infinite horizon optimal control of stochastic delay...

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 791786 14 pageshttpdxdoiorg1011552013791786

Research ArticleInfinite Horizon Optimal Control of Stochastic Delay EvolutionEquations in Hilbert Spaces

Xueping Zhu1 and Jianjun Zhou2

1 School of Astronautics Northwestern Polytechnical University Xirsquoan Shaanxi 710072 China2 College of Science Northwest AampF University Yangling Shaanxi 712100 China

Correspondence should be addressed to Jianjun Zhou zhoujj198310163com

Received 6 September 2012 Accepted 16 December 2012

Academic Editor Juan J Trujillo

Copyright copy 2013 X Zhu and J Zhou 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

The aim of the present paper is to study an infinite horizon optimal control problem in which the controlled state dynamicsis governed by a stochastic delay evolution equation in Hilbert spaces The existence and uniqueness of the optimal controlare obtained by means of associated infinite horizon backward stochastic differential equations without assuming the Gateauxdifferentiability of the drift coefficient and the diffusion coefficient An optimal control problem of stochastic delay partialdifferential equations is also given as an example to illustrate our results

1 Introduction

In this paper we consider a controlled stochastic evolutionequation of the following form

119889119883119906

(119904) = 119860119883119906

(119904) 119889119904 + 119865 (119904 119883119906

119904) 119889119904

+ 119866 (119904 119883119906

119904) 119877 (119904 119883

119906

119904 119906 (119904)) 119889119904 + 119866 (119904 119883

119906

119904) 119889119882 (119904)

119904 ge 119905

119883119906

119905= 119909

(1)

where

119883119906

119904(119897) = 119883

119906

(119904 + 119897) 119897 isin [minus120591 0] 119909 isin 119862 ([minus120591 0] 119867) (2)

119906 is the control process in a measurable space (119880U)and119882 is a cylindrical Wiener process in a Hilbert space Ξ 119860is the generator of a strongly continuous semigroup ofbounded linear operator in another Hilbert space 119867 andthe coefficients 119865 and 119866 defined on [0infin) times 119862 ([minus120591 0]119867)are assumed to satisfy Lipschitz conditions with respect toappropriate norms We introduce the cost function

119869 (119906) = 119864int

infin

119905

119890minus120582119904

119892 (119904 119883119906

119904 119906 (119904)) 119889119904 (3)

Here 119892 is a given real function 120582 is large enough and thecontrol problem is understood in the weak sense We wish tominimize the cost function over all admissible controls

The particular form of the control system is essentialfor our results but it covers numerous interesting cases Forexample in the particular cases119880 = 119867 and 119877(119905 119909 119906) = 119906 theterm 119906(119905)119889119905 + 119889119882(119905) in the state equation can be consideredas a control affected by noise

The stochastic optimal control problem was consideredin 1977 by Bismut [1] The optimal control problem forstochastic partial differential equations in the framework ofa compact control state space has been studied in [2ndash5]Buckdahn and Rascanu [6] considered an optimal controlproblem for a semilinear parabolic stochastic differentialequation with a nonlinear diffusion coefficient and theexistence of a quasioptimal (nonrelaxed) control is showedwithout assuming convexity of the coefficients In [7ndash11] theauthors provided a direct (classical or mild) solution of theHamilton-Jacobi-Bellman equation for the value functionwhich is then used to prove that the optimal control isrelated to the corresponding optimal trajectory by a feedbacklaw In Gozzi [10 11] the existence and uniqueness of amild solution of the associated Hamilton-Jacobi-Bellmanequation are proved when the diffusion term only satisfiesweak nondegeneracy conditions The proofs are based on

2 Abstract and Applied Analysis

the corresponding regularity properties of the transitionsemigroup of the associated Ornstein-Uhlenbeck process

The main tools for the control problem are techniquesfrom the theory of backward stochastic differential equations(BSDEs) in the sense of Pardoux and Peng first consideredin the nonlinear case in [12] see [13 14] as general referencesBSDEs have been successfully applied to control problemssee for example [15 16] and we also refer the reader to[17ndash20] Fuhrman and Tessiture [19] considered the optimalcontrol problem for stochastic differential equation in thestrong form assuming Lipschitz conditions and allowingdegeneracy of the diffusion coefficient under some structuralconstraint on the state equation Existence of an optimalcontrol for stochastic systems in infinite dimensional spacesalso has been obtained in [21ndash27] In [21] Fuhrman and Tes-sitore showed the regularity with respect to parameters andthe regularity in the Malliavin spaces for the solution of thebackward-forward system and defined the feedback law byMalliavin calculus Finally the optimal control is obtained bythe feedback Appealing to the Malliavin calculus comparedwith Fuhrman et al [23] the existence of optimal controlfor stochastic differential equations with delay is proved bythe feedback law Fuhrman and Tessiture [24] dealt with aninfinite horizon optimal control problem for the stochas-tic evolution equation in Hilbert space and the optimalcontrol is showed by means of infinite horizon backwardstochastic differential equation in infinite dimensional spacesand Malliavin calculus In Masiero [25] the infinite horizonoptimal control problem for stochastic evolution equationis also studied by means of the Hamilton-Jacobi-Bellmanequation In Fuhrman [26] a class of optimal control prob-lems governed by stochastic evolution equations in Hilbertspaces which includes state constraints is considered andthe optimal control is obtained by the Fleming logarithmictransformation Masiero [27] studied stochastic evolutionequations evolving in a Banach space where 119866 is a constantand characterized the optimal control via a feedback law byavoiding use of Malliavin calculus Since there is a lack ofregularity of 119865 and 119866 Malliavin calculus is not available inthis case the method in [27] also cannot be used as 119866 isnot a constant but we can prove a theorem similar to [26Proposition 32] which will be used to define the feedbacklaw

In the present paper we study the infinite horizon optimalcontrol problem for stochastic delay evolution equations inHilbert spaces and by usingTheorem 10 the optimal controlis obtained Since we do not relate the optimal feedback lawwith the gradient of the value function and do not considerthe associated Hamilton-Jacobi-Bellman equation we candrop the Gateaux differentiability of the drift term and thediffusion term

The plan of the paper is as follows In the next sectionsome notations are fixed and the stochastic delay evolutionequations are considered with an infinite horizon in particu-lar continuous dependence on initial value (119905 119909) is proved InSection 3 we give the proof of Theorem 10 which is the keyof many subsequent results The addressed optimal controlproblem is considered and the fundamental relation betweenthe optimal control problem and BSDEs is established in

Section 4 Section 5 is devoted to proving the existence anduniqueness of optimal control in the weak sense Finally anapplication is given in Section 6

2 Preliminaries

We list some notations that are used in this paper We usethe symbol | sdot | to denote the norm in a Banach space 119865with a subscript if necessary Let Ξ 119867 and 119870 denote realseparable Hilbert spaces with scalar products (sdot sdot)

Ξ (sdot sdot)

119867

and (sdot sdot)119870 respectively For fixed 120591 gt 0 C = 119862([minus120591 0]119867)

denotes the space of continuous functions from [minus120591 0] to119867endowed with the usual norm |119891|

119862= sup

120579isin[minus1205910]|119891(120579)|

119867 Let

Ξlowast denote the dual space of Ξ with scalar product (sdot sdot)

Ξlowast and

let 119871(Ξ119867) denote the space of all bounded linear operatorsfrom Ξ into 119867 the subspace of Hilbert-Schmidt operatorswith the Hilbert-Schmidt norm is denoted by 119871

2(Ξ119867)

Let (ΩF 119875) be a complete space with a filtration F119905119905ge0

which satisfies the usual condition By a cylindrical Wienerprocess with values in aHilbert spaceΞ defined on (ΩF 119875)we mean a family 119882(119905) 119905 ge 0 of linear mappings Ξ rarr

1198712

(Ω) such that for every 120585 120578 isin Ξ 119882(119905)120585 119905 ge 0 is areal Wiener process and 119864(119882(119905)120585 sdot 119882(119905)120578) = (120585 120578)

Ξ In

the following 119882(119905) 119905 ge 0 is a cylindrical Wiener processadapted to the filtration F

119905119905ge0

In this section and the next section F

119905119905ge0

will denotethe natural filtration of 119882 augmented with the family Nof 119875-null of F The filtration F

119905 119905 ge 0 satisfies the usual

conditions For [119886 119887] [119886infin) sub [0infin) we also use thefollowing notations

F[119886119887]

= 120590 (119882 (119904) minus 119882 (119886) 119904 isin [119886 119887]) orN

F[119886infin)

= 120590 (119882 (119904) minus 119882 (119886) 119904 isin [ainfin)) orN

(4)

By P we denote the predictable 120590-algebra and by B(Λ) wedenote the Borel 120590-algebra of any topological space Λ

Similar to [24] we define several classes of stochasticprocesses with values in a Banach space 119865 as follows

(i) 1198712P(Ω times [119905infin) 119865) denotes the space of equivalenceclasses of processes 119884 isin 119871

2

(Ω times [119905infin) 119865) admittinga predictable version 1198712P(Ω times [119905infin) 119865) is endowedwith the norm

|119884|2

= 119864int

infin

119905

|119884(119904)|2

119889119904 (5)

(ii) 119871119901P(Ω 119871

119902

120573([119905infin) 119865)) defined for 120573 isin 119877 and 119901 119902 isin

[1infin) denotes the space of equivalence classes ofprocesses 119884(119904) 119904 ge 119905 with values in 119865 such that thenorm

|119884|119901

= 119864(int

infin

119905

119890119902120573119904

|119884 (119904)|119902

119889119904)

119901119902

(6)

is finite and 119884 admits a predictable version(iii) K119901

120573(119905) denotes the space 119871

119901

P(Ω 119871

2

120573([119905infin) 119865)) times

119871119901

P(Ω 119871

2

120573([119905infin) 119871

2(Ξ 119865))) The norm of an element

Abstract and Applied Analysis 3

(119884 119885) isinK119901

120573is |(119884 119885)| = |119884|+ |119885| Here 119865 is a Hilbert

space(iv) 119871119901

P(Ω 119862([119905 119879] 119865)) defined for 119879 gt 119905 ge 0 and

119901 isin [1infin) denotes the space of predictable processes119884(119904) 119904 isin [119905 119879]with continuous paths in 119865 such thatthe norm

|119884|119901

= 119864 sup119904isin[119905119879]

|119884 (119904)|119901

(7)

is finite Elements of 119871119901P(Ω 119862([119905 119879] 119865)) are identified

up to indistinguishability

(v) 119871119902P(Ω 119862

120578([119905infin) 119865)) defined for 120578 isin 119877 and 119902 isin

[1infin) denotes the space of predictable processes119884(119904) 119904 ge 119905 with continuous paths in 119865 such that thenorm

|119884|119902

= 119864 sup119904ge119905

119890120578119902119904

|119884 (119904)|119902

(8)

is finite Elements of 119871119902P(Ω 119862

120578(119865)) are identified up

to indistinguishability(vi) Finally for 120578 isin 119877 and 119902 isin [1infin) we

defined Hq120578(119905) as the space 119871119902

P(Ω 119871

119902

120578([119905infin) 119865)) cap

119871119902

P(Ω 119862

120578([119905infin) 119865)) endowed with the norm

|119884|H119902

120578

= |119884|119871119902

P(Ω119871119902

120578([119905infin)119865))

+ |119884|119871119902

P(Ω119862120578([119905infin)119865))

(9)

For simplicity we denote 119871119901

P(Ω 119871

119902

120573([0infin) 119865)) 119871119902

P(Ω

119862120578([0infin) 119865)) H119902

120578(0) and K

119901

120573(0) by 119871119901

P(Ω 119871

119902

120573(119865)) 119871119902

P(Ω

119862120578(119865))H119902

120578 andK

119901

120573 respectively

Now for every fixed 119905 ge 0 we consider the followingstochastic delay evolution equation

119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119905 119883119904) 119889119904 + 119866 (119904 119883

119904) 119889119882 (119904)

119904 isin [119905infin)

119883119905= 119909 isin C

(10)

We make the following assumptions

Hypothesis 1 (i)The operator119860 is the generator of a stronglycontinuous semigroup 119890

119905119860

119905 ge 0 of bounded linearoperators in the Hilbert space119867 We denote by119872 and 120596 twoconstants such that |119890119905119860| le 119872119890120596119905 for 119905 ge 0

(ii) The mapping 119865 [0infin) timesC rarr 119867 is measurable andsatisfies for some constant 119871 gt 0 and 0 le 120579 lt 1

10038161003816100381610038161003816119890119904119860

119865 (119905 119909)10038161003816100381610038161003816le 119871119890

120596119904

119904minus120579

(1 + |119909|119862)

10038161003816100381610038161003816119890119904119860

119865 (119905 119909) minus 119890119904119860

119865 (119905 119910)10038161003816100381610038161003816le 119871119890

120596119904

119904minus1205791003816100381610038161003816119909 minus 119910

1003816100381610038161003816119862

119904 gt 0 119905 isin [0 +infin) 119909 119910 isin C

(11)

(iii) 119866 is a mapping [0infin) timesC rarr 119871(Ξ119867) such that forevery 119907 isin Ξ the map 119866119907 [0infin) times C rarr 119867 is measurable

119890119904119860

119866(119905 119909) isin 1198712(Ξ119867) for every s gt 0 119905 isin [0infin) and 119909 isin C

and

10038161003816100381610038161003816119890119904119860

119866 (119905 119909)100381610038161003816100381610038161198712(Ξ119867)

le 119871119890120596119904

119904minus120574

(1 + |119909|119862)

10038161003816100381610038161003816119890119904119860

119866 (119905 119909) minus 119890119904119860

119866 (119905 119910)100381610038161003816100381610038161198712(Ξ119867)

le 119871119890120596119904

119904minus120574

(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816119862)


(12)

for some constants 119871 gt 0 and 120574 isin [0 12)We say that119883 is amild solution of (10) if it is a continuous

F119905119905ge0

-predictable process with values in 119867 and it satisfies119875-as

119883 (119904) = 119890(119904minus119905)119860

119909 (0) + int

119904

119905

119890(119904minus120590)119860

119865 (120590119883120590) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883120590) 119889119882 (120590) 119904 isin [119905infin)

119883119905= 119909 isin C

(13)

To stress dependence on initial data we denote the solutionby 119883(119904 119905 119909) Note that 119883(119904 119905 119909) is F

[119905119904]measurable hence

independent ofF119905

We first recall a well-known result on solvability of (10)on bounded interval

Theorem 1 Assume that Hypothesis 1 holds Then for all119902 isin [2infin) and 119879 gt 0 there exists a unique process 119883 isin

119871119902

P(Ω 119862([119905 119879]119867)) as mild solution of (10) Moreover

119864 sup119904isin[119905119879]

|119883 (119904)|119902

le 119862(1 + |119909|119862)119902

(14)

for some constant C depending only on 119902 120574 120579T 120591 L 120596 andM

By Theorem 1 and the arbitrariness of 119879 in its statementthe solution is defined for every 119904 ge 119905 We have the followingresult

Theorem 2 Assume that Hypothesis 1 holds and the process119883(sdot 119905 119909) is mild solution of (10) with initial value (119905 119909) isin

[0infin) timesC Then for every 119902 isin [1infin) there exists a constant120578(119902) such that the process 119883

sdot(119905 119909) isin H

119902

120578(119902)(119905) Moreover for a

suitable constant 119862 gt 0 one has

119864supsget119890120578(119902)1199021199041003816100381610038161003816119883119904

1003816100381610038161003816119902

119862+ 119864int

infin

119905

119890120578(119902)1199021199041003816100381610038161003816119883119904

1003816100381610038161003816119902

119862119889119904 le 119862(1 + |119909|

119862)119902

(15)

with the constant 120578(119902) depending only on 119902 120574 120579 120591 119871 120596 and119872


Proof We define a mapping Φ from H119902

120578(119905) times [0infin) times C to

H119902

120578(119905) by the formula

Φ(119883sdot 119905 119909)

119904(119897) = 119890

(119904+119897minus119905)119860

119909 (0) + int

119904+119897

119905

119890(119904+119897minus120590)119860

119865 (120590119883120590) 119889120590

+ int

119904+119897

119905

119890(119904+119897minus120590)119860

119866 (120590119883120590) 119889119882 (120590)

119904 isin [119905infin) 119897 isin [minus120591 0] 119904 + 119897 ge 119905

Φ(119883sdot 119905 119909)

119904(119897) = 119909 (119904 + 119897 minus 119905)

119904 isin [119905infin) 119897 isin [minus120591 0] 119904 + 119897 lt 119905

(16)

We are going to show that provided 120578 is suitably chosenΦ(sdot 119905 119909) is well defined and that it is a contraction inH119902

120578(119905)

that is there exists 119888 lt 1 such that10038161003816100381610038161003816Φ (119883

1

sdot 119905 119909) minus Φ (119883

1

sdot 119905 119909)

10038161003816100381610038161003816H119902

120578(119905)

le 119888100381610038161003816100381610038161198831

sdotminus 119883

2

sdot

10038161003816100381610038161003816H119902

120578(119905)

1198831

sdot 119883

2

sdotisinH

119902

120578(119905)

(17)

For simplicity we set 119905 = 0 and we treat only the case 119865 = 0the general case being handled in a similar way We will usethe so called factorization method see [28 Theorem 525]Let us take 119902 gt 1 and 120572 isin (0 1) such that 1119902 lt 120572 lt (12) minus120574 and let 119888minus1

120572= int

s120590

(119904 minus 119903)120572minus1

(119903 minus 120590)minus120572

119889119903By the stochastic Fubini theorem

Φ(119883sdot 0 119909)

119904(119897) = 119890

(119904+119897)119860

119909 (0)

+ 119888120572int

119904+119897

0

int

119904+119897

120590

(119904 + 119897 minus 119903)120572minus1

(119903 minus 120590)minus120572

times 119890(119904+119897minus119903)119860

119890(119903minus120590)119860

119889119903119866 (120590119883120590) 119889119882 (120590)

= 119890(119904+119897)119860

119909 (0) + Φ1015840

(119883119904) (119897)


Φ(119883sdot 0 119909)

119904(119897) = 119909 (119904 + 119897)


(18)

where

Φ1015840

(119883sdot)119904(119897) = 119888

120572int

119904+119897

0

(119904 + 119897 minus 119903)120572minus1

119890(119904+119897minus119903)119860

119884 (119903) 119889119903

119884 (119903) = int

119903

0

(119903 minus 120590)minus120572

119890(119903minus120590)119860

119866 (120590119883120590) 119889119882 (120590)

(19)

Since supminus120591le119897le0

|119890(119904+119897)119860

119909(0)| le 119872119890120596119904

|119909|119862 the process 119890(119904+sdot)119860

119909(0) 119904 ge 0 belongs to H119902

120578provided 120596 + 120578 lt 0 Next we

estimate Φ1015840

(119883sdot) where

10038161003816100381610038161003816Φ1015840

(119883sdot)119904(119897)10038161003816100381610038161003816le 119888

120572int

119904+119897

0

(119904 + 119897 minus 119903)120572minus1

119872119890(119904+119897minus119903)120596

|119884 (119903)| 119889119903

(20)

setting 1199021015840 = 119902(119902 minus 1) so that

11989011990212057811990410038161003816100381610038161003816Φ1015840

(119883sdot)119904

10038161003816100381610038161003816

119902

le 119888119902

120572119872

119902 supminus120591le119897le0

119890119902120578119904

(int

119904+119897

0

(119904+119897minus 119903)120572minus1

119890120596(119904+119897minus119903)

|119884 (119903)| 119889119903)

119902

le 119888119902

120572119872

119902 supminus120591le119897le0

(int

119904+119897

0

(119904 + 119897 minus 119903)120572minus1

119890((120596+120578)119902

1015840

)(119904+119897minus119903)

times119890((120596+120578)119902)(119904minus119903)

119890120578119903

|119884 (119903)| 119889119903)

119902

le 119888119902

120572119872

119902 supminus120591le119897le0

(int

119904+119897

0

119890(120578+120596)(119904+119897minus119903)

(119904 + 119897 minus 119903)(120572minus1)119902

1015840

119889119903)

1199021199021015840

times int

119904+119897

0

119890(120578+120596)(119904minus119903)

119890119902120578119903

|119884 (119903)|119902

119889119903

le 119888119902

120572119872

119902

(int

119904

0

119890(120578+120596)119903

1199031199021015840

(120572minus1)

119889119903)

1199021199021015840

times int

119904

0

119890(120578+120596)(119904minus119903)

119890119902120578119903

|119884 (119903)|119902

119889119903

(21)

Applying the Young inequality for convolutions we have

int

infin

0

11989011990212057811990410038161003816100381610038161003816Φ1015840

(119883sdot)119904

10038161003816100381610038161003816

119902

119889119904 le 119888119902

120572119872

119902

(int

infin

0

119890(120578+120596)119904

1199041199021015840

(120572minus1)

