 Examples



in (14) with

Γ = (

minus1 0 1 0 0

1 minus2 1 0 0

2 1 minus3 0 0

0 0 0 minus1 1

0 0 0 1 minus1

) (89)

Γ = (

minus1 0 1 0 0

0 minus1 0 1 0

0 0 minus1 0 1

0 1 0 minus1 0

1 0 0 0 minus1

) (90)


(1205871

1205872

1205873)(

minus1 0 1

1 minus2 1

2 1 minus3

) = 0 (91)

and 1205871+ 120587

2+ 120587

3= 1 we get 1205831


(1205874

1205875) (

minus1 1

1 minus1) = 0 (92)

and 1205874+ 120587

5= 1 we have 120583

2

= (12 12) So

Γ = 120583Γ1 = (

5

8

1

8

1

40 0

0 0 01

2

1

2

)

times (

minus1 0 1 0 0

0 minus1 0 1 0

0 0 minus1 0 1

0 1 0 minus1 0

1 0 0 0 minus1

)(

1 0

1 0

1 0

0 1

0 1

)

= (minus3

8

3

81 minus1

)

(93)


119889 [120593120576

(0) minus 119863 (120593120576

119903120576

(119905))]

= 119891 (120593120576

119903120576

(119905)) 119889119905 + 119892 (120593120576

119903120576

(119905)) 119889119908 (119905)

(94)

with

119863(120593 11990411) = minus06 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990412) = minus02 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990413) = minus04 int

0

minus1

120593 (120579) 119889120579

119891 (120593 11990411) = minus16120593 (0) minus 8 cos [120593 (0)]

119891 (120593 11990412) = 8120593 (0) + 4 cos [120593 (0)]

119891 (120593 11990413) = 16120593 (0)

119892 (120593 11990411) =

radic10

10int

0

minus1

120593 (120579) 119889120579 cos [int0

minus1

120593 (120579) 119889120579]

119892 (120593 11990412) = minus

radic2

2int

0

minus1

120593 (120579) 119889120579 sin [int

0

minus1

120593 (120579) 119889120579]

119892 (120593 11990413) =

radic3

2int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990421) = 05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 11990422) = 05 int

0

minus1

120593 (120579) 119889120579

119891 (120593 11990421) = minus2120593 (0) 119891 (120593 119904

22) = minus2120593 (0)


119892 (120593 11990421) =

int

0

minus1

120593 (120579) 119889120579 sin [int

0

minus1

120593 (120579) 119889120579]

4radic2

119892 (120593 11990422) =

int

0

minus1

120593 (120579) 119889120579 cos [int0

minus1

120593 (120579) 119889120579]

4radic2

(95)

For any 120593 isin 1198712

F119905([minus1 0]R) and 120581 = max06 02 04 = 06


E1003816100381610038161003816119863 (120593 119894)

10038161003816100381610038162


119890]120579E

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

le 0361003817100381710038171003817120593

10038171003817100381710038172

(96)


119889 [120593 (0) minus 119863 (120593 119903 (119905))] = 119891 (120593 119903 (119905)) 119889119905 + 119892 (120593 119903 (119905)) 119889119908 (119905)

(97)


119863(120593 1) = minus05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 2) = 05 int

0

minus1

120593 (120579) 119889120579

119891 (120593 1) = minus5120593 (0) 119891 (120593 2) = minus2120593 (0)

119892 (120593 1) =1

2int

0

minus1

120593 (120579) 119889120579 119892 (120593 2) =1

4int

0

minus1

120593 (120579) 119889120579

(98)

We define 119881(119909 1) = 21199092 119881(119909 2) = 119909


L119881 (120593 1) le minus203

81205932

(0) +13

32

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

L119881 (120593 2) le minus5

21205932

(0) +13

16

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

(99)

Consequently

max119894=12

L119881 (120593 119894) le minus5

21205932

(0) +13

16

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

= minus5

4[max119894=12

119881 (119909 119894)] +13

16[min119894=12

119881 (119909 119894)]

(100)


2


119894isinS120601(120579)] le


E [max119894isinS

L119881 (120593 119894)] le minus (5

4minus

13119902

16)E [max

119894=12

119881 (119909 119894)] (101)




Γ120576

=1

120576Γ + Γ =

1

120576(

minus2 0 2 0

1 minus2 0 1

0 2 minus2 0

0 1 1 minus2

) (102)



119889 [120593120576

(0) minus 119863 (120593120576

119903120576

(119905))]

= 119891 (120593120576

119903120576

(119905)) 119889119905 + int

infin

0

120590 (119903120576

(119905) 119911) 119909120576

119905minus (119889119905 119889119911)

(103)

with

119863(120593 1) = minus09 int

0

minus1

120593 (120579) 119889120579

119863 (120593 2) = minus04 int

0

minus1

120593 (120579) 119889120579

119863 (120593 3) = minus05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 4) = minus03 int

0

minus1

120593 (120579) 119889120579

119891 (120593 1) = 2 sin [120593 (0)] 119891 (120593 2) = minus11

2120593 (0)

119891 (120593 3) = minus11

3120593 (0) 119891 (120593 4) = minus2 sin [120593 (0)]

(104)

Let

120572 (119911) =2

11120590 (1 119911) +

4

11120590 (2 119911) +

3

11120590 (3 119911) +

2

11120590 (4 119911)

int

infin

0

1205722

(119911) 120582 (119889119911) lt 2

(105)