119889119904)

1199021199021015840

times int

infin

0

119890(120578+120596)119904

119889119904int

infin

0

119890119902120578119904

|119884 (119904)|119902

119889119904

(22)

and we conclude that10038161003816100381610038161003816Φ1015840

(119883sdot)10038161003816100381610038161003816119871119902

P(Ω119871119902

120578(C))

le 119888120572119872|119884|

119871119902

P(Ω119871119902

120578(119867))

times (int

infin

0

119890(120578+120596)119904

1199041199021015840

(120572minus1)

119889119904)

11199021015840

times (int

infin

0

119890(120578+120596)119904

119889119904)

1119902

(23)

If we start again from (20) and apply theHolder inequality weobtain

10038161003816100381610038161003816119890120578(119904+119897)

Φ1015840

(119883sdot)119904(119897)10038161003816100381610038161003816le 119888

120572119872(int

119904+119897

0

119903(120572minus1)119902

1015840

119890(120596+120578)119903119902

1015840

119889119903)

11199021015840

times (int

119904+119897

0

119890120578119903119902

|119884 (119903)|119902

119889119903)

1119902

10038161003816100381610038161003816119890120578119904

Φ1015840

(119883sdot)119904

10038161003816100381610038161003816le 119888

120572119872(int

119904

0

119903(120572minus1)119902

1015840

119890(120596+120578)119903119902

1015840

119889119903)

11199021015840

times (int

119904

0

119890120578119903119902

|119884(119903)|119902

119889119903)

1119902

(24)


So we conclude that10038161003816100381610038161003816Φ1015840

(119883sdot)10038161003816100381610038161003816119871119902

P(Ω119862120578(C))

le 119888120572119872|119884|

119871119902

P(Ω119871119902

120578(119867))

times (int

infin

0

119903(120572minus1)119902

1015840

119890(120596+120578)119903119902

1015840

119889119903)

11199021015840

(25)

On the other hand by the Burkholder-Davis-Gundy inequal-ities for some constant 119888

119902depending only on 119902 we have

119864|119884 (119903)|119902

le 119888119902119864(int

119903

0

(119903 minus 120590)minus212057210038161003816100381610038161003816119890(119903minus120590)119860

119866 (120590119883120590)10038161003816100381610038161003816

2

1198712(Ξ119867)

119889120590)

1199022

le 119871119902

119888119902119864

times (int

119903

0

(119903 minus 120590)minus2120572minus2120574

1198902120596(119903minus120590)

(1 +1003816100381610038161003816119883120590

10038161003816100381610038162

119862) 119889120590)

1199022

(26)

which implies that

[119864|119884 (119903)|119902

]2119902

le 1198712

1198882119902

119902int

119903

0


times 1198902120596(119903minus120590)

[119864(1 +1003816100381610038161003816119883120590

1003816100381610038161003816119862)119902

]2119902

119889120590

(27)

so that

1198902120578119903

[119864|119884 (119903)|119902

]2119902

le 1198621int

119903

0


1198902(120596+120578)(119903minus120590)

1198902120578120590

119889120590

+ 1198622int

119903

0


1198902(120596+120578)(119903minus120590)

times 1198902120578120590

[1198641003816100381610038161003816119883120590

1003816100381610038161003816119902

119862]2119902

119889120590

(28)

for suitable constants 1198621 119862

2 Applying the Young inequality

for convolutions we obtain

int

infin

0

119890119902120578119903

119864|119884 (119903)|119902

119889119904le 1198621(int

infin

0

119904minus2120572minus2120574

1198902(120596+120578)119904

119889119904)

1199022

int

infin

0

119890119902120578119904

119889119904

+ 1198622(int

infin

0

119904minus2120572minus2120574

1198902(120596+120578)119904

119889119904)

1199022

times int

infin

0

119890119902120578119904

1198641003816100381610038161003816119883119904

1003816100381610038161003816119902

119862119889119904

(29)

This shows that |119884|119871119902

P(Ω119871119902

120578(119867))

is finite provided we assumethat 120578 lt 0 and 120596 + 120578 lt 0 so the map is well defined

If 1198831

sdot 119883

2

sdotare processes belonging to H119902

120578and 1198841 1198842 are

defined accordingly the entirely analogous passages showthat100381610038161003816100381610038161198841

minus 119884210038161003816100381610038161003816119871119902

P(Ω119871119902

120578(119867))

le 1198711198881119902

120572

100381610038161003816100381610038161198831

sdotminus 119883

2

sdot

10038161003816100381610038161003816119871119902

P(Ω119871119902

120578(C))

times (int

infin

0

119904minus2120572minus2120574

1198902(120596+120578)119904

119889119904)

12

(30)

Recalling the inequalities (23) and (25) and noting that themap Y rarr Φ

1015840

(Xsdot) is linear we obtain an explicit expression

for the constant 119888 in (17) and it is immediate to verify that119888 lt 1 provided 120578 lt 0 is chosen sufficiently large We fixsuch a value of 120578(119902) The first result is a consequence of thecontraction principle The estimate (15) also follows from thecontraction property ofΦ(sdot 119905 119909)

For investigating the dependence of the solution119883(119904 119905 119909)on the initial data 119909 and 119905 we reformulate (13) as an equationon [0infin) We set

119878 (119904) = 119890119904119860

for 119904 ge 0 119878 (119904) = 119868 for 119904 lt 0 (31)

and we consider the equation

119883(119904) = 119878 (119904 minus 119905) 119909 ((0 and (119904 minus 119905)) or (minus120591))

+ int

119904

0

119868[119905infin)

(120590) 119878 (119904 minus 120590) 119865 (120590119883120590) 119889120590

+ int

119904

0

119868[119905infin)

(120590) 119878 (119904 minus 120590)

times 119866 (120590119883120590) 119889119882 (120590) 119904 isin [0infin)

1198830(120579) = 119909 ((minus119905 + 120579) or (minus120591)) 120579 isin [minus120591 0]

(32)

Under the assumptions of Hypothesis 1 by Theorem 2 it iseasy to prove that equation (32) has a unique solution 119883 and119883sdotisin H

119902

120578(119902)for every 119902 isin [2infin) It clearly satisfies 119883(119904) =

119909((119904 minus 119905) or (minus120591)) for 119904 isin [minus120591 119905) and its restriction to the timeinternal [119905infin) is the unique mild solution of (10) From nowon we denote by119883(119904 119905 119909) 119904 isin [0infin) the solution of (32)

We need the following parameter-depending contractionprinciple which is stated in the following lemma and provedin [29 Theorems 101 and 102]

Lemma3 (ParameterDependingContraction Principle) Let119861119863 denote Banach spaces Let ℎ 119861times119863 rarr 119861 be a continuousmapping satisfying

1003816100381610038161003816ℎ (1199091 119910) minus ℎ (1199092 119910)1003816100381610038161003816 le 120572

10038161003816100381610038161199091 minus 11990921003816100381610038161003816 (33)

for some 120572 isin [0 1) and every 1199091 119909

2isin 119861 y isin 119863 Let 120601(119910)

denote the unique fixed point of the mapping ℎ(sdot 119910) 119861 rarr 119861Then 120601 119863 rarr 119861 is continuous

Theorem 4 Assume that Hypothesis 1 holds true Then forevery 119902 isin [1infin) the map (119905 119909) rarr 119883

sdot(119905 119909) is continuous

from [0infin) timesC toHq120578(q)

Proof Clearly it is enough to prove the claim for 119902 large Letus consider the map Φ defined in the proof of Theorem 2 In


our present notationΦ can be seen as a mapping fromH119902

120578times

[0infin) timesC toH119902

120578as follows

Φ(119883sdot 119905 119909)

119904(119897) = 119878 (119904 + 119897 minus 119905) 119909 (0)

+ int

119904+119897

0

119868[119905infin)

(120590) 119878 (119904 + 119897 minus 120590) 119865 (120590119883120590) 119889120590

+ int

119904+119897

0

119868[119905infin)

(120590) 119878 (119904 + 119897 minus 120590)

times 119866 (120590119883120590) 119889119882 (120590)


Φ(119883sdot 119905 119909)

119904(119897) = 119909 ((119904 + 119897 minus 119905) or (minus120591))

119904 isin [0infin) 119897 isin [minus120591 0] 119904 + 119897 le 119905

(34)

By the arguments of the proof of Theorem 2 Φ(sdot 119905 119909)is a contraction in H119902

120578uniformly with respect to 119905 119909

The process 119883sdot(119905 119909) is the unique fixed point of Φ(sdot 119905 119909)

So by the parameter-depending contraction principle(Lemma 3) it suffices to show that Φ is continuous fromH119902

120578times [0infin) times C to H119902

120578 From the contraction property

of Φ(sdot 119905 119909) mentioned earlier we have that Φ(sdot 119905 119909) iscontinuous uniformly in 119905 119909 Moreover for fixed 119883

sdot it is

easy to verify that Φ(119883sdot sdot sdot) is continuous from [0infin) times C

toH119902

120578 The proof is finished

Remark 5 By similar passages we can show that for fixed119905 Theorem 4 still holds true for 119902 large enough if the spaces[0infin) times C and H119902

120578are replaced by the spaces 119871119902(ΩCF

119905)

and H119902

120578(119905) respectively where 119871119902(ΩCF

119905) denotes that the

space of F119905-measurable function with value in C such that

the norm

|119909|119902

= 119864|119909|119902

119862 (35)

is finite

3 The Backward-Forward System

In this section we consider the system of stochastic differen-tial equations 119875-as

119883 (119904) = 119890(119904minus119905)119860

119909 (0) + int

119904

119905

119890(119904minus120590)119860

119865 (120590119883120590) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883120590) 119889119882 (120590) 119904 isin [119905infin)

119883119905= 119909 isin C

119884 (119904) minus 119884 (119879) + int

119879

119904

119885 (120590) 119889119882 (120590) + 120582int

119879

119904

119884 (120590) 119889120590

= int

119879

119904

120595 (120590119883120590 119884 (120590) 119885 (120590)) 119889120590 0 le 119904 le 119879 lt infin

(36)

for 119904 varying on the time interval [119905infin) sub [0infin) Asin Section 2 we extend the domain of the solution setting119883(119904 119905 119909) = 119909((119904 minus 119905) or (minus120591)) for 119904 isin [minus120591 119905)


Hypothesis 2 Themapping 120595 [0infin)timesCtimes119870times1198712(Ξ 119870) rarr

119870 is Borelmeasurable such that for all 119905 isin [0infin)120595(119905 sdot) Ctimes119870 times 119871

2(Ξ 119870) rarr 119870 is continuous and for some 119871

119910 119871

119911gt 0

120583 isin 119877 and119898 ge 11003816100381610038161003816120595 (119904 119909 1199101 1199111) minus 120595 (119904 119909 1199102 1199112)

1003816100381610038161003816

le 119871119910

10038161003816100381610038161199101 minus 11991021003816100381610038161003816 + 119871119911

10038161003816100381610038161199111 minus 11991121003816100381610038161003816

1003816100381610038161003816120595 (119904 119909 119910 119911)1003816100381610038161003816 le 119871 (1 + |119909|

119898

119862+10038161003816100381610038161199101003816100381610038161003816 + |119911|)

⟨120595 (119904 119909 1199101 119911) minus 120595 (119904 119909 119910

2 119911) 119910

1minus 119910

2⟩119870ge 120583

10038161003816100381610038161199101 minus 119910210038161003816100381610038162

(37)

for every 119904 isin [0infin) 119909 isin C 119910 1199101 119910

2isin 119870 119911 119911

1 and 119911

2isin

1198712(Ξ 119870)We note that the third inequality in (37) follows from the

first one taking 120583 = minus119871119910but that the third inequalitymay also

hold for different values of 120583Firstly we consider the backward stochastic differential

equation

119884 (119904) minus 119884 (119879) + int

119879

119904

119885 (120590) 119889119882 (120590) + 120582int

119879

119904

119884 (120590) 119889120590

= int

119879

119904

120595 (120590119883120590 119884 (120590) 119885 (120590)) 119889120590 0 le 119904 le 119879 lt infin

(38)

119870 is a Hilbert space the mapping 120595 [0infin) times C times 119870 times

1198712(Ξ 119870) rarr 119870 is a given measurable function 119883

sdotis a

predictable process with values in another Banach space Cand 120582 is a real number

Theorem 6 Assume that Hypothesis 2 holds Let 119901 gt 2 and120575 lt 0 be given and choose

119902 ge 119898119901 120578 gt120575

119898 (39)

Then the following hold

(i) For 119883sdotisin 119871

119902

P(Ω 119871

119902

120578(C)) and 120582 gt minus(120575 + 120583 minus (119871

2

1199112))

(38) has a unique solution in Kp120575that will be denoted

by (119884(119883sdot)(119904) 119885(119883

sdot)(119904)) 119904 ge 0

(ii) The estimate

119864sup119904ge0

(119884 (119883sdot) (119904))

119901

119890119901120575119904

+ 119864(int

infin

0

11989021205751205901003816100381610038161003816119884(119883sdot

)(120590)10038161003816100381610038162

119889120590)

1199012

+ 119864(int

infin

0

11989021205751205901003816100381610038161003816119885 (119883sdot

) (120590)10038161003816100381610038162

119889120590)

1199012

le 119888(1 +1003816100381610038161003816119883sdot

1003816100381610038161003816119898

119871119902

P(Ω119871119902

120578(C))

)119901

(40)


holds for a suitable constant 119888 In particular 119884(119883sdot) isin

119871119901

P(Ω 119862

120575(119870))

(iii) The map 119883sdotrarr (119884(119883

sdot) 119885(119883

sdot)) is continuous from

119871119902

P(Ω 119871

119902

120578(C)) toK119901

120575 and 119883

sdotrarr 119884(119883

sdot) is continuous

from 119871119902

P(Ω 119871

119902

120578(C)) to 119871119901

P(Ω 119862

120575(119870))

(iv) The statements of points (i) (ii) and (iii) still holdtrue if the space 119871119902

P(Ω 119871

119902

120578(C)) is replaced by the space

119871119902

P(Ω 119862

120578(C))

Proof The theorem is very similar to Proposition 311 in [24]The only minor difference is that the mapping 120595 [0infin) times

Ctimes119870times1198712(Ξ 119870) rarr 119870 is a givenmeasurable function while

in [24] the measurable function 120595 is from119867 times119870 times 1198712(Ξ 119870)

to 119870 however the same arguments apply

Theorem 7 Assume that Hypothesis 1 holds and thatHypothesis 2 holds true in the particular case 119870 = 119877 Thenfor every 119901 gt 2 119902 120575 lt 0 satisfying (39) with 120578 = 120578(119902)and for every 120582 gt 120582

1015840

= minus(120575 + 120583 minus (1198712

1199112)) there exists a

unique solution in H119902

120578(119902)times K

119901

120575of (36) that will be denoted

by (119883(sdot 119905 119909) 119884(sdot 119905 119909) 119885(sdot 119905 119909)) Moreover 119884(sdot 119905 119909) isin

119871119901

P(Ω 119862

120575(119877)) The map (119905 119909) rarr (119884(sdot 119905 119909) 119885(sdot 119905 119909)) is con-

tinuous from [0infin)timesC toK119901

120575 and themap (119905 119909) rarr 119884(sdot 119905 119909)

is continuous from [0infin) timesC to 119871119901P(Ω 119862

120575(119877))

Proof We first notice that the system is decoupled the firstdoes not contain the solution (119884 119885) of the second oneThere-fore under the assumption of Hypothesis 1 by Theorem 2there exists a unique solution 119883(sdot 119905 119909) and 119883

sdot(119905 119909) isin H

119902

120578(119902)

of the first equation Moreover from Theorem 4 it followsthat the map (119905 119909) rarr 119883

sdot(119905 119909) is continuous from [0infin)timesC

toH119902

120578(119902)

Let 119870 = 119877 from Theorem 6 we have that thereexists a unique solution (119884(sdot 119905 119909) 119885(sdot 119905 119909)) isin K

119901

120575of the

second equation and the map 119883sdotrarr (119884(119883

sdot) 119885(119883

sdot)) is

continuous from H119902

120578(119902)to K

119901

120575while X

sdotrarr (Y(X

sdot)) is

continuous fromH119902

120578(119902)to119871119901

P(Ω 119862

120575(119877))We have proved that

(119883(sdot 119905 119909) 119884(sdot 119905 119909) 119885(sdot 119905 119909)) isin H119902

120578(119902)times K

119901

120575is the unique

solution of (36) and the other assertions follow from com-position

Remark 8 From Remark 5 by similar passages we can showthat for fixed 119905 and for 119902 large enough under the assumptionsof Theorem 7 the map 119909 rarr (119884(sdot 119905 119909) 119885(sdot 119905 119909)) is continu-ous from 119871

119902

(ΩCF119905) toK119901

120575(119905)

We also remark that the process 119883(sdot 119905 119909) is F[119905infin)

measurable since C is separable Banach space we have that119883sdot(119905 119909) is F

[119905infin)measurable So that 119884(119905) is measurable

with respect to both F[119905infin)

and F119905 it follows that 119884(119905) is

deterministicFor later use we notice three useful identities for 119905 le 119904 lt

infin the equality 119875-as

119883119897(119904 119883

119904(119905 119909)) = 119883

119897(119905 119909) 119897 isin [119904infin) (41)

is a consequence of the uniqueness of the solution of (13)Since the solution of the backward equation is uniquely

determined on an interval [119904infin) by the values of the process119883sdoton the same interval for 119905 le 119904 lt infin we have 119875-as

119884 (119897 119904 119883119904(119905 119909)) = 119884 (119897 119905 119909) for 119897 isin [119904infin)

119885 (119897 119904 119883119904(119905 119909)) = 119885 (119897 119905 119909) for aa 119897 isin [119904infin)

(42)

Lemma 9 (see [30]) Let 119864 be a metric space with metric 119889and let 119891 Ω rarr 119864 be strongly measurable Then thereexists a sequence 119891

119899 119899 isin 119873 of simple 119864-valued functions

(ie 119891119899isFB(E)measurable and takes only a finite number

of values) such that for arbitrary 120596 isin Ω the sequence119889(119891

119899(120596) 119891(120596)) 119899 isin 119873 is monotonically decreasing to zero

Let now 119891 isin 119871119902

(ΩC) By Lemma 9 we get the existenceof a sequence of simple function 119891

119899 119899 isin 119873 such that

1003816100381610038161003816119891119899 (120596) minus 119891 (120596)1003816100381610038161003816 darr 0 for all 120596 isin Ω as 119899 rarr infin (43)

Hence 119891119899

rarr 119891 in | sdot |119871119902(ΩC) by Lebesguersquos dominated

convergence theoremWe are now in a position of showing the main result in

this section

Theorem 10 Assume that Hypothesis 1 holds true and thatHypothesis 2 holds in the particular case 119870 = 119877 Then thereexist two Borel measurable deterministic functions 120592 [tinfin) times

C rarr 119877 and 120577 [119905infin) times C rarr Ξlowast

= 119871(Ξ 119877) =

1198712(Ξ 119877) such that for 119905 isin [0infin) and x isin C the solution

(119883(119905 119909) 119884(119905 119909) 119885(119905 119909)) of (36) satisfies

119884 (119904 119905 119909) = 120592 (119904 119883119904(119905 119909)) 119885 (119904 119905 119909) = 120577 (119904 119883

119904(119905 119909))

119875-as for aa 119904 isin [119905infin)

(44)

Proof We apply the techniques introduced in [26 Proposi-tion 32] Let 119890

119894 be a basis of Ξlowast and let us define 119885119894119873

=

((119885 119890119894)Ξlowast and119873) or (minus119873) Then for every 0 le 119905

1lt 119905

2lt infin Δ gt

0 and 1199091 119909

2isin C we have that

100381610038161003816100381610038161003816100381610038161003816

119864 int

1199051+Δ

1199051

119885119894119873

(119904 1199051 119909

1) 119889119904 minus 119864int

1199052+Δ

1199052

119885119894119873

(119904 1199052 119909

2) 119889119904

100381610038161003816100381610038161003816100381610038161003816

le 119864int

1199052

1199051

10038161003816100381610038161003816119885119894119873

(119904 1199051 119909

1)10038161003816100381610038161003816119889119904

+ 119864int

1199051+Δ

1199052

10038161003816100381610038161003816119885119894119873

(119904 1199051 119909

1) minus 119885

119894119873

(119904 1199052 119909

2)10038161003816100381610038161003816119889119904

+ 119864int

1199052+Δ

1199051+Δ

10038161003816100381610038161003816119885119894119873

(119904 1199052 119909

2)10038161003816100381610038161003816119889119904


le 211987310038161003816100381610038161199052 minus 1199051

1003816100381610038161003816 + Δ12

119890minus120575(1199051+Δ)

times(119864(int

infin

0

119890212057511990410038161003816100381610038161003816119885119894119873

(119904 1199051 119909

1)minus119885

119894119873

(119904 1199052 119909

2)10038161003816100381610038161003816

2

119889119904)

1199012

)

1119901

le 211987310038161003816100381610038161199052 minus 1199051

1003816100381610038161003816 + Δ12

119890minus120575(1199051+Δ)

times (119864(int

infin

0

11989021205751199041003816100381610038161003816119885 (119904 1199051 1199091)minus119885 (119904 1199052 1199092)

10038161003816100381610038162

119889119904)

1199012

)

1119901

(45)

From Theorem 7 we have that the map (119905 119909) rarr int119905+Δ

119905

119885119894119873

(119904 119905 119909)119889119904 is continuous from [0infin) times C to 119877 ByRemark 8 we also have that for fixed 119905 the map 119909 rarr

119864int119905+Δ

119905

119864119885119894119873

(119904 119905 119909)119889119904 is continuous from 119871119902

(ΩCF119905) to 119877

for 119902 large enough Let us define

120577119894119873

(119905 119909) = lim inf119899rarrinfin

119899119864int

119905+(1119899)

119905

119885119894119873

(119904 119905 119909) 119889119904

119905 isin [0infin) 119909 isin C

(46)

It is clear that 120577119894119873 [0infin) timesC rarr 119877 is a Borel functionWe fix 119909 and 0 le 119905 le 119904 lt infin For 119897 isin [119904infin) we

denote 119864[119885119894119873

(119897 119904 119910)]|119910=119883119904(119905119909)

the random variable obtainedby composing119883

119904(119905 119909) with the map 119910 rarr 119864[119885

119894119873

(119897 119904 119910)]By Lemma 9 there exists a sequence of C-valued F

119904-

measurable simple functions

119891119898 Ω 997888rarr C 119891

119898=

119873119898

sum

119896=1

ℎ(119898)

119896119868119891119898=ℎ(119898)

119896 119873

119898isin 119873 (47)

where ℎ(119898)1 ℎ

(119898)

119898are pairwise distinct andΩ = ⋃

119873119898

119896=1119891

119898=

ℎ(119898)

119896 such that

1003816100381610038161003816119891119898 (120596) minus 119883119904(120596)

1003816100381610038161003816 darr 0 for all 120596 isin Ω as 119899 997888rarr infin (48)

For any 119861 isin F119904 we have

int119861

int

119904+(1119899)

119904

119885119894119873

(119897 119905 119909) 119889119897119889119875

= int119861

int

119904+(1119899)

119904

119885119894119873

(119897 119904 119883119904) 119889119897119889119875

= 119864119868119861int

119904+(1119899)

119904

119885119894119873

(119897 119904 119883119904) 119889119897

= lim119898rarrinfin

119864(119868119861int

119904+(1119899)

119904

119885119894119873

(119897 119904 119891119898) 119889119897)


119873119898

sum

119896=1

119864(119868119861119868119891119898=ℎ(119898)

119896int

119904+(1119899)

119904

119885119894119873

(119897 119904 ℎ(119898)

119896) 119889119897)


119864(119868119861

119873119898

sum

119896=1

119868119891119898=ℎ(119898)

119896)119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 ℎ(119898)

119896) 119889119897


119864119868119861(119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 119910) 119889119897|119910=119891119898

)


int119861

(119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 119910) 119889119897|119910=119891119898

)119889119875

= int119861

(119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 119910) 119889119897|119910=119883119904

)119889119875

(49)

and we get that

120577119894119873

(119904 119883119904(119905 119909)) = lim inf

119899rarrinfin

119899

times [119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 119910) 119889119897|119910=119883119904(119905119909)

]

= lim inf119899rarrinfin

119899119864[int

119904+(1119899)

119904

119885119894119873

(119897 119905 119909) 119889119897

100381610038161003816100381610038161003816100381610038161003816

F119904]

119875-as(50)

Fix 119905 and 119909 Recalling that |119885119894119873

| le 119873 by the Lebesguetheorem on differentiation it follows that 119875-as

lim119899rarrinfin

119899int

119904+(1119899)

119904

119885119894119873

(119897 119905 119909) 119889119897 = 119885119894119873

(119904 119905 119909)

for aa 119904 isin [119905infin)

(51)

By the boundedness of 119885119894119873 applying the dominated conver-gence theorem we get that

120577119894119873

(119904 119883119904(119905 119909)) = 119864 [119885

119894119873

(119904 119905 119909)10038161003816100381610038161003816F

119904] = 119885

119894119873

(119904 119905 119909)


(52)

Now we have proved that for every 119905 119909

120577119894119873

(119904 119883119904(119905 119909)) = 119885

119894119873

(119904 119905 119909)


(53)

for every 119894 119873 Let 119862 sub [0infin) times C denote the set ofpairs (119905 119909) such that lim

119873rarrinfin120577119894119873

(119905 119909) exists and the seriessuminfin

119894=1(lim

119873rarrinfin120577119894119873

(119905 119909))119890119894converges in Ξlowast We define

120577 (119905 119909) =

infin

sum

119894=1

( lim119873rarrinfin

120577119894119873

(119905 119909)) 119890119894 (119905 119909) isin 119862

120577 (119905 119909) = 0 (119905 119909) notin 119862

(54)


Since 119885 satisfies

119885 (120596 119904 119905 119909) =

infin

sum

119894=1


119885119894119873

(120596 119904 119905 119909)) 119890119894 (55)

for every 120596 119904 119905 119909 From (53) it follows that for every 119905 119909 wehave (119904 119883

119904(120596 119905 119909)) isin 119862 119875-as for almost all 119904 isin [119905infin) and

119885(119904 119905 119909) = 120577(119904 119883119904(119905 119909)) 119875-as for aa 119904 isin [119905infin)

We define 120592(119905 119909) = 119884(119905 119905 119909) since 119884(119905 119905 119909) is deter-ministic so the map (119905 119909) rarr 120592(119905 119909) can be written as acomposition 120592(119905 119909) = Γ

3(Γ2(119905 Γ

1(119905 119909))) with

Γ1 [0infin) timesC 997888rarr 119871

119901

P(Ω 119862

120575(119877))

Γ1(119905 119909) = 119884 (sdot 119905 119909)

Γ2 [0infin) times 119871

119901

P(Ω 119862

120575(119877)) 997888rarr 119871

119901

(Ω 119877)

Γ2(119905 119881) = 119881 (119905)

Γ3 119871

119901

(Ω 119877) 997888rarr 119877 Γ3120585 = 119864120585

(56)

FromTheorem 7 it follows that Γ1is continuous By

|119881(119905) minus 119880(119904)|119871119901(Ω119877)

le |119881 (119905) minus 119881 (119904)|119871119901(Ω119877)

+ 119890minus120575119901119904

|119881 minus 119880||119871

119901

P(Ω119862120575(119877))

(57)

we have that Γ2is continuous It is clear that Γ

3is continuous

Then themap (119905 119909) rarr 120592(119905 119909) is continuous from [0infin)timesCto 119877 therefore 120592(119905 119909)is a Borel measurable function Fromuniqueness of the solution of (36) it follows that 119884(119904 119905 119909) =120592(119904 119883

119904(119905 119909)) 119875-as for aa 119904 isin [119905infin)

4 The Fundamental Relation

Let (ΩF 119875) be a given complete probability space with afiltration F

119905119905ge0

satisfying the usual conditions 119882(119905) 119905 ge 0

is a cylindrical Wiener process in Ξ with respect to F119905119905ge0

We will say that an F

119905ge0-predictable process 119906 with values

in a given measurable space (119880U) is an admissible controlThe function 119877 [0infin) times C times 119880 rarr Ξ is measurableand bounded We consider the following controlled stateequation

119889119883119906

(119904) = 119860119883119906

(119904) 119889119904 + 119865 (119904 119883119906

119904) 119889119904

+ 119866 (119904 119883119906

119904) 119877 (119904 119883

119906

119904 119906 (119904)) 119889119904 + 119866 (119904 119883

119906

119904) 119889119882 (119904)

119904 isin [119905infin)

119883119906

119905= 119909

(58)

Here we assume that there exists a mild solution of (58)which will be denoted by 119883119906

(119904 119905 119909) or simply by 119883119906

(119904) Weconsider a cost function of the form

119869 (119906) = 119864int

infin

119905

119890minus120582119904

119892 (119904 119883119906

119904 119906 (119904)) 119889119904 (59)

Here 119892 is function on [0infin) times C times 119880 with real values Ourpurpose is to minimize the function 119869 over all admissiblecontrols

We define in a classical way the Hamiltonian functionrelative to the previous problem for all 119905 isin [0infin) 119909 isin

C and 119911 isin Ξlowast

120595 (119905 119909 119911) = inf 119892 (119905 119909 119906) + 119911119877 (119905 119909 119906) 119906 isin 119880

(60)

and the corresponding possibly empty set of minimizers

Γ (119905 119909 119911) = 119906 isin 119880 119892 (119905 119909 119906) + 119911119877 (119905 119909 119906) = 120595 (119905 119909 119911)

(61)

We are now ready to formulate the assumptions we need

Hypothesis 3 (i) 119860 119865 and G verify Hypothesis 1(ii) (119880U) is ameasurable spaceThemap 119892 [0infin)timesCtimes

119880 rarr 119877 is continuous and satisfies |119892(119905 119909 119906)| le 119870119892(1+|119909|

119898119892

119862)

for suitable constants 119870119892gt 0 119898

119892gt 0 and all 119909 isin C119906 isin

119880 The map 119877 [0infin) times C times 119880 rarr Ξ is measurable and|119877(119905 119904 119906)| le 119871

119877for a suitable constant 119870

119877gt 0 and all 119909 isin

C119906 isin 119880 and119911 isin Ξlowast

(iii)TheHamiltonian120595 defined in (60) satisfies the requi-rements of Hypothesis 2 (with119870 = 119877)

(iv) We fix here 119901 gt 2 q and 120575 lt 0 satisfying (39) with120578 = 120578(119902) and such that 119902 gt 119898

119892

We are in a position to prove the main result of thissection

Theorem11 Assume thatHypothesis 3 holds and suppose that120582 verifies

120582 gt (minus120575 minus 120583 +1198712

119911

2) or (minus120575 +

1198712

119877

2 (119901 minus 1))

or (1198712

119877119898119892

2 (119902 minus 119898119892)minus 120578 (119902)119898

119892)

(62)

Let 120592 120577 denote the function in the statement of Theorem 10Then for every admissible control 119906 and for the correspondingtrajectory119883 starting at (119905 119909) one has

119869 (119906) = 120592 (119905 119909) + 119864int

infin

119905

119890minus120582119904

[minus120595 (119904 119883119906

119904 120577 (119904 119883

119906

119904)) + 120577 (119904 119883

119906

119904)

times 119877 (119904 119883119906

119904 119906 (119904)) + 119892 (119904 119883

119906

119904 119906 (119904))] 119889119904

(63)

Proof Consider (58) in the probability space (ΩF 119875) withfiltration F

119905119905ge0

and with an F119905119905ge0

-cylindrical Wienerprocess 119882(119905) 119905 ge 0 Let us define

119882119906

(119904) = 119882 (119904) + int

119904

119905and119904

119877 (120590119883119906

120590 119906 (120590)) 119889120590 119904 isin [0infin)

120588 (119879) = exp(int119879

119905

minus119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882 (119904)

minus1

2int

119879

119905

1003816100381610038161003816119877 (119904 119883119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)

(64)


Let 119875119906 be the unique probability onF[0infin)

such that

119875119906

|F119879

= 120588 (119879) 119875|F119879

(65)

We notice that under 119875119906 the process119882119906 is aWiener processLet us denote by F119906

119905119905ge0

the filtration generated by119882119906 andcompleted in the usual way Relatively to 119882119906 (58) can berewritten as

119889119883119906

(119904) = 119860119883119906

(119904) 119889119904 + 119865 (119904 119883119906

119904) 119889119904

+ 119866 (119904 119883119906

119904) 119889119882

119906

(119904) 119904 isin [119905infin)

119883119906

119905= 119909

(66)

In the space (ΩF[0infin)

F119906

119905119905ge0 119875

119906

) we consider the follow-ing system of forward-backward equations

119883119906

(119904) = 119890(119904minus119905)119860

119909 (0) + int

119904

119905

119890(119904minus120590)119860

119865 (120590119883119906

120590) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883119906

120590) 119889119882

119906

(120590) 119904 isin [119905infin)

119883119906

119905= 119909 isin C

119884119906

(119904) minus 119884119906

(119879) + int

119879

119904

119885119906

(120590) 119889119882119906

(120590) + 120582int

119879

119904

119884119906

(120590) 119889120590

= int

119879

119904

120595 (120590119883119906

120590 119885

119906

(120590)) 119889120590 0 le 119904 le 119879 lt infin

(67)

Applying the Ito formula to 119890minus120582119904119884119906(119904) and writing the back-ward equation in (67) with respect to the process119882 we get

119884119906

(119904) + int

119879

119904

119890minus120582120590

119885119906

(120590) 119889119882 (120590)

= int

119879

119904

119890minus120582120590

[120595 (120590119883119906

120590 119885

119906

(120590))

minus119885119906

(120590) 119877 (120590119883119906

120590 119906 (120590))] 119889120590

+ 119890minus120582119879

119884119906

(119879)

(68)

Recalling that 119877 is bounded we get for all 119903 ge 1 and someconstant 119862

119864119906

[120588(119879)minus119903

] = 119864119906

[exp 119903 (int119879

119905

119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882

119906

(119904)

minus1

2int

119879

119905

1003816100381610038161003816119877 (119904 119883119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)]

= 119864119906

[exp(int119879

119905

119903119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882

119906

(119904)

minus1

2int

119879

119905

11990321003816100381610038161003816119877 (119904 119883

119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)

times exp 119903 (119903 minus 1)2

int

119879

119905

1003816100381610038161003816119877 (119904 119883119906

119904 119906 (119904))

10038161003816100381610038162

119889119904]

le 119890(12)119903(119903minus1)119879119871

2

119877119864119906

times exp(int119879

119905

2119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882

119906

(119904)

minus1

2int

119879

119905

41003816100381610038161003816119877 (119904 119883

119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)

= 119890(12)119903(119903minus1)119879119871

2

119877

(69)

It follows that

119864(int

119879

119905

|119890minus120582119904

119885119906

(119904)|2

119889119904)

12

= 119864119906

[(int

119879

119905

10038161003816100381610038161003816119890minus120582119904

119885119906

(119904)10038161003816100381610038161003816

2

119889119904)

12

120588minus1

]

le (119864119906

int

119879

119905

10038161003816100381610038161003816119890minus120582119904

119885119906

(119904)10038161003816100381610038161003816

2

119889119904)

12

times (119864119906

120588minus2

)12

lt infin

(70)

We conclude that the stochastic integral in (68) has zeroexpectation If we set 119904 = 119905 in (68) and we take expectationwith respect to 119875 we obtain

119890minus120582119879

119864119884119906

(119879) minus 119884119906

(119905)

= 119864int

119879

119905

119890minus120582120590

[minus120595 (120590119883119906

120590 119885

119906

(120590))

+119885119906

(120590) 119877 (120590119883119906

120590 119906 (120590))] 119889120590

(71)

ByTheorem 7 119884119906(sdot 119905 119909) isin 119871119901P(Ω 119862

120575(119877)) so that

119864119906

|119884(119879 119905 119909)|119901

le 119862 exp (minus119901120575119879) (72)

By the Holder inequality we have that for suitable constant119862 gt 0

119864 |119884 (119879 119905 119909)| = 119864119906

(120588minus1

(119879) |119884 (119879 119905 119909)|)

le 119864(120588minus119901(119901minus1)

)(119901minus1)119901

119864(|119884 (119879 119905 119909)|119901

)1119901

le 119862119890((1198712

1198772(119901minus1))minus120575))119879

(73)

From Theorem 2 we obtain 119864119906sup119904ge119905119890120578119902119904

|119883119906

119904|119902

lt infin by thesimilar process we get that

1198641003816100381610038161003816119883

119906

119879

1003816100381610038161003816119898119892

le 119862119890(1198712

119877119898119892(2119902minus2119898

119892)minus1

minus120578(119902)119898119892)119879

(74)


for suitable constant 119862 gt 0 and

119864int

infin

119905

119890minus120582120590 1003816100381610038161003816119892 (120590119883

119906

120590 119906 (120590))

1003816100381610038161003816 119889120590 lt infin (75)

Since 119884119906(119905 119905 119909) = 120592(119905 119909) and 119885119906

(119904 119905 119909) = 120577(119904 119883119906

119904(119905 119909)) 119875-

as for aa 119904 isin [119905infin) we have that

119890minus120582119879

119864119884119906

(119879) minus 119907 (119905 119909)

= 119864int

119879

119905

119890minus120582120590

[minus120595 (120590119883119906

120590 120577 (120590 119883

119906

120590))

+120577 (120590119883119906

120590) 119877 (120590119883

119906

120590 119906 (120590))] 119889120590

(76)

Thus adding and subtracting119864intinfin119905

119890minus120582120590

119892(120590119883119906

120590 119906(120590))119889120590 and

letting 119879 rarr infin we conclude that

119869 (119906) = 120592 (119905 119909)

+ 119864int

infin

119905

119890minus120582119904

[minus120595 (119904 119883119906

119904 120577 (119904 119883

119906

119904)) + 120577 (119904 119883

119906

119904) 119877

times (119904 119883119906

119904 119906 (119904)) + 119892 (119904 119883

119906

119904 119906 (119904))] 119889119904

(77)

The proof is finished

We immediately deduce the following consequences

Theorem 12 Let 119905 isin [0infin) and 119909 isin C be fixed assumethat the set-valued map Γ has nonempty values and it admitsa measurable selection Γ

0 [0infin)timesCtimesΞlowast rarr 119880 and assume

that a control 119906(sdot) satisfies

119906 (119904) = Γ0(119904 119883

119906

119904 120577 (119904 119883

119906

119904))

119875-as for almost every 119904 isin [119905infin)

(78)

Then 119869(119905 119909 119906) = 120592(119905 119909) and the pair (119906(sdot) 119883) is optimal forthe control problem starting from 119909 at time 119905

Such a control can be shown to exist if there exists a solutionfor the so-called closed-loop equation as follows

119889119883 (119904)=119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904+119866 (119904 119883

119904)

times(119877 (119904 119883119904 Γ

0(119904 119883

119904 120577 (119904 119883

119904))) 119889119904 + 119889119882 (119904))

119904 isin [119905infin)

119883119905= 119909

(79)

since in this case we can define an optimal control setting

119906 (119904) = Γ0(119904 119883

119904 120577 (119904 119883

119904)) (80)

However under the present assumptions we cannot guaranteethat the closed-loop equation has a solution in the mildsense To circumvent this difficulty we will revert to a weakformulation of the optimal control problem

5 Existence of Optimal Control

We formulate the optimal control problem in the weak sensefollowing the approach of [31]Themain advantage is that wewill be able to solve the closed-loop equation in a weak senseand hence to find an optimal control even if the feedbacklaw is nonsmooth

We call (ΩF F119905119905ge0 119875119882) an admissible setup if

(ΩF F119905119905ge0 119875) is a filtered probability space satisfying the

usual conditions and 119882 is a cylindrical 119875-Wiener processwith values in Ξ with respect to the filtration F

119905119905ge0

By an admissible control system we mean (ΩF

F119905119905ge0 119875119882 119906119883

119906

) where (ΩF F119905119905ge0 119875 119882) is an

admissible setup 119906 is an F119905-predictable process with values

in119880 and119883119906 is a mild solution of (58) An admissible controlsystem will be briefly denoted by (119882 119906119883119906

) in the followingOur purpose is to minimize the cost functional

119869 (119906) = 119864int

infin

119905

119890minus120582119904

119892 (119904 119883119906

119904 119906 (119904)) 119889119904 (81)

over all the admissible control systemOur main result in this section is based on the solvability

of the closed-loop equation

119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904 + 119866 (119904 119883

119904)

times (119877 (119904 119883119904 Γ

0(119904 119883

119904 120577 (119904 119883

119904))) 119889119904 +119889119882 (119904))

119904 isin [119905infin)

119883119905= 119909

(82)

In the following sense we say that 119883 is a weak solution of(82) if there exists an admissible setup (ΩF F

119905119905ge0 119875119882)

and anF119905-adapted continuous process119883(119905)with values in119867

which solves the equation in the mild sense namely 119875-as

119883 (119904) = 119890(119904minus119905)119860

119909 (0) + int

119904

119905

119890(119904minus120590)119860

119865 (120590119883120590) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883120590) 119877

times (120590119883120590 Γ

0(120590119883

120590 120577 (120590 119883

120590))) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883120590) 119889119882

120590 119904 isin [119905infin)

(83)