For any 120593 isin 1198712

F119905([minus1 0]R) and 120581 = max09 04 05 03 =


E1003816100381610038161003816119863 (120593 119894)

10038161003816100381610038162


119890]120579E

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

le 0811003817100381710038171003817120593

10038171003817100381710038172

(106)


119889 [120593 (0) + 05 int

0

minus1

120593 (120579) 119889120579]

= minus3120593 (0) 119889119905 + int

infin

0

120572 (119911) 119909119905minus (119889119905 119889119911)

(107)


Let 119881(119909) = 1199092 then

L119881 (120593 119894) le minus61205932

(0) + int

infin

0

1205722

(119911) 120582 (119889119911)

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

(108)


1198712


119894isinS120601(120579)] le 119902E[max

119894isinS120601(0)]


E [max119894isinS


119881 (119909 119894)] (109)





Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119892 (120593 11990421) =

int

0

minus1

120593 (120579) 119889120579 sin [int

0

minus1

120593 (120579) 119889120579]

4radic2

119892 (120593 11990422) =

int

0

minus1

120593 (120579) 119889120579 cos [int0

minus1

120593 (120579) 119889120579]

4radic2

(95)

For any 120593 isin 1198712

F119905([minus1 0]R) and 120581 = max06 02 04 = 06


E1003816100381610038161003816119863 (120593 119894)

10038161003816100381610038162


119890]120579E

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

le 0361003817100381710038171003817120593

10038171003817100381710038172

(96)


119889 [120593 (0) minus 119863 (120593 119903 (119905))] = 119891 (120593 119903 (119905)) 119889119905 + 119892 (120593 119903 (119905)) 119889119908 (119905)

(97)


119863(120593 1) = minus05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 2) = 05 int

0

minus1

120593 (120579) 119889120579

119891 (120593 1) = minus5120593 (0) 119891 (120593 2) = minus2120593 (0)

119892 (120593 1) =1

2int

0

minus1

120593 (120579) 119889120579 119892 (120593 2) =1

4int

0

minus1

120593 (120579) 119889120579

(98)

We define 119881(119909 1) = 21199092 119881(119909 2) = 119909


L119881 (120593 1) le minus203

81205932

(0) +13

32

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

L119881 (120593 2) le minus5

21205932

(0) +13

16

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

(99)

Consequently

max119894=12

L119881 (120593 119894) le minus5

21205932

(0) +13

16

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

= minus5

4[max119894=12

119881 (119909 119894)] +13

16[min119894=12

119881 (119909 119894)]

(100)


2


119894isinS120601(120579)] le


E [max119894isinS

L119881 (120593 119894)] le minus (5

4minus

13119902

16)E [max

119894=12

119881 (119909 119894)] (101)




Γ120576

=1

120576Γ + Γ =

1

120576(

minus2 0 2 0

1 minus2 0 1

0 2 minus2 0

0 1 1 minus2

) (102)



119889 [120593120576

(0) minus 119863 (120593120576

119903120576

(119905))]

= 119891 (120593120576

119903120576

(119905)) 119889119905 + int

infin

0

120590 (119903120576

(119905) 119911) 119909120576

119905minus (119889119905 119889119911)

(103)

with

119863(120593 1) = minus09 int

0

minus1

120593 (120579) 119889120579

119863 (120593 2) = minus04 int

0

minus1

120593 (120579) 119889120579

119863 (120593 3) = minus05 int

0

minus1

120593 (120579) 119889120579

119863 (120593 4) = minus03 int

0

minus1

120593 (120579) 119889120579

119891 (120593 1) = 2 sin [120593 (0)] 119891 (120593 2) = minus11

2120593 (0)

119891 (120593 3) = minus11

3120593 (0) 119891 (120593 4) = minus2 sin [120593 (0)]

(104)

Let

120572 (119911) =2

11120590 (1 119911) +

4

11120590 (2 119911) +

3

11120590 (3 119911) +

2

11120590 (4 119911)

int

infin

0

1205722

(119911) 120582 (119889119911) lt 2

(105)

For any 120593 isin 1198712

F119905([minus1 0]R) and 120581 = max09 04 05 03 =


E1003816100381610038161003816119863 (120593 119894)

10038161003816100381610038162


119890]120579E

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

le 0811003817100381710038171003817120593

10038171003817100381710038172

(106)


119889 [120593 (0) + 05 int

0

minus1

120593 (120579) 119889120579]

= minus3120593 (0) 119889119905 + int

infin

0

120572 (119911) 119909119905minus (119889119905 119889119911)

(107)


Let 119881(119909) = 1199092 then

L119881 (120593 119894) le minus61205932

(0) + int

infin

0

1205722

(119911) 120582 (119889119911)

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

(108)


1198712


119894isinS120601(120579)] le 119902E[max

119894isinS120601(0)]


E [max119894isinS


119881 (119909 119894)] (109)





Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Let 119881(119909) = 1199092 then

L119881 (120593 119894) le minus61205932

(0) + int

infin

0

1205722

(119911) 120582 (119889119911)

100381610038161003816100381610038161003816100381610038161003816

int

0

minus1

120593 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161003816

2

(108)


1198712


119894isinS120601(120579)] le 119902E[max

119894isinS120601(0)]


E [max119894isinS


119881 (119909 119894)] (109)





Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article exponential stability of neutral stochastic...

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