119883119905= 119909 (84)

Theorem 13 Assume that Hypothesis 3 holds Then thereexists a weak solution of the closed-loop equation (82) whichis unique in law


Proof (uniqueness) Let 119883 be a weak solution of (82) in anadmissible setup (ΩF F

119905119905ge0 119875119882)We define

120588 (119879) = exp(int119879

119905

minus119877lowast

(120590 119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590))) 119889119882 (120590)

minus1

2int

119879

119905

1003816100381610038161003816119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590)))10038161003816100381610038162

119889120590)

(85)

Since 119877 is bounded the Girsanov theorem ensures that thereexists a probability measure 1198750 such that the process

1198820

(119904) = 119882 (119904) + int

119904

119905and119904

119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590))) 119889120590

119904 isin [0infin)

(86)

is a 1198750-Wiener process and

1198750

|F119879

= 120588 (119879) 119875|F119879

(87)

Let us denote by F0

119905119905ge0

the filtration generated by1198820 andcompleted in the usual way In (ΩF

[0infin) F0

119905119905ge0 119875

0

) 119883 isa mild solution of

119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119905 119883119904) 119889119904

+ 119866 (119904 119883119904) 119889119882

0

(119904) 119904 isin [119905infin)

119883119905= 119909

120588 (119879) = exp(int119879

119905

minus119877lowast

(120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590)) 119889119882

0

(120590)

+ 12int

119879

119905

1003816100381610038161003816119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590)))10038161003816100381610038162

119889120590)

(88)

By Hypothesis 3 the joint law of 119883 and 1198820 is uniquely

determined by 119860 119865 119866 and 119909 Taking into account the lastdisplayed formula we conclude that the joint law of 119883 and120588(119879) under1198750 is also uniquely determined and consequentlyso is the law of 119883 under 119875 This completes the proof of theuniqueness part

Proof (existence) Let (ΩF 119875) be a given complete probabil-ity space 119882(119905) 119905 ge 0 is a cylindrical Wiener process on(ΩF 119875)with values inΞ and F

119905119905ge0

is the natural filtrationof 119882(119905) 119905 ge 0 augmented with the family of119875-null sets Let119883(sdot) be the mild solution of

119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904

+ 119866 (119904 119883119904) 119889119882 (119904) 119904 isin [119905infin)

119883119905= 119909

(89)

and by the Girsanov theorem let 1198751 be the probability on Ωunder which

1198821

(119904) = 119882 (119904) minus int

119904

119905and119904

119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590))) 119889120590

(90)

is aWiener process (notice that 119877 is bounded)Then119883 is theweak solution of (82) relatively to the probability 1198751 and theWiener process1198821

Now we can state the main result of this section

Corollary 14 Assume that Hypothesis 3 holds true and 120582

verifies (62) Also assume that the set-valued map Γ hasnonempty values and it admits a measurable selection Γ

0

[0infin) timesC times Ξlowast

rarr 119880 Then for every 119905 isin [0infin) and x isin Cand for all admissible control system (119882 119906119883

119906

) one has

119869 (119906 119905 119909) ge 120592 (119905 119909) (91)

and the equality holds if

119906 (119904) = Γ0(119904 119883

119906

119904 120577 (119904 119883

119906

119904))

119875 minus 119886119904 119891119900119903 119886119897119898119900119904119905 119890119907119890119903119910 119904 isin [119905infin)

(92)

Moreover from Theorem 13 it follows that the closed-loop equation (82) admits a weak solution (ΩFF

119905119905ge0 119875119882119883) which is unique in law and setting

119906 (119904) = Γ0(119904 119883

119904 120577 (119904 119883

119904)) (93)

we obtain an optimal admissible control system (119882 119906119883)

6 Applications

In this section we present a simple application of the previousresults We consider the stochastic delay partial differentialequation in the bounded domain 119861 sub 119877

119899 with smoothboundary 120597119861 as follows

119889119911119906

(119905 120585) = Δ119911119906

(119905 120585) 119889119905 + 119891 (119905 119911119906

119905(120585)) 119889119905

+

119889

sum

119894=1

119892119894(119905 119911

119906

119905(120585)) [119903

119894

(120585) 119906119894

(119905) 119889119905 + 119889119882119894

(119905)]

119911119906

0(120579 120585) = 119909 (120579 120585) 120585 isin 119861 120579 isin [minus1 0]

119911119906

(119905 120585) = 0 119905 isin [0infin) 120585 isin 120597119861

(94)

Here119882 = (1198821

1198822

119882119889

) is a standardWiener process in119877119889 and the functions 119891 [0 +infin) times 119862([minus1 0] 119877) rarr 119877 and119892119894 [0 +infin) times 119862([minus1 0] 119877) rarr 119877 are Lipschitz continuous

and bounded Setting 119880 as a bounded subset of 119877119889 Ξ = 119877119889

119867 = 1198712

(119861) and 119909 isin 119862([minus1 0]119867) We define 119865 and 119866 asfollowing

119865 (119905 119909) (120585) = 119891 (119905 119909 (120585))

(119866 (119905 119909) 119911) (120585) =

119889

sum

119894=1

119892119894(119905 119909 (120585)) 119911

119894

(120585)

120585 isin 119861 119909 isin 119862 ([minus1 0] 119867) 119911 isin 119871 (Ξ119867)

(95)

and let 119860 denote the Laplace operator Δ in 1198712

(119861) withdomain11988222

(119861)⋂11988212

0(119861) then (94) has the form (58) and

Hypothesis 1 holds


Let us consider the optimal control problem associatedwith the cost

119869 (119906) = 119864int

infin

0

119890minus120582119905

[int119861

120590 (120585 119911119906

119905(120585)) 119889120585 + 119906

2

(119905)] 119889119905 (96)

where 120582 verifies (62) and 120590 119862([minus1 0] 119877) times 119880 rarr [0infin) isa bounded measurable function Define 119892 119862([minus1 0]119867) times

119880 rarr [0infin) and 119877 119862([minus1 0]119867) times 119880 rarr Ξ by119892(119910 119906) = int

119861

120590(119905 119910(120585) 119906)119889120585 + 1199062 and 119877(119910 119906) = (int

119861

1199031

(120585)1199061

119889120585

int119861

1199032

(120585)1199062

119889120585 int119861

119903119889

(120585)119906119889

119889120585) for 119910 isin 119862([minus1 0]119867) 119906 =

(1199061

1199062

119906119889

) isin 119880 respectively It can be easily verifiedthat Hypothesis 3 holds true and the set-valued map Γ hasnonempty values and it admits a measurable selection Γ

0

[0infin) times C times Ξlowast

rarr 119880 Then the closed-loop equation(82) admits a weak solution (ΩF F

119905119905ge0 119875119882 119906 119911

sdot(sdot)) and

setting

119906 (119904) = Γ0(119904 119911

119904(sdot) 120577 (119904 119911

119904(sdot))) (97)

we obtain an optimal admissible control system (119882 119906 119911(sdot))

References

[1] J Bismut ldquoOn optimal control of linear stochastic equationswith a linear-quadratic criterionrdquo SIAM Journal on Control andOptimization vol 15 no 3 pp 1ndash4 1977

[2] N Nagase ldquoOn the existence of optimal control for controlledstochastic partial differential equationsrdquo Nagoya MathematicsJournal vol 115 pp 73ndash85 1989

[3] N El Karoui D Huu Nguyen and M Jeanblanc-Pique ldquoCom-pactification methods in the control of degenerate diffusionsrdquoStochastics vol 20 pp 169ndash219 1987

[4] M Nisio ldquoOptimal control for stochastic partial differentialequations and viscosity solutions of Bellman equationsrdquoNagoyaMathematics Journal vol 123 pp 13ndash37 1991

[5] M Nisio ldquoOn sensitive control for stochastic partial differentialequationsrdquo in Stochastic Analysis on Infinite Dimensional SpacesProceedings of the US Japan Bilateral Seminar H Kunita et alEd vol 310 of Pitman Research Notes Mathematical Series pp231ndash241 Longman Scientific and Technical Baton Rouge LaUSA January 1994

[6] R Buckdahn and A Rascanu ldquoOn the existence of stochasticoptimal control of distributed state systemrdquoNonlinear AnalysisTheory Methods and Applications vol 52 no 4 pp 1153ndash11842003

[7] V Barbu and G Da Prato Equations in Hilbert Spaces vol 86 ofPitman Research Notes in Mathematics Pitman 1983

[8] P Cannarsa and G Da Prato ldquoSecond-order Hamilton-Jacobiequations in infinite dimensionsrdquo SIAM Journal on Control andOptimization vol 29 no 2 pp 474ndash492 1991

[9] P Cannarsa and G Da Prato ldquoDirect solution of a second-orderHamilton-Jacobi equations in Hilbert spacesrdquo in StochasticPartial Differential Equations and Applications G Da Pratoand L Tubaro Eds vol 268 of Pitman Research Notes inMathematics Pitman 1992

[10] F Gozzi ldquoRegularity of solutions of second order Hamilton-Jacobi equations and application to a control problemrdquoCommu-nications in Partial Differential Equations vol 20 pp 775ndash8261995

[11] F Gozzi ldquoGlobal regular solutions of second order Hamilton-Jacobi equations in Hilbert spaces with locally Lipschitz non-linearitiesrdquo Journal of Mathematical Analysis and Applicationsvol 198 no 2 pp 399ndash443 1996

[12] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems and Control Lettersvol 14 no 1 pp 55ndash61 1990

[13] N El Karoui and LMazliak Eds Backward Stochastic Differen-tial Equations vol 364 of Pitman ResearchNotes inMathematicsSeries Longman 1997

[14] E Pardoux and BSDEs ldquoweak convergence and homogeneiza-tion of semilinear PDEsrdquo in Non- Linear Analysis DifferentialEquations and Control F H Clarke and R J Stern Eds pp503ndash549 Kluwer Dordrecht The Netherlands 1999

[15] S Peng ldquoA generalized dynamic programming principle andHamilton-Jacobi-Bellman equationrdquo Stochastics and StochasticsReports vol 38 pp 119ndash134 1992

[16] N E Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquo Mathematical Finance vol 7no 1 pp 1ndash71 1997

[17] SHamad120583ene and J P Lepeltier ldquoBackward equations stochas-tic control and zero-sum stochastic differential gamesrdquo Stochas-tics and Stochastics Reports vol 54 pp 221ndash231 1995

[18] N El-Karoui and S Hamadene ldquoBSDEs and risk-sensitive con-trol zero-sum and nonzero-sum game problems of stochasticfunctional differential equationsrdquo Stochastic Processes and theirApplications vol 107 no 1 pp 145ndash169 2003

[19] M Fuhrman and G Tessiture ldquoExistence of optimal stochasticcontrols and global solutions of forward-backward stochasticdifferential equationsrdquo SIAM Journal on Control and Optimiza-tion vol 43 no 3 pp 813ndash830 2005

[20] M Fuhrman Y Hu and G Tessitore ldquoOn a class of stochasticoptimal control problems related to bsdes with quadraticgrowthrdquo SIAM Journal on Control and Optimization vol 45 no4 pp 1279ndash1296 2006

[21] M Fuhrman and G Tessitore ldquoNonlinear kolmogorov equa-tions in infinite dimensional spaces the backward stochasticdifferential equations approach and applications to optimalcontrolrdquoAnnals of Probability vol 30 no 3 pp 1397ndash1465 2002

[22] F Masiero ldquoSemilinear kolmogorov equations and applicationsto stochastic optimal controlrdquo Applied Mathematics and Opti-mization vol 51 no 1 pp 201ndash250 2005

[23] M Fuhrman FMasiero andG Tessitore ldquoStochastic equationswith delay optimal control via BSDEs and regular solutions ofHamilton-jacobi-bellman equationsrdquo SIAM Journal on Controland Optimization vol 48 no 7 pp 4624ndash4651 2010

[24] M Fuhrman and G Tessiture ldquoInfinite horizon backwardstochastic differential equations and elliptic equations in hilbertspacesrdquo Annals of Probability vol 32 no 1 pp 607ndash660 2004

[25] F Masiero ldquoInfinite horizon stochastic optimal control prob-lems with degenerate noise and elliptic equations in Hilbertspacesrdquo Applied Mathematics and Optimization vol 55 no 3pp 285ndash326 2007

[26] M Fuhrman ldquoA class of stochastic optimal control problemsin Hilbert spaces BSDEs and optimal control laws stateconstraints conditioned processesrdquo Stochastic Processes andtheir Applications vol 108 no 2 pp 263ndash298 2003

[27] F Masiero ldquoStochastic optimal control problems and parabolicequations in banach spacesrdquo SIAM Journal on Control andOptimization vol 47 no 1 pp 251ndash300 2008

[28] G Da Prato and J Zabczyk Ergodicity For Infinite-DimensionalSystems Cambridge University Press 1996


[29] J Zabczyk ldquoParabolic equations on Hilbert spacesrdquo in StochaS-tic PDErsquoS and Kolmogorov Equations in Infinite Dimensionsvol 1715 of Lecture Notes in Math pp 117ndash213 Springer BerlinGermany 1999

[30] G Da Prato and J Zabczyk Stochstic Equations in InfiniteDimensions Cambridge University Press 1992

[31] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 of Applications of MathematicsSpringer New York NY USA 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014


Stochastic AnalysisInternational Journal of


the corresponding regularity properties of the transitionsemigroup of the associated Ornstein-Uhlenbeck process

The main tools for the control problem are techniquesfrom the theory of backward stochastic differential equations(BSDEs) in the sense of Pardoux and Peng first consideredin the nonlinear case in [12] see [13 14] as general referencesBSDEs have been successfully applied to control problemssee for example [15 16] and we also refer the reader to[17ndash20] Fuhrman and Tessiture [19] considered the optimalcontrol problem for stochastic differential equation in thestrong form assuming Lipschitz conditions and allowingdegeneracy of the diffusion coefficient under some structuralconstraint on the state equation Existence of an optimalcontrol for stochastic systems in infinite dimensional spacesalso has been obtained in [21ndash27] In [21] Fuhrman and Tes-sitore showed the regularity with respect to parameters andthe regularity in the Malliavin spaces for the solution of thebackward-forward system and defined the feedback law byMalliavin calculus Finally the optimal control is obtained bythe feedback Appealing to the Malliavin calculus comparedwith Fuhrman et al [23] the existence of optimal controlfor stochastic differential equations with delay is proved bythe feedback law Fuhrman and Tessiture [24] dealt with aninfinite horizon optimal control problem for the stochas-tic evolution equation in Hilbert space and the optimalcontrol is showed by means of infinite horizon backwardstochastic differential equation in infinite dimensional spacesand Malliavin calculus In Masiero [25] the infinite horizonoptimal control problem for stochastic evolution equationis also studied by means of the Hamilton-Jacobi-Bellmanequation In Fuhrman [26] a class of optimal control prob-lems governed by stochastic evolution equations in Hilbertspaces which includes state constraints is considered andthe optimal control is obtained by the Fleming logarithmictransformation Masiero [27] studied stochastic evolutionequations evolving in a Banach space where 119866 is a constantand characterized the optimal control via a feedback law byavoiding use of Malliavin calculus Since there is a lack ofregularity of 119865 and 119866 Malliavin calculus is not available inthis case the method in [27] also cannot be used as 119866 isnot a constant but we can prove a theorem similar to [26Proposition 32] which will be used to define the feedbacklaw

In the present paper we study the infinite horizon optimalcontrol problem for stochastic delay evolution equations inHilbert spaces and by usingTheorem 10 the optimal controlis obtained Since we do not relate the optimal feedback lawwith the gradient of the value function and do not considerthe associated Hamilton-Jacobi-Bellman equation we candrop the Gateaux differentiability of the drift term and thediffusion term

The plan of the paper is as follows In the next sectionsome notations are fixed and the stochastic delay evolutionequations are considered with an infinite horizon in particu-lar continuous dependence on initial value (119905 119909) is proved InSection 3 we give the proof of Theorem 10 which is the keyof many subsequent results The addressed optimal controlproblem is considered and the fundamental relation betweenthe optimal control problem and BSDEs is established in

Section 4 Section 5 is devoted to proving the existence anduniqueness of optimal control in the weak sense Finally anapplication is given in Section 6

2 Preliminaries

We list some notations that are used in this paper We usethe symbol | sdot | to denote the norm in a Banach space 119865with a subscript if necessary Let Ξ 119867 and 119870 denote realseparable Hilbert spaces with scalar products (sdot sdot)

Ξ (sdot sdot)

119867

and (sdot sdot)119870 respectively For fixed 120591 gt 0 C = 119862([minus120591 0]119867)

denotes the space of continuous functions from [minus120591 0] to119867endowed with the usual norm |119891|

119862= sup

120579isin[minus1205910]|119891(120579)|

119867 Let

Ξlowast denote the dual space of Ξ with scalar product (sdot sdot)

Ξlowast and

let 119871(Ξ119867) denote the space of all bounded linear operatorsfrom Ξ into 119867 the subspace of Hilbert-Schmidt operatorswith the Hilbert-Schmidt norm is denoted by 119871

2(Ξ119867)

Let (ΩF 119875) be a complete space with a filtration F119905119905ge0

which satisfies the usual condition By a cylindrical Wienerprocess with values in aHilbert spaceΞ defined on (ΩF 119875)we mean a family 119882(119905) 119905 ge 0 of linear mappings Ξ rarr

1198712

(Ω) such that for every 120585 120578 isin Ξ 119882(119905)120585 119905 ge 0 is areal Wiener process and 119864(119882(119905)120585 sdot 119882(119905)120578) = (120585 120578)

Ξ In

the following 119882(119905) 119905 ge 0 is a cylindrical Wiener processadapted to the filtration F

119905119905ge0

In this section and the next section F

119905119905ge0

will denotethe natural filtration of 119882 augmented with the family Nof 119875-null of F The filtration F

119905 119905 ge 0 satisfies the usual

conditions For [119886 119887] [119886infin) sub [0infin) we also use thefollowing notations

F[119886119887]

= 120590 (119882 (119904) minus 119882 (119886) 119904 isin [119886 119887]) orN

F[119886infin)

= 120590 (119882 (119904) minus 119882 (119886) 119904 isin [ainfin)) orN

(4)

By P we denote the predictable 120590-algebra and by B(Λ) wedenote the Borel 120590-algebra of any topological space Λ

Similar to [24] we define several classes of stochasticprocesses with values in a Banach space 119865 as follows

(i) 1198712P(Ω times [119905infin) 119865) denotes the space of equivalenceclasses of processes 119884 isin 119871

2

(Ω times [119905infin) 119865) admittinga predictable version 1198712P(Ω times [119905infin) 119865) is endowedwith the norm

|119884|2

= 119864int

infin

119905

|119884(119904)|2

119889119904 (5)

(ii) 119871119901P(Ω 119871

119902

120573([119905infin) 119865)) defined for 120573 isin 119877 and 119901 119902 isin

[1infin) denotes the space of equivalence classes ofprocesses 119884(119904) 119904 ge 119905 with values in 119865 such that thenorm

|119884|119901

= 119864(int

infin

119905

119890119902120573119904

|119884 (119904)|119902

119889119904)

119901119902

(6)

is finite and 119884 admits a predictable version(iii) K119901

120573(119905) denotes the space 119871

119901

P(Ω 119871

2

120573([119905infin) 119865)) times

119871119901

P(Ω 119871

2

120573([119905infin) 119871

2(Ξ 119865))) The norm of an element


(119884 119885) isinK119901


space(iv) 119871119901



|119884|119901

= 119864 sup119904isin[119905119879]

|119884 (119904)|119901

(7)



(v) 119871119902P(Ω 119862



|119884|119902

= 119864 sup119904ge119905

119890120578119902119904

|119884 (119904)|119902

(8)





P(Ω 119871

119902

120578([119905infin) 119865)) cap

119871119902

P(Ω 119862


|119884|H119902

120578

= |119884|119871119902

P(Ω119871119902

120578([119905infin)119865))

+ |119884|119871119902

P(Ω119862120578([119905infin)119865))

(9)


P(Ω 119871

119902

120573([0infin) 119865)) 119871119902

P(Ω

119862120578([0infin) 119865)) H119902

120578(0) and K

119901

120573(0) by 119871119901

P(Ω 119871

119902

120573(119865)) 119871119902

P(Ω

119862120578(119865))H119902

120578 andK

119901

120573 respectively


119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119905 119883119904) 119889119904 + 119866 (119904 119883

119904) 119889119882 (119904)

119904 isin [119905infin)

119883119905= 119909 isin C

(10)



119905119860



10038161003816100381610038161003816119890119904119860

119865 (119905 119909)10038161003816100381610038161003816le 119871119890

120596119904

119904minus120579

(1 + |119909|119862)

10038161003816100381610038161003816119890119904119860

119865 (119905 119909) minus 119890119904119860

119865 (119905 119910)10038161003816100381610038161003816le 119871119890

120596119904

119904minus1205791003816100381610038161003816119909 minus 119910

1003816100381610038161003816119862


(11)


119890119904119860


and

10038161003816100381610038161003816119890119904119860

119866 (119905 119909)100381610038161003816100381610038161198712(Ξ119867)

le 119871119890120596119904

119904minus120574

(1 + |119909|119862)

10038161003816100381610038161003816119890119904119860

119866 (119905 119909) minus 119890119904119860

119866 (119905 119910)100381610038161003816100381610038161198712(Ξ119867)

le 119871119890120596119904

119904minus120574

(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816119862)


(12)


F119905119905ge0


119883 (119904) = 119890(119904minus119905)119860

119909 (0) + int

119904

119905

119890(119904minus120590)119860

119865 (120590119883120590) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883120590) 119889119882 (120590) 119904 isin [119905infin)

119883119905= 119909 isin C

(13)






119871119902


119864 sup119904isin[119905119879]

|119883 (119904)|119902

le 119862(1 + |119909|119862)119902

(14)





sdot(119905 119909) isin H

119902

120578(119902)(119905) Moreover for a


119864supsget119890120578(119902)1199021199041003816100381610038161003816119883119904

1003816100381610038161003816119902

119862+ 119864int

infin

119905

119890120578(119902)1199021199041003816100381610038161003816119883119904

1003816100381610038161003816119902

119862119889119904 le 119862(1 + |119909|

119862)119902

(15)





H119902


Φ(119883sdot 119905 119909)

119904(119897) = 119890

(119904+119897minus119905)119860

119909 (0) + int

119904+119897

119905

119890(119904+119897minus120590)119860

119865 (120590119883120590) 119889120590

+ int

119904+119897

119905

119890(119904+119897minus120590)119860

119866 (120590119883120590) 119889119882 (120590)


Φ(119883sdot 119905 119909)

119904(119897) = 119909 (119904 + 119897 minus 119905)


(16)


120578(119905)


1

sdot 119905 119909) minus Φ (119883

1

sdot 119905 119909)

10038161003816100381610038161003816H119902

120578(119905)

le 119888100381610038161003816100381610038161198831

sdotminus 119883

2

sdot

10038161003816100381610038161003816H119902

120578(119905)

1198831

sdot 119883

2

sdotisinH

119902

120578(119905)

(17)


120572= int

s120590

(119904 minus 119903)120572minus1

(119903 minus 120590)minus120572


Φ(119883sdot 0 119909)

119904(119897) = 119890

(119904+119897)119860

119909 (0)

+ 119888120572int

119904+119897

0

int

119904+119897

120590

(119904 + 119897 minus 119903)120572minus1

(119903 minus 120590)minus120572

times 119890(119904+119897minus119903)119860

119890(119903minus120590)119860

119889119903119866 (120590119883120590) 119889119882 (120590)

= 119890(119904+119897)119860

119909 (0) + Φ1015840

(119883119904) (119897)


Φ(119883sdot 0 119909)

119904(119897) = 119909 (119904 + 119897)


(18)

where

Φ1015840

(119883sdot)119904(119897) = 119888

120572int

119904+119897

0

(119904 + 119897 minus 119903)120572minus1

119890(119904+119897minus119903)119860

119884 (119903) 119889119903

119884 (119903) = int

119903

0

(119903 minus 120590)minus120572

119890(119903minus120590)119860

119866 (120590119883120590) 119889119882 (120590)

(19)


|119890(119904+119897)119860

119909(0)| le 119872119890120596119904

|119909|119862 the process 119890(119904+sdot)119860

119909(0) 119904 ge 0 belongs to H119902


estimate Φ1015840

(119883sdot) where

10038161003816100381610038161003816Φ1015840

(119883sdot)119904(119897)10038161003816100381610038161003816le 119888

120572int

119904+119897

0

(119904 + 119897 minus 119903)120572minus1

119872119890(119904+119897minus119903)120596

|119884 (119903)| 119889119903

(20)


11989011990212057811990410038161003816100381610038161003816Φ1015840

(119883sdot)119904

10038161003816100381610038161003816

119902

le 119888119902

120572119872

119902 supminus120591le119897le0

119890119902120578119904

(int

119904+119897

0

(119904+119897minus 119903)120572minus1

119890120596(119904+119897minus119903)

|119884 (119903)| 119889119903)

119902

le 119888119902

120572119872

119902 supminus120591le119897le0

(int

119904+119897

0

(119904 + 119897 minus 119903)120572minus1

119890((120596+120578)119902

1015840

)(119904+119897minus119903)

times119890((120596+120578)119902)(119904minus119903)

119890120578119903

|119884 (119903)| 119889119903)

119902

le 119888119902

120572119872

119902 supminus120591le119897le0

(int

119904+119897

0

119890(120578+120596)(119904+119897minus119903)

(119904 + 119897 minus 119903)(120572minus1)119902

1015840

119889119903)

1199021199021015840

times int

119904+119897

0

119890(120578+120596)(119904minus119903)

119890119902120578119903

|119884 (119903)|119902

119889119903

le 119888119902

120572119872

119902

(int

119904

0

119890(120578+120596)119903

1199031199021015840

(120572minus1)

119889119903)

1199021199021015840

times int

119904

0

119890(120578+120596)(119904minus119903)

119890119902120578119903

|119884 (119903)|119902

119889119903

(21)


int

infin

0

11989011990212057811990410038161003816100381610038161003816Φ1015840

(119883sdot)119904

10038161003816100381610038161003816

119902

119889119904 le 119888119902

120572119872

119902

(int

infin

0

119890(120578+120596)119904

1199041199021015840

(120572minus1)

119889119904)

1199021199021015840

times int

infin

0

119890(120578+120596)119904

119889119904int

infin

0

119890119902120578119904

|119884 (119904)|119902

119889119904

(22)


(119883sdot)10038161003816100381610038161003816119871119902

P(Ω119871119902

120578(C))

le 119888120572119872|119884|

119871119902

P(Ω119871119902

120578(119867))

times (int

infin

0

119890(120578+120596)119904

1199041199021015840

(120572minus1)

119889119904)

11199021015840

times (int

infin

0

119890(120578+120596)119904

119889119904)

1119902

(23)


10038161003816100381610038161003816119890120578(119904+119897)

Φ1015840

(119883sdot)119904(119897)10038161003816100381610038161003816le 119888

120572119872(int

119904+119897

0

119903(120572minus1)119902

1015840

119890(120596+120578)119903119902

1015840

119889119903)

11199021015840

times (int

119904+119897

0

119890120578119903119902

|119884 (119903)|119902

119889119903)

1119902

10038161003816100381610038161003816119890120578119904

Φ1015840

(119883sdot)119904

10038161003816100381610038161003816le 119888

120572119872(int

119904

0

119903(120572minus1)119902

1015840

119890(120596+120578)119903119902

1015840

119889119903)

11199021015840

times (int

119904

0

119890120578119903119902

|119884(119903)|119902

119889119903)

1119902

(24)



(119883sdot)10038161003816100381610038161003816119871119902

P(Ω119862120578(C))

le 119888120572119872|119884|

119871119902

P(Ω119871119902

120578(119867))

times (int

infin

0

119903(120572minus1)119902

1015840

119890(120596+120578)119903119902

1015840

119889119903)

11199021015840

(25)



119864|119884 (119903)|119902

le 119888119902119864(int

119903

0


119866 (120590119883120590)10038161003816100381610038161003816

2

1198712(Ξ119867)

119889120590)

1199022

le 119871119902

119888119902119864

times (int

119903

0


1198902120596(119903minus120590)

(1 +1003816100381610038161003816119883120590

10038161003816100381610038162

119862) 119889120590)

1199022

(26)

which implies that

[119864|119884 (119903)|119902

]2119902

le 1198712

1198882119902

119902int

119903

0


times 1198902120596(119903minus120590)

[119864(1 +1003816100381610038161003816119883120590

1003816100381610038161003816119862)119902

]2119902

119889120590

(27)

so that

1198902120578119903

[119864|119884 (119903)|119902

]2119902

le 1198621int

119903

0


1198902(120596+120578)(119903minus120590)

1198902120578120590

119889120590

+ 1198622int

119903

0


1198902(120596+120578)(119903minus120590)

times 1198902120578120590

[1198641003816100381610038161003816119883120590

1003816100381610038161003816119902

119862]2119902

119889120590

(28)




int

infin

0

119890119902120578119903

119864|119884 (119903)|119902

119889119904le 1198621(int

infin

0

119904minus2120572minus2120574

1198902(120596+120578)119904

119889119904)

1199022

int

infin

0

119890119902120578119904

119889119904

+ 1198622(int

infin

0

119904minus2120572minus2120574

1198902(120596+120578)119904

119889119904)

1199022

times int

infin

0

119890119902120578119904

1198641003816100381610038161003816119883119904

1003816100381610038161003816119902

119862119889119904

(29)

This shows that |119884|119871119902

P(Ω119871119902

120578(119867))


If 1198831

sdot 119883

2


120578and 1198841 1198842 are


minus 119884210038161003816100381610038161003816119871119902

P(Ω119871119902

120578(119867))

le 1198711198881119902

120572

100381610038161003816100381610038161198831

sdotminus 119883

2

sdot

10038161003816100381610038161003816119871119902

P(Ω119871119902

120578(C))

times (int

infin

0

119904minus2120572minus2120574

1198902(120596+120578)119904

119889119904)

12

(30)


1015840




119878 (119904) = 119890119904119860

for 119904 ge 0 119878 (119904) = 119868 for 119904 lt 0 (31)



+ int

119904

0

119868[119905infin)

(120590) 119878 (119904 minus 120590) 119865 (120590119883120590) 119889120590

+ int

119904

0

119868[119905infin)

(120590) 119878 (119904 minus 120590)

times 119866 (120590119883120590) 119889119882 (120590) 119904 isin [0infin)


(32)


119902





1003816100381610038161003816ℎ (1199091 119910) minus ℎ (1199092 119910)1003816100381610038161003816 le 120572

10038161003816100381610038161199091 minus 11990921003816100381610038161003816 (33)


2isin 119861 y isin 119863 Let 120601(119910)








120578times


120578as follows

Φ(119883sdot 119905 119909)

119904(119897) = 119878 (119904 + 119897 minus 119905) 119909 (0)

+ int

119904+119897

0

119868[119905infin)

(120590) 119878 (119904 + 119897 minus 120590) 119865 (120590119883120590) 119889120590

+ int

119904+119897

0

119868[119905infin)

(120590) 119878 (119904 + 119897 minus 120590)

times 119866 (120590119883120590) 119889119882 (120590)


Φ(119883sdot 119905 119909)

119904(119897) = 119909 ((119904 + 119897 minus 119905) or (minus120591))


(34)








sdot it is


toH119902




119905)

and H119902




the norm

|119909|119902

= 119864|119909|119902

119862 (35)

is finite



119883 (119904) = 119890(119904minus119905)119860

119909 (0) + int

119904

119905

119890(119904minus120590)119860

119865 (120590119883120590) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883120590) 119889119882 (120590) 119904 isin [119905infin)

119883119905= 119909 isin C

119884 (119904) minus 119884 (119879) + int

119879

119904

119885 (120590) 119889119882 (120590) + 120582int

119879

119904

119884 (120590) 119889120590

= int

119879

119904

120595 (120590119883120590 119884 (120590) 119885 (120590)) 119889120590 0 le 119904 le 119879 lt infin

(36)






119910 119871

119911gt 0

120583 isin 119877 and119898 ge 11003816100381610038161003816120595 (119904 119909 1199101 1199111) minus 120595 (119904 119909 1199102 1199112)

1003816100381610038161003816

le 119871119910

10038161003816100381610038161199101 minus 11991021003816100381610038161003816 + 119871119911

10038161003816100381610038161199111 minus 11991121003816100381610038161003816

1003816100381610038161003816120595 (119904 119909 119910 119911)1003816100381610038161003816 le 119871 (1 + |119909|

119898

119862+10038161003816100381610038161199101003816100381610038161003816 + |119911|)

⟨120595 (119904 119909 1199101 119911) minus 120595 (119904 119909 119910

2 119911) 119910

1minus 119910

2⟩119870ge 120583

10038161003816100381610038161199101 minus 119910210038161003816100381610038162

(37)


2isin 119870 119911 119911

1 and 119911

2isin




equation

119884 (119904) minus 119884 (119879) + int

119879

119904

119885 (120590) 119889119882 (120590) + 120582int

119879

119904

119884 (120590) 119889120590

= int

119879

119904

120595 (120590119883120590 119884 (120590) 119885 (120590)) 119889120590 0 le 119904 le 119879 lt infin

(38)



sdotis a



119902 ge 119898119901 120578 gt120575

119898 (39)



119902

P(Ω 119871

119902


2

1199112))


by (119884(119883sdot)(119904) 119885(119883

sdot)(119904)) 119904 ge 0

(ii) The estimate

119864sup119904ge0

(119884 (119883sdot) (119904))

119901

119890119901120575119904

+ 119864(int

infin

0

11989021205751205901003816100381610038161003816119884(119883sdot

)(120590)10038161003816100381610038162

119889120590)

1199012

+ 119864(int

infin

0

11989021205751205901003816100381610038161003816119885 (119883sdot

) (120590)10038161003816100381610038162

119889120590)

1199012

le 119888(1 +1003816100381610038161003816119883sdot

1003816100381610038161003816119898

119871119902

P(Ω119871119902

120578(C))

)119901

(40)



119871119901

P(Ω 119862

120575(119870))


sdot) 119885(119883


119871119902

P(Ω 119871

119902

120578(C)) toK119901

120575 and 119883

sdotrarr 119884(119883

sdot) is continuous

from 119871119902

P(Ω 119871

119902

120578(C)) to 119871119901

P(Ω 119862

120575(119870))


P(Ω 119871

119902


119871119902

P(Ω 119862

120578(C))






1015840

= minus(120575 + 120583 minus (1198712



120578(119902)times K

119901



119871119901

P(Ω 119862





120575(119877))


sdot(119905 119909) isin H

119902

120578(119902)



toH119902

120578(119902)


119901

120575of the


sdot) 119885(119883

sdot)) is


120578(119902)to K

119901

120575while X

sdotrarr (Y(X

sdot)) is


120578(119902)to119871119901

P(Ω 119862



120578(119902)times K

119901

120575is the unique



119902

(ΩCF119905) toK119901

120575(119905)








119883119897(119904 119883

119904(119905 119909)) = 119883

119897(119905 119909) 119897 isin [119904infin) (41)



119884 (119897 119904 119883119904(119905 119909)) = 119884 (119897 119905 119909) for 119897 isin [119904infin)

119885 (119897 119904 119883119904(119905 119909)) = 119885 (119897 119905 119909) for aa 119897 isin [119904infin)

(42)






Let now 119891 isin 119871119902


119899 119899 isin 119873 such that


Hence 119891119899



this section



= 119871(Ξ 119877) =


(119883(119905 119909) 119884(119905 119909) 119885(119905 119909)) of (36) satisfies

119884 (119904 119905 119909) = 120592 (119904 119883119904(119905 119909)) 119885 (119904 119905 119909) = 120577 (119904 119883

119904(119905 119909))


(44)



=


1lt 119905

2lt infin Δ gt

0 and 1199091 119909


100381610038161003816100381610038161003816100381610038161003816

119864 int

1199051+Δ

1199051

119885119894119873

(119904 1199051 119909

1) 119889119904 minus 119864int

1199052+Δ

1199052

119885119894119873

(119904 1199052 119909

2) 119889119904

100381610038161003816100381610038161003816100381610038161003816

le 119864int

1199052

1199051

10038161003816100381610038161003816119885119894119873

(119904 1199051 119909

1)10038161003816100381610038161003816119889119904

+ 119864int

1199051+Δ

1199052

10038161003816100381610038161003816119885119894119873

(119904 1199051 119909

1) minus 119885

119894119873

(119904 1199052 119909

2)10038161003816100381610038161003816119889119904

+ 119864int

1199052+Δ

1199051+Δ

10038161003816100381610038161003816119885119894119873

(119904 1199052 119909

2)10038161003816100381610038161003816119889119904


le 211987310038161003816100381610038161199052 minus 1199051

1003816100381610038161003816 + Δ12

119890minus120575(1199051+Δ)

times(119864(int

infin

0

119890212057511990410038161003816100381610038161003816119885119894119873

(119904 1199051 119909

1)minus119885

119894119873

(119904 1199052 119909

2)10038161003816100381610038161003816

2

119889119904)

1199012

)

1119901

le 211987310038161003816100381610038161199052 minus 1199051

1003816100381610038161003816 + Δ12

119890minus120575(1199051+Δ)

times (119864(int

infin

0

11989021205751199041003816100381610038161003816119885 (119904 1199051 1199091)minus119885 (119904 1199052 1199092)

10038161003816100381610038162

119889119904)

1199012

)

1119901

(45)


119905

119885119894119873


119864int119905+Δ

119905

119864119885119894119873


(ΩCF119905) to 119877


120577119894119873


119899119864int

119905+(1119899)

119905

119885119894119873

(119904 119905 119909) 119889119904


(46)


denote 119864[119885119894119873

(119897 119904 119910)]|119910=119883119904(119905119909)


119904(119905 119909) with the map 119910 rarr 119864[119885

119894119873


119904-


119891119898 Ω 997888rarr C 119891

119898=

119873119898

sum

119896=1

ℎ(119898)

119896119868119891119898=ℎ(119898)

119896 119873

119898isin 119873 (47)

where ℎ(119898)1 ℎ

(119898)


119873119898

119896=1119891

119898=

ℎ(119898)

119896 such that

1003816100381610038161003816119891119898 (120596) minus 119883119904(120596)



int119861

int

119904+(1119899)

119904

119885119894119873

(119897 119905 119909) 119889119897119889119875

= int119861

int

119904+(1119899)

119904

119885119894119873

(119897 119904 119883119904) 119889119897119889119875

= 119864119868119861int

119904+(1119899)

119904

119885119894119873

(119897 119904 119883119904) 119889119897


119864(119868119861int

119904+(1119899)

119904

119885119894119873

(119897 119904 119891119898) 119889119897)


119873119898

sum

119896=1

119864(119868119861119868119891119898=ℎ(119898)

119896int

119904+(1119899)

119904

119885119894119873

(119897 119904 ℎ(119898)

119896) 119889119897)


119864(119868119861

119873119898

sum

119896=1

119868119891119898=ℎ(119898)

119896)119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 ℎ(119898)

119896) 119889119897


119864119868119861(119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 119910) 119889119897|119910=119891119898

)


int119861

(119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 119910) 119889119897|119910=119891119898

)119889119875

= int119861

(119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 119910) 119889119897|119910=119883119904

)119889119875

(49)

and we get that

120577119894119873

(119904 119883119904(119905 119909)) = lim inf

119899rarrinfin

119899

times [119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 119910) 119889119897|119910=119883119904(119905119909)

]


119899119864[int

119904+(1119899)

119904

119885119894119873

(119897 119905 119909) 119889119897

100381610038161003816100381610038161003816100381610038161003816

F119904]

119875-as(50)



lim119899rarrinfin

119899int

119904+(1119899)

119904

119885119894119873

(119897 119905 119909) 119889119897 = 119885119894119873

(119904 119905 119909)


(51)


120577119894119873

(119904 119883119904(119905 119909)) = 119864 [119885

119894119873

(119904 119905 119909)10038161003816100381610038161003816F

119904] = 119885

119894119873

(119904 119905 119909)


(52)


120577119894119873

(119904 119883119904(119905 119909)) = 119885

119894119873

(119904 119905 119909)


(53)


119873rarrinfin120577119894119873


119894=1(lim

119873rarrinfin120577119894119873


120577 (119905 119909) =

infin

sum

119894=1


120577119894119873

(119905 119909)) 119890119894 (119905 119909) isin 119862

120577 (119905 119909) = 0 (119905 119909) notin 119862

(54)



119885 (120596 119904 119905 119909) =

infin

sum

119894=1


119885119894119873

(120596 119904 119905 119909)) 119890119894 (55)





3(Γ2(119905 Γ

1(119905 119909))) with


119901

P(Ω 119862

120575(119877))

Γ1(119905 119909) = 119884 (sdot 119905 119909)


119901

P(Ω 119862

120575(119877)) 997888rarr 119871

119901

(Ω 119877)

Γ2(119905 119881) = 119881 (119905)

Γ3 119871

119901

(Ω 119877) 997888rarr 119877 Γ3120585 = 119864120585

(56)


|119881(119905) minus 119880(119904)|119871119901(Ω119877)

le |119881 (119905) minus 119881 (119904)|119871119901(Ω119877)

+ 119890minus120575119901119904

|119881 minus 119880||119871

119901

P(Ω119862120575(119877))

(57)


3is continuous





119905119905ge0






119889119883119906

(119904) = 119860119883119906

(119904) 119889119904 + 119865 (119904 119883119906

119904) 119889119904

+ 119866 (119904 119883119906

119904) 119877 (119904 119883

119906

119904 119906 (119904)) 119889119904 + 119866 (119904 119883

119906

119904) 119889119882 (119904)

119904 isin [119905infin)

119883119906

119905= 119909

(58)


(119904 119905 119909) or simply by 119883119906


119869 (119906) = 119864int

infin

119905

119890minus120582119904

119892 (119904 119883119906

119904 119906 (119904)) 119889119904 (59)




120595 (119905 119909 119911) = inf 119892 (119905 119909 119906) + 119911119877 (119905 119909 119906) 119906 isin 119880

(60)


Γ (119905 119909 119911) = 119906 isin 119880 119892 (119905 119909 119906) + 119911119877 (119905 119909 119906) = 120595 (119905 119909 119911)

(61)




119898119892

119862)









119892



120582 gt (minus120575 minus 120583 +1198712

119911

2) or (minus120575 +

1198712

119877

2 (119901 minus 1))

or (1198712

119877119898119892

2 (119902 minus 119898119892)minus 120578 (119902)119898

119892)

(62)


119869 (119906) = 120592 (119905 119909) + 119864int

infin

119905

119890minus120582119904

[minus120595 (119904 119883119906

119904 120577 (119904 119883

119906

119904)) + 120577 (119904 119883

119906

119904)

times 119877 (119904 119883119906

119904 119906 (119904)) + 119892 (119904 119883

119906

119904 119906 (119904))] 119889119904

(63)


119905119905ge0



119882119906

(119904) = 119882 (119904) + int

119904

119905and119904

119877 (120590119883119906

120590 119906 (120590)) 119889120590 119904 isin [0infin)

120588 (119879) = exp(int119879

119905

minus119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882 (119904)

minus1

2int

119879

119905

1003816100381610038161003816119877 (119904 119883119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)

(64)



such that

119875119906

|F119879

= 120588 (119879) 119875|F119879

(65)


119905119905ge0


119889119883119906

(119904) = 119860119883119906

(119904) 119889119904 + 119865 (119904 119883119906

119904) 119889119904

+ 119866 (119904 119883119906

119904) 119889119882

119906

(119904) 119904 isin [119905infin)

119883119906

119905= 119909

(66)


F119906

119905119905ge0 119875

119906


119883119906

(119904) = 119890(119904minus119905)119860

119909 (0) + int

119904

119905

119890(119904minus120590)119860

119865 (120590119883119906

120590) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883119906

120590) 119889119882

119906

(120590) 119904 isin [119905infin)

119883119906

119905= 119909 isin C

119884119906

(119904) minus 119884119906

(119879) + int

119879

119904

119885119906

(120590) 119889119882119906

(120590) + 120582int

119879

119904

119884119906

(120590) 119889120590

= int

119879

119904

120595 (120590119883119906

120590 119885

119906

(120590)) 119889120590 0 le 119904 le 119879 lt infin

(67)


119884119906

(119904) + int

119879

119904

119890minus120582120590

119885119906

(120590) 119889119882 (120590)

= int

119879

119904

119890minus120582120590

[120595 (120590119883119906

120590 119885

119906

(120590))

minus119885119906

(120590) 119877 (120590119883119906

120590 119906 (120590))] 119889120590

+ 119890minus120582119879

119884119906

(119879)

(68)


119864119906

[120588(119879)minus119903

] = 119864119906

[exp 119903 (int119879

119905

119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882

119906

(119904)

minus1

2int

119879

119905

1003816100381610038161003816119877 (119904 119883119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)]

= 119864119906

[exp(int119879

119905

119903119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882

119906

(119904)

minus1

2int

119879

119905

11990321003816100381610038161003816119877 (119904 119883

119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)

times exp 119903 (119903 minus 1)2

int

119879

119905

1003816100381610038161003816119877 (119904 119883119906

119904 119906 (119904))

10038161003816100381610038162

119889119904]

le 119890(12)119903(119903minus1)119879119871

2

119877119864119906

times exp(int119879

119905

2119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882

119906

(119904)

minus1

2int

119879

119905

41003816100381610038161003816119877 (119904 119883

119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)

= 119890(12)119903(119903minus1)119879119871

2

119877

(69)

It follows that

119864(int

119879

119905

|119890minus120582119904

119885119906

(119904)|2

119889119904)

12

= 119864119906

[(int

119879

119905

10038161003816100381610038161003816119890minus120582119904

119885119906

(119904)10038161003816100381610038161003816

2

119889119904)

12

120588minus1

]

le (119864119906

int

119879

119905

10038161003816100381610038161003816119890minus120582119904

119885119906

(119904)10038161003816100381610038161003816

2

119889119904)

12

times (119864119906

120588minus2

)12

lt infin

(70)


119890minus120582119879

119864119884119906

(119879) minus 119884119906

(119905)

= 119864int

119879

119905

119890minus120582120590

[minus120595 (120590119883119906

120590 119885

119906

(120590))

+119885119906

(120590) 119877 (120590119883119906

120590 119906 (120590))] 119889120590

(71)


120575(119877)) so that

119864119906

|119884(119879 119905 119909)|119901

le 119862 exp (minus119901120575119879) (72)


119864 |119884 (119879 119905 119909)| = 119864119906

(120588minus1

(119879) |119884 (119879 119905 119909)|)

le 119864(120588minus119901(119901minus1)

)(119901minus1)119901

119864(|119884 (119879 119905 119909)|119901

)1119901

le 119862119890((1198712

1198772(119901minus1))minus120575))119879

(73)


|119883119906

119904|119902


1198641003816100381610038161003816119883

119906

119879

1003816100381610038161003816119898119892

le 119862119890(1198712

119877119898119892(2119902minus2119898

119892)minus1

minus120578(119902)119898119892)119879

(74)



119864int

infin

119905

119890minus120582120590 1003816100381610038161003816119892 (120590119883

119906

120590 119906 (120590))

1003816100381610038161003816 119889120590 lt infin (75)

Since 119884119906(119905 119905 119909) = 120592(119905 119909) and 119885119906

(119904 119905 119909) = 120577(119904 119883119906

119904(119905 119909)) 119875-


119890minus120582119879

119864119884119906

(119879) minus 119907 (119905 119909)

= 119864int

119879

119905

119890minus120582120590

[minus120595 (120590119883119906

120590 120577 (120590 119883

119906

120590))

+120577 (120590119883119906

120590) 119877 (120590119883

119906

120590 119906 (120590))] 119889120590

(76)


119890minus120582120590

119892(120590119883119906

120590 119906(120590))119889120590 and


119869 (119906) = 120592 (119905 119909)

+ 119864int

infin

119905

119890minus120582119904

[minus120595 (119904 119883119906

119904 120577 (119904 119883

119906

119904)) + 120577 (119904 119883

119906

119904) 119877

times (119904 119883119906

119904 119906 (119904)) + 119892 (119904 119883

119906

119904 119906 (119904))] 119889119904

(77)






119906 (119904) = Γ0(119904 119883

119906

119904 120577 (119904 119883

119906

119904))


(78)



119889119883 (119904)=119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904+119866 (119904 119883

119904)

times(119877 (119904 119883119904 Γ

0(119904 119883

119904 120577 (119904 119883

119904))) 119889119904 + 119889119882 (119904))

119904 isin [119905infin)

119883119905= 119909

(79)


119906 (119904) = Γ0(119904 119883

119904 120577 (119904 119883

119904)) (80)







119905119905ge0


F119905119905ge0 119875119882 119906119883

119906

) where (ΩF F119905119905ge0 119875 119882) is an




119869 (119906) = 119864int

infin

119905

119890minus120582119904

119892 (119904 119883119906

119904 119906 (119904)) 119889119904 (81)



119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904 + 119866 (119904 119883

119904)

times (119877 (119904 119883119904 Γ

0(119904 119883

119904 120577 (119904 119883

119904))) 119889119904 +119889119882 (119904))

119904 isin [119905infin)

119883119905= 119909

(82)


119905119905ge0 119875119882)



119883 (119904) = 119890(119904minus119905)119860

119909 (0) + int

119904

119905

119890(119904minus120590)119860

119865 (120590119883120590) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883120590) 119877

times (120590119883120590 Γ

0(120590119883

120590 120577 (120590 119883

120590))) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883120590) 119889119882

120590 119904 isin [119905infin)

(83)

119883119905= 119909 (84)




119905119905ge0 119875119882)We define

120588 (119879) = exp(int119879

119905

minus119877lowast

(120590 119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590))) 119889119882 (120590)

minus1

2int

119879

119905

1003816100381610038161003816119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590)))10038161003816100381610038162

119889120590)

(85)


1198820

(119904) = 119882 (119904) + int

119904

119905and119904

119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590))) 119889120590


(86)


1198750

|F119879

= 120588 (119879) 119875|F119879

(87)

Let us denote by F0

119905119905ge0


[0infin) F0

119905119905ge0 119875

0


119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119905 119883119904) 119889119904

+ 119866 (119904 119883119904) 119889119882

0

(119904) 119904 isin [119905infin)

119883119905= 119909

120588 (119879) = exp(int119879

119905

minus119877lowast

(120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590)) 119889119882

0

(120590)

+ 12int

119879

119905

1003816100381610038161003816119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590)))10038161003816100381610038162

119889120590)

(88)




119905119905ge0


119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904

+ 119866 (119904 119883119904) 119889119882 (119904) 119904 isin [119905infin)

119883119905= 119909

(89)


1198821

(119904) = 119882 (119904) minus int

119904

119905and119904

119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590))) 119889120590

(90)





0



119906

) one has

119869 (119906 119905 119909) ge 120592 (119905 119909) (91)


119906 (119904) = Γ0(119904 119883

119906

119904 120577 (119904 119883

119906

119904))

119875 minus 119886119904 119891119900119903 119886119897119898119900119904119905 119890119907119890119903119910 119904 isin [119905infin)

(92)



119906 (119904) = Γ0(119904 119883

119904 120577 (119904 119883

119904)) (93)


6 Applications



119889119911119906

(119905 120585) = Δ119911119906

(119905 120585) 119889119905 + 119891 (119905 119911119906

119905(120585)) 119889119905

+

119889

sum

119894=1

119892119894(119905 119911

119906

119905(120585)) [119903

119894

(120585) 119906119894

(119905) 119889119905 + 119889119882119894

(119905)]

119911119906

0(120579 120585) = 119909 (120579 120585) 120585 isin 119861 120579 isin [minus1 0]

119911119906

(119905 120585) = 0 119905 isin [0infin) 120585 isin 120597119861

(94)

Here119882 = (1198821

1198822

119882119889



119867 = 1198712


119865 (119905 119909) (120585) = 119891 (119905 119909 (120585))

(119866 (119905 119909) 119911) (120585) =

119889

sum

119894=1

119892119894(119905 119909 (120585)) 119911

119894

(120585)


(95)


(119861) withdomain11988222

(119861)⋂11988212


Hypothesis 1 holds



119869 (119906) = 119864int

infin

0

119890minus120582119905

[int119861

120590 (120585 119911119906

119905(120585)) 119889120585 + 119906

2

(119905)] 119889119905 (96)



119861

120590(119905 119910(120585) 119906)119889120585 + 1199062 and 119877(119910 119906) = (int

119861

1199031

(120585)1199061

119889120585

int119861

1199032

(120585)1199062

119889120585 int119861

119903119889

(120585)119906119889

119889120585) for 119910 isin 119862([minus1 0]119867) 119906 =

(1199061

1199062

119906119889


0



119905119905ge0 119875119882 119906 119911

sdot(sdot)) and

setting

119906 (119904) = Γ0(119904 119911

119904(sdot) 120577 (119904 119911

119904(sdot))) (97)


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014




(119884 119885) isinK119901


space(iv) 119871119901



|119884|119901

= 119864 sup119904isin[119905119879]

|119884 (119904)|119901

(7)



(v) 119871119902P(Ω 119862



|119884|119902

= 119864 sup119904ge119905

119890120578119902119904

|119884 (119904)|119902

(8)





P(Ω 119871

119902

120578([119905infin) 119865)) cap

119871119902

P(Ω 119862


|119884|H119902

120578

= |119884|119871119902

P(Ω119871119902

120578([119905infin)119865))

+ |119884|119871119902

P(Ω119862120578([119905infin)119865))

(9)


P(Ω 119871

119902

120573([0infin) 119865)) 119871119902

P(Ω

119862120578([0infin) 119865)) H119902

120578(0) and K

119901

120573(0) by 119871119901

P(Ω 119871

119902

120573(119865)) 119871119902

P(Ω

119862120578(119865))H119902

120578 andK

119901

120573 respectively


119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119905 119883119904) 119889119904 + 119866 (119904 119883

119904) 119889119882 (119904)

119904 isin [119905infin)

119883119905= 119909 isin C

(10)



119905119860



10038161003816100381610038161003816119890119904119860

119865 (119905 119909)10038161003816100381610038161003816le 119871119890

120596119904

119904minus120579

(1 + |119909|119862)

10038161003816100381610038161003816119890119904119860

119865 (119905 119909) minus 119890119904119860

119865 (119905 119910)10038161003816100381610038161003816le 119871119890

120596119904

119904minus1205791003816100381610038161003816119909 minus 119910

1003816100381610038161003816119862


(11)


119890119904119860


and

10038161003816100381610038161003816119890119904119860

119866 (119905 119909)100381610038161003816100381610038161198712(Ξ119867)

le 119871119890120596119904

119904minus120574

(1 + |119909|119862)

10038161003816100381610038161003816119890119904119860

119866 (119905 119909) minus 119890119904119860

119866 (119905 119910)100381610038161003816100381610038161198712(Ξ119867)

le 119871119890120596119904

119904minus120574

(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816119862)


(12)


F119905119905ge0


119883 (119904) = 119890(119904minus119905)119860

119909 (0) + int

119904

119905

119890(119904minus120590)119860

119865 (120590119883120590) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883120590) 119889119882 (120590) 119904 isin [119905infin)

119883119905= 119909 isin C

(13)






119871119902


119864 sup119904isin[119905119879]

|119883 (119904)|119902

le 119862(1 + |119909|119862)119902

(14)





sdot(119905 119909) isin H

119902

120578(119902)(119905) Moreover for a


119864supsget119890120578(119902)1199021199041003816100381610038161003816119883119904

1003816100381610038161003816119902

119862+ 119864int

infin

119905

119890120578(119902)1199021199041003816100381610038161003816119883119904

1003816100381610038161003816119902

119862119889119904 le 119862(1 + |119909|

119862)119902

(15)





H119902


Φ(119883sdot 119905 119909)

119904(119897) = 119890

(119904+119897minus119905)119860

119909 (0) + int

119904+119897

119905

119890(119904+119897minus120590)119860

119865 (120590119883120590) 119889120590

+ int

119904+119897

119905

119890(119904+119897minus120590)119860

119866 (120590119883120590) 119889119882 (120590)


Φ(119883sdot 119905 119909)

119904(119897) = 119909 (119904 + 119897 minus 119905)


(16)


120578(119905)


1

sdot 119905 119909) minus Φ (119883

1

sdot 119905 119909)

10038161003816100381610038161003816H119902

120578(119905)

le 119888100381610038161003816100381610038161198831

sdotminus 119883

2

sdot

10038161003816100381610038161003816H119902

120578(119905)

1198831

sdot 119883

2

sdotisinH

119902

120578(119905)

(17)


120572= int

s120590

(119904 minus 119903)120572minus1

(119903 minus 120590)minus120572


Φ(119883sdot 0 119909)

119904(119897) = 119890

(119904+119897)119860

119909 (0)

+ 119888120572int

119904+119897

0

int

119904+119897

120590

(119904 + 119897 minus 119903)120572minus1

(119903 minus 120590)minus120572

times 119890(119904+119897minus119903)119860

119890(119903minus120590)119860

119889119903119866 (120590119883120590) 119889119882 (120590)

= 119890(119904+119897)119860

119909 (0) + Φ1015840

(119883119904) (119897)


Φ(119883sdot 0 119909)

119904(119897) = 119909 (119904 + 119897)


(18)

where

Φ1015840

(119883sdot)119904(119897) = 119888

120572int

119904+119897

0

(119904 + 119897 minus 119903)120572minus1

119890(119904+119897minus119903)119860

119884 (119903) 119889119903

119884 (119903) = int

119903

0

(119903 minus 120590)minus120572

119890(119903minus120590)119860

119866 (120590119883120590) 119889119882 (120590)

(19)


|119890(119904+119897)119860

119909(0)| le 119872119890120596119904

|119909|119862 the process 119890(119904+sdot)119860

119909(0) 119904 ge 0 belongs to H119902


estimate Φ1015840

(119883sdot) where

10038161003816100381610038161003816Φ1015840

(119883sdot)119904(119897)10038161003816100381610038161003816le 119888

120572int

119904+119897

0

(119904 + 119897 minus 119903)120572minus1

119872119890(119904+119897minus119903)120596

|119884 (119903)| 119889119903

(20)


11989011990212057811990410038161003816100381610038161003816Φ1015840

(119883sdot)119904

10038161003816100381610038161003816

119902

le 119888119902

120572119872

119902 supminus120591le119897le0

119890119902120578119904

(int

119904+119897

0

(119904+119897minus 119903)120572minus1

119890120596(119904+119897minus119903)

|119884 (119903)| 119889119903)

119902

le 119888119902

120572119872

119902 supminus120591le119897le0

(int

119904+119897

0

(119904 + 119897 minus 119903)120572minus1

119890((120596+120578)119902

1015840

)(119904+119897minus119903)

times119890((120596+120578)119902)(119904minus119903)

119890120578119903

|119884 (119903)| 119889119903)

119902

le 119888119902

120572119872

119902 supminus120591le119897le0

(int

119904+119897

0

119890(120578+120596)(119904+119897minus119903)

(119904 + 119897 minus 119903)(120572minus1)119902

1015840

119889119903)

1199021199021015840

times int

119904+119897

0

119890(120578+120596)(119904minus119903)

119890119902120578119903

|119884 (119903)|119902

119889119903

le 119888119902

120572119872

119902

(int

119904

0

119890(120578+120596)119903

1199031199021015840

(120572minus1)

119889119903)

1199021199021015840

times int

119904

0

119890(120578+120596)(119904minus119903)

119890119902120578119903

|119884 (119903)|119902

119889119903

(21)


int

infin

0

11989011990212057811990410038161003816100381610038161003816Φ1015840

(119883sdot)119904

10038161003816100381610038161003816

119902

119889119904 le 119888119902

120572119872

119902

(int

infin

0

119890(120578+120596)119904

1199041199021015840

(120572minus1)

119889119904)

1199021199021015840

times int

infin

0

119890(120578+120596)119904

119889119904int

infin

0

119890119902120578119904

|119884 (119904)|119902

119889119904

(22)


(119883sdot)10038161003816100381610038161003816119871119902

P(Ω119871119902

120578(C))

le 119888120572119872|119884|

119871119902

P(Ω119871119902

120578(119867))

times (int

infin

0

119890(120578+120596)119904

1199041199021015840

(120572minus1)

119889119904)

11199021015840

times (int

infin

0

119890(120578+120596)119904

119889119904)

1119902

(23)


10038161003816100381610038161003816119890120578(119904+119897)

Φ1015840

(119883sdot)119904(119897)10038161003816100381610038161003816le 119888

120572119872(int

119904+119897

0

119903(120572minus1)119902

1015840

119890(120596+120578)119903119902

1015840

119889119903)

11199021015840

times (int

119904+119897

0

119890120578119903119902

|119884 (119903)|119902

119889119903)

1119902

10038161003816100381610038161003816119890120578119904

Φ1015840

(119883sdot)119904

10038161003816100381610038161003816le 119888

120572119872(int

119904

0

119903(120572minus1)119902

1015840

119890(120596+120578)119903119902

1015840

119889119903)

11199021015840

times (int

119904

0

119890120578119903119902

|119884(119903)|119902

119889119903)

1119902

(24)



(119883sdot)10038161003816100381610038161003816119871119902

P(Ω119862120578(C))

le 119888120572119872|119884|

119871119902

P(Ω119871119902

120578(119867))

times (int

infin

0

119903(120572minus1)119902

1015840

119890(120596+120578)119903119902

1015840

119889119903)

11199021015840

(25)



119864|119884 (119903)|119902

le 119888119902119864(int

119903

0


119866 (120590119883120590)10038161003816100381610038161003816

2

1198712(Ξ119867)

119889120590)

1199022

le 119871119902

119888119902119864

times (int

119903

0


1198902120596(119903minus120590)

(1 +1003816100381610038161003816119883120590

10038161003816100381610038162

119862) 119889120590)

1199022

(26)

which implies that

[119864|119884 (119903)|119902

]2119902

le 1198712

1198882119902

119902int

119903

0


times 1198902120596(119903minus120590)

[119864(1 +1003816100381610038161003816119883120590

1003816100381610038161003816119862)119902

]2119902

119889120590

(27)

so that

1198902120578119903

[119864|119884 (119903)|119902

]2119902

le 1198621int

119903

0


1198902(120596+120578)(119903minus120590)

1198902120578120590

119889120590

+ 1198622int

119903

0


1198902(120596+120578)(119903minus120590)

times 1198902120578120590

[1198641003816100381610038161003816119883120590

1003816100381610038161003816119902

119862]2119902

119889120590

(28)




int

infin

0

119890119902120578119903

119864|119884 (119903)|119902

119889119904le 1198621(int

infin

0

119904minus2120572minus2120574

1198902(120596+120578)119904

119889119904)

1199022

int

infin

0

119890119902120578119904

119889119904

+ 1198622(int

infin

0

119904minus2120572minus2120574

1198902(120596+120578)119904

119889119904)

1199022

times int

infin

0

119890119902120578119904

1198641003816100381610038161003816119883119904

1003816100381610038161003816119902

119862119889119904

(29)

This shows that |119884|119871119902

P(Ω119871119902

120578(119867))


If 1198831

sdot 119883

2


120578and 1198841 1198842 are


minus 119884210038161003816100381610038161003816119871119902

P(Ω119871119902

120578(119867))

le 1198711198881119902

120572

100381610038161003816100381610038161198831

sdotminus 119883

2

sdot

10038161003816100381610038161003816119871119902

P(Ω119871119902

120578(C))

times (int

infin

0

119904minus2120572minus2120574

1198902(120596+120578)119904

119889119904)

12

(30)


1015840




119878 (119904) = 119890119904119860

for 119904 ge 0 119878 (119904) = 119868 for 119904 lt 0 (31)



+ int

119904

0

119868[119905infin)

(120590) 119878 (119904 minus 120590) 119865 (120590119883120590) 119889120590

+ int

119904

0

119868[119905infin)

(120590) 119878 (119904 minus 120590)

times 119866 (120590119883120590) 119889119882 (120590) 119904 isin [0infin)


(32)


119902





1003816100381610038161003816ℎ (1199091 119910) minus ℎ (1199092 119910)1003816100381610038161003816 le 120572

10038161003816100381610038161199091 minus 11990921003816100381610038161003816 (33)


2isin 119861 y isin 119863 Let 120601(119910)








120578times


120578as follows

Φ(119883sdot 119905 119909)

119904(119897) = 119878 (119904 + 119897 minus 119905) 119909 (0)

+ int

119904+119897

0

119868[119905infin)

(120590) 119878 (119904 + 119897 minus 120590) 119865 (120590119883120590) 119889120590

+ int

119904+119897

0

119868[119905infin)

(120590) 119878 (119904 + 119897 minus 120590)

times 119866 (120590119883120590) 119889119882 (120590)


Φ(119883sdot 119905 119909)

119904(119897) = 119909 ((119904 + 119897 minus 119905) or (minus120591))


(34)








sdot it is


toH119902




119905)

and H119902




the norm

|119909|119902

= 119864|119909|119902

119862 (35)

is finite



119883 (119904) = 119890(119904minus119905)119860

119909 (0) + int

119904

119905

119890(119904minus120590)119860

119865 (120590119883120590) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883120590) 119889119882 (120590) 119904 isin [119905infin)

119883119905= 119909 isin C

119884 (119904) minus 119884 (119879) + int

119879

119904

119885 (120590) 119889119882 (120590) + 120582int

119879

119904

119884 (120590) 119889120590

= int

119879

119904

120595 (120590119883120590 119884 (120590) 119885 (120590)) 119889120590 0 le 119904 le 119879 lt infin

(36)






119910 119871

119911gt 0

120583 isin 119877 and119898 ge 11003816100381610038161003816120595 (119904 119909 1199101 1199111) minus 120595 (119904 119909 1199102 1199112)

1003816100381610038161003816

le 119871119910

10038161003816100381610038161199101 minus 11991021003816100381610038161003816 + 119871119911

10038161003816100381610038161199111 minus 11991121003816100381610038161003816

1003816100381610038161003816120595 (119904 119909 119910 119911)1003816100381610038161003816 le 119871 (1 + |119909|

119898

119862+10038161003816100381610038161199101003816100381610038161003816 + |119911|)

⟨120595 (119904 119909 1199101 119911) minus 120595 (119904 119909 119910

2 119911) 119910

1minus 119910

2⟩119870ge 120583

10038161003816100381610038161199101 minus 119910210038161003816100381610038162

(37)


2isin 119870 119911 119911

1 and 119911

2isin




equation

119884 (119904) minus 119884 (119879) + int

119879

119904

119885 (120590) 119889119882 (120590) + 120582int

119879

119904

119884 (120590) 119889120590

= int

119879

119904

120595 (120590119883120590 119884 (120590) 119885 (120590)) 119889120590 0 le 119904 le 119879 lt infin

(38)



sdotis a



119902 ge 119898119901 120578 gt120575

119898 (39)



119902

P(Ω 119871

119902


2

1199112))


by (119884(119883sdot)(119904) 119885(119883

sdot)(119904)) 119904 ge 0

(ii) The estimate

119864sup119904ge0

(119884 (119883sdot) (119904))

119901

119890119901120575119904

+ 119864(int

infin

0

11989021205751205901003816100381610038161003816119884(119883sdot

)(120590)10038161003816100381610038162

119889120590)

1199012

+ 119864(int

infin

0

11989021205751205901003816100381610038161003816119885 (119883sdot

) (120590)10038161003816100381610038162

119889120590)

1199012

le 119888(1 +1003816100381610038161003816119883sdot

1003816100381610038161003816119898

119871119902

P(Ω119871119902

120578(C))

)119901

(40)



119871119901

P(Ω 119862

120575(119870))


sdot) 119885(119883


119871119902

P(Ω 119871

119902

120578(C)) toK119901

120575 and 119883

sdotrarr 119884(119883

sdot) is continuous

from 119871119902

P(Ω 119871

119902

120578(C)) to 119871119901

P(Ω 119862

120575(119870))


P(Ω 119871

119902


119871119902

P(Ω 119862

120578(C))






1015840

= minus(120575 + 120583 minus (1198712



120578(119902)times K

119901



119871119901

P(Ω 119862





120575(119877))


sdot(119905 119909) isin H

119902

120578(119902)



toH119902

120578(119902)


119901

120575of the


sdot) 119885(119883

sdot)) is


120578(119902)to K

119901

120575while X

sdotrarr (Y(X

sdot)) is


120578(119902)to119871119901

P(Ω 119862



120578(119902)times K

119901

120575is the unique



119902

(ΩCF119905) toK119901

120575(119905)








119883119897(119904 119883

119904(119905 119909)) = 119883

119897(119905 119909) 119897 isin [119904infin) (41)



119884 (119897 119904 119883119904(119905 119909)) = 119884 (119897 119905 119909) for 119897 isin [119904infin)

119885 (119897 119904 119883119904(119905 119909)) = 119885 (119897 119905 119909) for aa 119897 isin [119904infin)

(42)






Let now 119891 isin 119871119902


119899 119899 isin 119873 such that


Hence 119891119899



this section



= 119871(Ξ 119877) =


(119883(119905 119909) 119884(119905 119909) 119885(119905 119909)) of (36) satisfies

119884 (119904 119905 119909) = 120592 (119904 119883119904(119905 119909)) 119885 (119904 119905 119909) = 120577 (119904 119883

119904(119905 119909))


(44)



=


1lt 119905

2lt infin Δ gt

0 and 1199091 119909


100381610038161003816100381610038161003816100381610038161003816

119864 int

1199051+Δ

1199051

119885119894119873

(119904 1199051 119909

1) 119889119904 minus 119864int

1199052+Δ

1199052

119885119894119873

(119904 1199052 119909

2) 119889119904

100381610038161003816100381610038161003816100381610038161003816

le 119864int

1199052

1199051

10038161003816100381610038161003816119885119894119873

(119904 1199051 119909

1)10038161003816100381610038161003816119889119904

+ 119864int

1199051+Δ

1199052

10038161003816100381610038161003816119885119894119873

(119904 1199051 119909

1) minus 119885

119894119873

(119904 1199052 119909

2)10038161003816100381610038161003816119889119904

+ 119864int

1199052+Δ

1199051+Δ

10038161003816100381610038161003816119885119894119873

(119904 1199052 119909

2)10038161003816100381610038161003816119889119904


le 211987310038161003816100381610038161199052 minus 1199051

1003816100381610038161003816 + Δ12

119890minus120575(1199051+Δ)

times(119864(int

infin

0

119890212057511990410038161003816100381610038161003816119885119894119873

(119904 1199051 119909

1)minus119885

119894119873

(119904 1199052 119909

2)10038161003816100381610038161003816

2

119889119904)

1199012

)

1119901

le 211987310038161003816100381610038161199052 minus 1199051

1003816100381610038161003816 + Δ12

119890minus120575(1199051+Δ)

times (119864(int

infin

0

11989021205751199041003816100381610038161003816119885 (119904 1199051 1199091)minus119885 (119904 1199052 1199092)

10038161003816100381610038162

119889119904)

1199012

)

1119901

(45)


119905

119885119894119873


119864int119905+Δ

119905

119864119885119894119873


(ΩCF119905) to 119877


120577119894119873


119899119864int

119905+(1119899)

119905

119885119894119873

(119904 119905 119909) 119889119904


(46)


denote 119864[119885119894119873

(119897 119904 119910)]|119910=119883119904(119905119909)


119904(119905 119909) with the map 119910 rarr 119864[119885

119894119873


119904-


119891119898 Ω 997888rarr C 119891

119898=

119873119898

sum

119896=1

ℎ(119898)

119896119868119891119898=ℎ(119898)

119896 119873

119898isin 119873 (47)

where ℎ(119898)1 ℎ

(119898)


119873119898

119896=1119891

119898=

ℎ(119898)

119896 such that

1003816100381610038161003816119891119898 (120596) minus 119883119904(120596)



int119861

int

119904+(1119899)

119904

119885119894119873

(119897 119905 119909) 119889119897119889119875

= int119861

int

119904+(1119899)

119904

119885119894119873

(119897 119904 119883119904) 119889119897119889119875

= 119864119868119861int

119904+(1119899)

119904

119885119894119873

(119897 119904 119883119904) 119889119897


119864(119868119861int

119904+(1119899)

119904

119885119894119873

(119897 119904 119891119898) 119889119897)


119873119898

sum

119896=1

119864(119868119861119868119891119898=ℎ(119898)

119896int

119904+(1119899)

119904

119885119894119873

(119897 119904 ℎ(119898)

119896) 119889119897)


119864(119868119861

119873119898

sum

119896=1

119868119891119898=ℎ(119898)

119896)119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 ℎ(119898)

119896) 119889119897


119864119868119861(119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 119910) 119889119897|119910=119891119898

)


int119861

(119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 119910) 119889119897|119910=119891119898

)119889119875

= int119861

(119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 119910) 119889119897|119910=119883119904

)119889119875

(49)

and we get that

120577119894119873

(119904 119883119904(119905 119909)) = lim inf

119899rarrinfin

119899

times [119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 119910) 119889119897|119910=119883119904(119905119909)

]


119899119864[int

119904+(1119899)

119904

119885119894119873

(119897 119905 119909) 119889119897

100381610038161003816100381610038161003816100381610038161003816

F119904]

119875-as(50)



lim119899rarrinfin

119899int

119904+(1119899)

119904

119885119894119873

(119897 119905 119909) 119889119897 = 119885119894119873

(119904 119905 119909)


(51)


120577119894119873

(119904 119883119904(119905 119909)) = 119864 [119885

119894119873

(119904 119905 119909)10038161003816100381610038161003816F

119904] = 119885

119894119873

(119904 119905 119909)


(52)


120577119894119873

(119904 119883119904(119905 119909)) = 119885

119894119873

(119904 119905 119909)


(53)


119873rarrinfin120577119894119873


119894=1(lim

119873rarrinfin120577119894119873


120577 (119905 119909) =

infin

sum

119894=1


120577119894119873

(119905 119909)) 119890119894 (119905 119909) isin 119862

120577 (119905 119909) = 0 (119905 119909) notin 119862

(54)



119885 (120596 119904 119905 119909) =

infin

sum

119894=1


119885119894119873

(120596 119904 119905 119909)) 119890119894 (55)





3(Γ2(119905 Γ

1(119905 119909))) with


119901

P(Ω 119862

120575(119877))

Γ1(119905 119909) = 119884 (sdot 119905 119909)


119901

P(Ω 119862

120575(119877)) 997888rarr 119871

119901

(Ω 119877)

Γ2(119905 119881) = 119881 (119905)

Γ3 119871

119901

(Ω 119877) 997888rarr 119877 Γ3120585 = 119864120585

(56)


|119881(119905) minus 119880(119904)|119871119901(Ω119877)

le |119881 (119905) minus 119881 (119904)|119871119901(Ω119877)

+ 119890minus120575119901119904

|119881 minus 119880||119871

119901

P(Ω119862120575(119877))

(57)


3is continuous





119905119905ge0






119889119883119906

(119904) = 119860119883119906

(119904) 119889119904 + 119865 (119904 119883119906

119904) 119889119904

+ 119866 (119904 119883119906

119904) 119877 (119904 119883

119906

119904 119906 (119904)) 119889119904 + 119866 (119904 119883

119906

119904) 119889119882 (119904)

119904 isin [119905infin)

119883119906

119905= 119909

(58)


(119904 119905 119909) or simply by 119883119906


119869 (119906) = 119864int

infin

119905

119890minus120582119904

119892 (119904 119883119906

119904 119906 (119904)) 119889119904 (59)




120595 (119905 119909 119911) = inf 119892 (119905 119909 119906) + 119911119877 (119905 119909 119906) 119906 isin 119880

(60)


Γ (119905 119909 119911) = 119906 isin 119880 119892 (119905 119909 119906) + 119911119877 (119905 119909 119906) = 120595 (119905 119909 119911)

(61)




119898119892

119862)









119892



120582 gt (minus120575 minus 120583 +1198712

119911

2) or (minus120575 +

1198712

119877

2 (119901 minus 1))

or (1198712

119877119898119892

2 (119902 minus 119898119892)minus 120578 (119902)119898

119892)

(62)


119869 (119906) = 120592 (119905 119909) + 119864int

infin

119905

119890minus120582119904

[minus120595 (119904 119883119906

119904 120577 (119904 119883

119906

119904)) + 120577 (119904 119883

119906

119904)

times 119877 (119904 119883119906

119904 119906 (119904)) + 119892 (119904 119883

119906

119904 119906 (119904))] 119889119904

(63)


119905119905ge0



119882119906

(119904) = 119882 (119904) + int

119904

119905and119904

119877 (120590119883119906

120590 119906 (120590)) 119889120590 119904 isin [0infin)

120588 (119879) = exp(int119879

119905

minus119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882 (119904)

minus1

2int

119879

119905

1003816100381610038161003816119877 (119904 119883119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)

(64)



such that

119875119906

|F119879

= 120588 (119879) 119875|F119879

(65)


119905119905ge0


119889119883119906

(119904) = 119860119883119906

(119904) 119889119904 + 119865 (119904 119883119906

119904) 119889119904

+ 119866 (119904 119883119906

119904) 119889119882

119906

(119904) 119904 isin [119905infin)

119883119906

119905= 119909

(66)


F119906

119905119905ge0 119875

119906


119883119906

(119904) = 119890(119904minus119905)119860

119909 (0) + int

119904

119905

119890(119904minus120590)119860

119865 (120590119883119906

120590) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883119906

120590) 119889119882

119906

(120590) 119904 isin [119905infin)

119883119906

119905= 119909 isin C

119884119906

(119904) minus 119884119906

(119879) + int

119879

119904

119885119906

(120590) 119889119882119906

(120590) + 120582int

119879

119904

119884119906

(120590) 119889120590

= int

119879

119904

120595 (120590119883119906

120590 119885

119906

(120590)) 119889120590 0 le 119904 le 119879 lt infin

(67)


119884119906

(119904) + int

119879

119904

119890minus120582120590

119885119906

(120590) 119889119882 (120590)

= int

119879

119904

119890minus120582120590

[120595 (120590119883119906

120590 119885

119906

(120590))

minus119885119906

(120590) 119877 (120590119883119906

120590 119906 (120590))] 119889120590

+ 119890minus120582119879

119884119906

(119879)

(68)


119864119906

[120588(119879)minus119903

] = 119864119906

[exp 119903 (int119879

119905

119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882

119906

(119904)

minus1

2int

119879

119905

1003816100381610038161003816119877 (119904 119883119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)]

= 119864119906

[exp(int119879

119905

119903119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882

119906

(119904)

minus1

2int

119879

119905

11990321003816100381610038161003816119877 (119904 119883

119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)

times exp 119903 (119903 minus 1)2

int

119879

119905

1003816100381610038161003816119877 (119904 119883119906

119904 119906 (119904))

10038161003816100381610038162

119889119904]

le 119890(12)119903(119903minus1)119879119871

2

119877119864119906

times exp(int119879

119905

2119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882

119906

(119904)

minus1

2int

119879

119905

41003816100381610038161003816119877 (119904 119883

119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)

= 119890(12)119903(119903minus1)119879119871

2

119877

(69)

It follows that

119864(int

119879

119905

|119890minus120582119904

119885119906

(119904)|2

119889119904)

12

= 119864119906

[(int

119879

119905

10038161003816100381610038161003816119890minus120582119904

119885119906

(119904)10038161003816100381610038161003816

2

119889119904)

12

120588minus1

]

le (119864119906

int

119879

119905

10038161003816100381610038161003816119890minus120582119904

119885119906

(119904)10038161003816100381610038161003816

2

119889119904)

12

times (119864119906

120588minus2

)12

lt infin

(70)


119890minus120582119879

119864119884119906

(119879) minus 119884119906

(119905)

= 119864int

119879

119905

119890minus120582120590

[minus120595 (120590119883119906

120590 119885

119906

(120590))

+119885119906

(120590) 119877 (120590119883119906

120590 119906 (120590))] 119889120590

(71)


120575(119877)) so that

119864119906

|119884(119879 119905 119909)|119901

le 119862 exp (minus119901120575119879) (72)


119864 |119884 (119879 119905 119909)| = 119864119906

(120588minus1

(119879) |119884 (119879 119905 119909)|)

le 119864(120588minus119901(119901minus1)

)(119901minus1)119901

119864(|119884 (119879 119905 119909)|119901

)1119901

le 119862119890((1198712

1198772(119901minus1))minus120575))119879

(73)


|119883119906

119904|119902


1198641003816100381610038161003816119883

119906

119879

1003816100381610038161003816119898119892

le 119862119890(1198712

119877119898119892(2119902minus2119898

119892)minus1

minus120578(119902)119898119892)119879

(74)



119864int

infin

119905

119890minus120582120590 1003816100381610038161003816119892 (120590119883

119906

120590 119906 (120590))

1003816100381610038161003816 119889120590 lt infin (75)

Since 119884119906(119905 119905 119909) = 120592(119905 119909) and 119885119906

(119904 119905 119909) = 120577(119904 119883119906

119904(119905 119909)) 119875-


119890minus120582119879

119864119884119906

(119879) minus 119907 (119905 119909)

= 119864int

119879

119905

119890minus120582120590

[minus120595 (120590119883119906

120590 120577 (120590 119883

119906

120590))

+120577 (120590119883119906

120590) 119877 (120590119883

119906

120590 119906 (120590))] 119889120590

(76)


119890minus120582120590

119892(120590119883119906

120590 119906(120590))119889120590 and


119869 (119906) = 120592 (119905 119909)

+ 119864int

infin

119905

119890minus120582119904

[minus120595 (119904 119883119906

119904 120577 (119904 119883

119906

119904)) + 120577 (119904 119883

119906

119904) 119877

times (119904 119883119906

119904 119906 (119904)) + 119892 (119904 119883

119906

119904 119906 (119904))] 119889119904

(77)






119906 (119904) = Γ0(119904 119883

119906

119904 120577 (119904 119883

119906

119904))


(78)



119889119883 (119904)=119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904+119866 (119904 119883

119904)

times(119877 (119904 119883119904 Γ

0(119904 119883

119904 120577 (119904 119883

119904))) 119889119904 + 119889119882 (119904))

119904 isin [119905infin)

119883119905= 119909

(79)


119906 (119904) = Γ0(119904 119883

119904 120577 (119904 119883

119904)) (80)







119905119905ge0


F119905119905ge0 119875119882 119906119883

119906

) where (ΩF F119905119905ge0 119875 119882) is an




119869 (119906) = 119864int

infin

119905

119890minus120582119904

119892 (119904 119883119906

119904 119906 (119904)) 119889119904 (81)



119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904 + 119866 (119904 119883

119904)

times (119877 (119904 119883119904 Γ

0(119904 119883

119904 120577 (119904 119883

119904))) 119889119904 +119889119882 (119904))

119904 isin [119905infin)

119883119905= 119909

(82)


119905119905ge0 119875119882)



119883 (119904) = 119890(119904minus119905)119860

119909 (0) + int

119904

119905

119890(119904minus120590)119860

119865 (120590119883120590) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883120590) 119877

times (120590119883120590 Γ

0(120590119883

120590 120577 (120590 119883

120590))) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883120590) 119889119882

120590 119904 isin [119905infin)

(83)

119883119905= 119909 (84)




119905119905ge0 119875119882)We define

120588 (119879) = exp(int119879

119905

minus119877lowast

(120590 119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590))) 119889119882 (120590)

minus1

2int

119879

119905

1003816100381610038161003816119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590)))10038161003816100381610038162

119889120590)

(85)


1198820

(119904) = 119882 (119904) + int

119904

119905and119904

119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590))) 119889120590


(86)


1198750

|F119879

= 120588 (119879) 119875|F119879

(87)

Let us denote by F0

119905119905ge0


[0infin) F0

119905119905ge0 119875

0


119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119905 119883119904) 119889119904

+ 119866 (119904 119883119904) 119889119882

0

(119904) 119904 isin [119905infin)

119883119905= 119909

120588 (119879) = exp(int119879

119905

minus119877lowast

(120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590)) 119889119882

0

(120590)

+ 12int

119879

119905

1003816100381610038161003816119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590)))10038161003816100381610038162

119889120590)

(88)




119905119905ge0


119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904

+ 119866 (119904 119883119904) 119889119882 (119904) 119904 isin [119905infin)

119883119905= 119909

(89)


1198821

(119904) = 119882 (119904) minus int

119904

119905and119904

119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590))) 119889120590

(90)





0



119906

) one has

119869 (119906 119905 119909) ge 120592 (119905 119909) (91)


119906 (119904) = Γ0(119904 119883

119906

119904 120577 (119904 119883

119906

119904))

119875 minus 119886119904 119891119900119903 119886119897119898119900119904119905 119890119907119890119903119910 119904 isin [119905infin)

(92)



119906 (119904) = Γ0(119904 119883

119904 120577 (119904 119883

119904)) (93)


6 Applications



119889119911119906

(119905 120585) = Δ119911119906

(119905 120585) 119889119905 + 119891 (119905 119911119906

119905(120585)) 119889119905

+

119889

sum

119894=1

119892119894(119905 119911

119906

119905(120585)) [119903

119894

(120585) 119906119894

(119905) 119889119905 + 119889119882119894

(119905)]

119911119906

0(120579 120585) = 119909 (120579 120585) 120585 isin 119861 120579 isin [minus1 0]

119911119906

(119905 120585) = 0 119905 isin [0infin) 120585 isin 120597119861

(94)

Here119882 = (1198821

1198822

119882119889



119867 = 1198712


119865 (119905 119909) (120585) = 119891 (119905 119909 (120585))

(119866 (119905 119909) 119911) (120585) =

119889

sum

119894=1

119892119894(119905 119909 (120585)) 119911

119894

(120585)


(95)


(119861) withdomain11988222

(119861)⋂11988212


Hypothesis 1 holds



119869 (119906) = 119864int

infin

0

119890minus120582119905

[int119861

120590 (120585 119911119906

119905(120585)) 119889120585 + 119906

2

(119905)] 119889119905 (96)



119861

120590(119905 119910(120585) 119906)119889120585 + 1199062 and 119877(119910 119906) = (int

119861

1199031

(120585)1199061

119889120585

int119861

1199032

(120585)1199062

119889120585 int119861

119903119889

(120585)119906119889

119889120585) for 119910 isin 119862([minus1 0]119867) 119906 =

(1199061

1199062

119906119889


0



119905119905ge0 119875119882 119906 119911

sdot(sdot)) and

setting

119906 (119904) = Γ0(119904 119911

119904(sdot) 120577 (119904 119911

119904(sdot))) (97)


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014






H119902


Φ(119883sdot 119905 119909)

119904(119897) = 119890

(119904+119897minus119905)119860

119909 (0) + int

119904+119897

119905

119890(119904+119897minus120590)119860

119865 (120590119883120590) 119889120590

+ int

119904+119897

119905

119890(119904+119897minus120590)119860

119866 (120590119883120590) 119889119882 (120590)


Φ(119883sdot 119905 119909)

119904(119897) = 119909 (119904 + 119897 minus 119905)


(16)


120578(119905)


1

sdot 119905 119909) minus Φ (119883

1

sdot 119905 119909)

10038161003816100381610038161003816H119902

120578(119905)

le 119888100381610038161003816100381610038161198831

sdotminus 119883

2

sdot

10038161003816100381610038161003816H119902

120578(119905)

1198831

sdot 119883

2

sdotisinH

119902

120578(119905)

(17)


120572= int

s120590

(119904 minus 119903)120572minus1

(119903 minus 120590)minus120572


Φ(119883sdot 0 119909)

119904(119897) = 119890

(119904+119897)119860

119909 (0)

+ 119888120572int

119904+119897

0

int

119904+119897

120590

(119904 + 119897 minus 119903)120572minus1

(119903 minus 120590)minus120572

times 119890(119904+119897minus119903)119860

119890(119903minus120590)119860

119889119903119866 (120590119883120590) 119889119882 (120590)

= 119890(119904+119897)119860

119909 (0) + Φ1015840

(119883119904) (119897)


Φ(119883sdot 0 119909)

119904(119897) = 119909 (119904 + 119897)


(18)

where

Φ1015840

(119883sdot)119904(119897) = 119888

120572int

119904+119897

0

(119904 + 119897 minus 119903)120572minus1

119890(119904+119897minus119903)119860

119884 (119903) 119889119903

119884 (119903) = int

119903

0

(119903 minus 120590)minus120572

119890(119903minus120590)119860

119866 (120590119883120590) 119889119882 (120590)

(19)


|119890(119904+119897)119860

119909(0)| le 119872119890120596119904

|119909|119862 the process 119890(119904+sdot)119860

119909(0) 119904 ge 0 belongs to H119902


estimate Φ1015840

(119883sdot) where

10038161003816100381610038161003816Φ1015840

(119883sdot)119904(119897)10038161003816100381610038161003816le 119888

120572int

119904+119897

0

(119904 + 119897 minus 119903)120572minus1

119872119890(119904+119897minus119903)120596

|119884 (119903)| 119889119903

(20)


11989011990212057811990410038161003816100381610038161003816Φ1015840

(119883sdot)119904

10038161003816100381610038161003816

119902

le 119888119902

120572119872

119902 supminus120591le119897le0

119890119902120578119904

(int

119904+119897

0

(119904+119897minus 119903)120572minus1

119890120596(119904+119897minus119903)

|119884 (119903)| 119889119903)

119902

le 119888119902

120572119872

119902 supminus120591le119897le0

(int

119904+119897

0

(119904 + 119897 minus 119903)120572minus1

119890((120596+120578)119902

1015840

)(119904+119897minus119903)

times119890((120596+120578)119902)(119904minus119903)

119890120578119903

|119884 (119903)| 119889119903)

119902

le 119888119902

120572119872

119902 supminus120591le119897le0

(int

119904+119897

0

119890(120578+120596)(119904+119897minus119903)

(119904 + 119897 minus 119903)(120572minus1)119902

1015840

119889119903)

1199021199021015840

times int

119904+119897

0

119890(120578+120596)(119904minus119903)

119890119902120578119903

|119884 (119903)|119902

119889119903

le 119888119902

120572119872

119902

(int

119904

0

119890(120578+120596)119903

1199031199021015840

(120572minus1)

119889119903)

1199021199021015840

times int

119904

0

119890(120578+120596)(119904minus119903)

119890119902120578119903

|119884 (119903)|119902

119889119903

(21)


int

infin

0

11989011990212057811990410038161003816100381610038161003816Φ1015840

(119883sdot)119904

10038161003816100381610038161003816

119902

119889119904 le 119888119902

120572119872

119902

(int

infin

0

119890(120578+120596)119904

1199041199021015840

(120572minus1)

119889119904)

1199021199021015840

times int

infin

0

119890(120578+120596)119904

119889119904int

infin

0

119890119902120578119904

|119884 (119904)|119902

119889119904

(22)


(119883sdot)10038161003816100381610038161003816119871119902

P(Ω119871119902

120578(C))

le 119888120572119872|119884|

119871119902

P(Ω119871119902

120578(119867))

times (int

infin

0

119890(120578+120596)119904

1199041199021015840

(120572minus1)

119889119904)

11199021015840

times (int

infin

0

119890(120578+120596)119904

119889119904)

1119902

(23)


10038161003816100381610038161003816119890120578(119904+119897)

Φ1015840

(119883sdot)119904(119897)10038161003816100381610038161003816le 119888

120572119872(int

119904+119897

0

119903(120572minus1)119902

1015840

119890(120596+120578)119903119902

1015840

119889119903)

11199021015840

times (int

119904+119897

0

119890120578119903119902

|119884 (119903)|119902

119889119903)

1119902

10038161003816100381610038161003816119890120578119904

Φ1015840

(119883sdot)119904

10038161003816100381610038161003816le 119888

120572119872(int

119904

0

119903(120572minus1)119902

1015840

119890(120596+120578)119903119902

1015840

119889119903)

11199021015840

times (int

119904

0

119890120578119903119902

|119884(119903)|119902

119889119903)

1119902

(24)



(119883sdot)10038161003816100381610038161003816119871119902

P(Ω119862120578(C))

le 119888120572119872|119884|

119871119902

P(Ω119871119902

120578(119867))

times (int

infin

0

119903(120572minus1)119902

1015840

119890(120596+120578)119903119902

1015840

119889119903)

11199021015840

(25)



119864|119884 (119903)|119902

le 119888119902119864(int

119903

0


119866 (120590119883120590)10038161003816100381610038161003816

2

1198712(Ξ119867)

119889120590)

1199022

le 119871119902

119888119902119864

times (int

119903

0


1198902120596(119903minus120590)

(1 +1003816100381610038161003816119883120590

10038161003816100381610038162

119862) 119889120590)

1199022

(26)

which implies that

[119864|119884 (119903)|119902

]2119902

le 1198712

1198882119902

119902int

119903

0


times 1198902120596(119903minus120590)

[119864(1 +1003816100381610038161003816119883120590

1003816100381610038161003816119862)119902

]2119902

119889120590

(27)

so that

1198902120578119903

[119864|119884 (119903)|119902

]2119902

le 1198621int

119903

0


1198902(120596+120578)(119903minus120590)

1198902120578120590

119889120590

+ 1198622int

119903

0


1198902(120596+120578)(119903minus120590)

times 1198902120578120590

[1198641003816100381610038161003816119883120590

1003816100381610038161003816119902

119862]2119902

119889120590

(28)




int

infin

0

119890119902120578119903

119864|119884 (119903)|119902

119889119904le 1198621(int

infin

0

119904minus2120572minus2120574

1198902(120596+120578)119904

119889119904)

1199022

int

infin

0

119890119902120578119904

119889119904

+ 1198622(int

infin

0

119904minus2120572minus2120574

1198902(120596+120578)119904

119889119904)

1199022

times int

infin

0

119890119902120578119904

1198641003816100381610038161003816119883119904

1003816100381610038161003816119902

119862119889119904

(29)

This shows that |119884|119871119902

P(Ω119871119902

120578(119867))


If 1198831

sdot 119883

2


120578and 1198841 1198842 are


minus 119884210038161003816100381610038161003816119871119902

P(Ω119871119902

120578(119867))

le 1198711198881119902

120572

100381610038161003816100381610038161198831

sdotminus 119883

2

sdot

10038161003816100381610038161003816119871119902

P(Ω119871119902

120578(C))

times (int

infin

0

119904minus2120572minus2120574

1198902(120596+120578)119904

119889119904)

12

(30)


1015840




119878 (119904) = 119890119904119860

for 119904 ge 0 119878 (119904) = 119868 for 119904 lt 0 (31)



+ int

119904

0

119868[119905infin)

(120590) 119878 (119904 minus 120590) 119865 (120590119883120590) 119889120590

+ int

119904

0

119868[119905infin)

(120590) 119878 (119904 minus 120590)

times 119866 (120590119883120590) 119889119882 (120590) 119904 isin [0infin)


(32)


119902





1003816100381610038161003816ℎ (1199091 119910) minus ℎ (1199092 119910)1003816100381610038161003816 le 120572

10038161003816100381610038161199091 minus 11990921003816100381610038161003816 (33)


2isin 119861 y isin 119863 Let 120601(119910)








120578times


120578as follows

Φ(119883sdot 119905 119909)

119904(119897) = 119878 (119904 + 119897 minus 119905) 119909 (0)

+ int

119904+119897

0

119868[119905infin)

(120590) 119878 (119904 + 119897 minus 120590) 119865 (120590119883120590) 119889120590

+ int

119904+119897

0

119868[119905infin)

(120590) 119878 (119904 + 119897 minus 120590)

times 119866 (120590119883120590) 119889119882 (120590)


Φ(119883sdot 119905 119909)

119904(119897) = 119909 ((119904 + 119897 minus 119905) or (minus120591))


(34)








sdot it is


toH119902




119905)

and H119902




the norm

|119909|119902

= 119864|119909|119902

119862 (35)

is finite



119883 (119904) = 119890(119904minus119905)119860

119909 (0) + int

119904

119905

119890(119904minus120590)119860

119865 (120590119883120590) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883120590) 119889119882 (120590) 119904 isin [119905infin)

119883119905= 119909 isin C

119884 (119904) minus 119884 (119879) + int

119879

119904

119885 (120590) 119889119882 (120590) + 120582int

119879

119904

119884 (120590) 119889120590

= int

119879

119904

120595 (120590119883120590 119884 (120590) 119885 (120590)) 119889120590 0 le 119904 le 119879 lt infin

(36)






119910 119871

119911gt 0

120583 isin 119877 and119898 ge 11003816100381610038161003816120595 (119904 119909 1199101 1199111) minus 120595 (119904 119909 1199102 1199112)

1003816100381610038161003816

le 119871119910

10038161003816100381610038161199101 minus 11991021003816100381610038161003816 + 119871119911

10038161003816100381610038161199111 minus 11991121003816100381610038161003816

1003816100381610038161003816120595 (119904 119909 119910 119911)1003816100381610038161003816 le 119871 (1 + |119909|

119898

119862+10038161003816100381610038161199101003816100381610038161003816 + |119911|)

⟨120595 (119904 119909 1199101 119911) minus 120595 (119904 119909 119910

2 119911) 119910

1minus 119910

2⟩119870ge 120583

10038161003816100381610038161199101 minus 119910210038161003816100381610038162

(37)


2isin 119870 119911 119911

1 and 119911

2isin




equation

119884 (119904) minus 119884 (119879) + int

119879

119904

119885 (120590) 119889119882 (120590) + 120582int

119879

119904

119884 (120590) 119889120590

= int

119879

119904

120595 (120590119883120590 119884 (120590) 119885 (120590)) 119889120590 0 le 119904 le 119879 lt infin

(38)



sdotis a



119902 ge 119898119901 120578 gt120575

119898 (39)



119902

P(Ω 119871

119902


2

1199112))


by (119884(119883sdot)(119904) 119885(119883

sdot)(119904)) 119904 ge 0

(ii) The estimate

119864sup119904ge0

(119884 (119883sdot) (119904))

119901

119890119901120575119904

+ 119864(int

infin

0

11989021205751205901003816100381610038161003816119884(119883sdot

)(120590)10038161003816100381610038162

119889120590)

1199012

+ 119864(int

infin

0

11989021205751205901003816100381610038161003816119885 (119883sdot

) (120590)10038161003816100381610038162

119889120590)

1199012

le 119888(1 +1003816100381610038161003816119883sdot

1003816100381610038161003816119898

119871119902

P(Ω119871119902

120578(C))

)119901

(40)



119871119901

P(Ω 119862

120575(119870))


sdot) 119885(119883


119871119902

P(Ω 119871

119902

120578(C)) toK119901

120575 and 119883

sdotrarr 119884(119883

sdot) is continuous

from 119871119902

P(Ω 119871

119902

120578(C)) to 119871119901

P(Ω 119862

120575(119870))


P(Ω 119871

119902


119871119902

P(Ω 119862

120578(C))






1015840

= minus(120575 + 120583 minus (1198712



120578(119902)times K

119901



119871119901

P(Ω 119862





120575(119877))


sdot(119905 119909) isin H

119902

120578(119902)



toH119902

120578(119902)


119901

120575of the


sdot) 119885(119883

sdot)) is


120578(119902)to K

119901

120575while X

sdotrarr (Y(X

sdot)) is


120578(119902)to119871119901

P(Ω 119862



120578(119902)times K

119901

120575is the unique



119902

(ΩCF119905) toK119901

120575(119905)








119883119897(119904 119883

119904(119905 119909)) = 119883

119897(119905 119909) 119897 isin [119904infin) (41)



119884 (119897 119904 119883119904(119905 119909)) = 119884 (119897 119905 119909) for 119897 isin [119904infin)

119885 (119897 119904 119883119904(119905 119909)) = 119885 (119897 119905 119909) for aa 119897 isin [119904infin)

(42)






Let now 119891 isin 119871119902


119899 119899 isin 119873 such that


Hence 119891119899



this section



= 119871(Ξ 119877) =


(119883(119905 119909) 119884(119905 119909) 119885(119905 119909)) of (36) satisfies

119884 (119904 119905 119909) = 120592 (119904 119883119904(119905 119909)) 119885 (119904 119905 119909) = 120577 (119904 119883

119904(119905 119909))


(44)



=


1lt 119905

2lt infin Δ gt

0 and 1199091 119909


100381610038161003816100381610038161003816100381610038161003816

119864 int

1199051+Δ

1199051

119885119894119873

(119904 1199051 119909

1) 119889119904 minus 119864int

1199052+Δ

1199052

119885119894119873

(119904 1199052 119909

2) 119889119904

100381610038161003816100381610038161003816100381610038161003816

le 119864int

1199052

1199051

10038161003816100381610038161003816119885119894119873

(119904 1199051 119909

1)10038161003816100381610038161003816119889119904

+ 119864int

1199051+Δ

1199052

10038161003816100381610038161003816119885119894119873

(119904 1199051 119909

1) minus 119885

119894119873

(119904 1199052 119909

2)10038161003816100381610038161003816119889119904

+ 119864int

1199052+Δ

1199051+Δ

10038161003816100381610038161003816119885119894119873

(119904 1199052 119909

2)10038161003816100381610038161003816119889119904


le 211987310038161003816100381610038161199052 minus 1199051

1003816100381610038161003816 + Δ12

119890minus120575(1199051+Δ)

times(119864(int

infin

0

119890212057511990410038161003816100381610038161003816119885119894119873

(119904 1199051 119909

1)minus119885

119894119873

(119904 1199052 119909

2)10038161003816100381610038161003816

2

119889119904)

1199012

)

1119901

le 211987310038161003816100381610038161199052 minus 1199051

1003816100381610038161003816 + Δ12

119890minus120575(1199051+Δ)

times (119864(int

infin

0

11989021205751199041003816100381610038161003816119885 (119904 1199051 1199091)minus119885 (119904 1199052 1199092)

10038161003816100381610038162

119889119904)

1199012

)

1119901

(45)


119905

119885119894119873


119864int119905+Δ

119905

119864119885119894119873


(ΩCF119905) to 119877


120577119894119873


119899119864int

119905+(1119899)

119905

119885119894119873

(119904 119905 119909) 119889119904


(46)


denote 119864[119885119894119873

(119897 119904 119910)]|119910=119883119904(119905119909)


119904(119905 119909) with the map 119910 rarr 119864[119885

119894119873


119904-


119891119898 Ω 997888rarr C 119891

119898=

119873119898

sum

119896=1

ℎ(119898)

119896119868119891119898=ℎ(119898)

119896 119873

119898isin 119873 (47)

where ℎ(119898)1 ℎ

(119898)


119873119898

119896=1119891

119898=

ℎ(119898)

119896 such that

1003816100381610038161003816119891119898 (120596) minus 119883119904(120596)



int119861

int

119904+(1119899)

119904

119885119894119873

(119897 119905 119909) 119889119897119889119875

= int119861

int

119904+(1119899)

119904

119885119894119873

(119897 119904 119883119904) 119889119897119889119875

= 119864119868119861int

119904+(1119899)

119904

119885119894119873

(119897 119904 119883119904) 119889119897


119864(119868119861int

119904+(1119899)

119904

119885119894119873

(119897 119904 119891119898) 119889119897)


119873119898

sum

119896=1

119864(119868119861119868119891119898=ℎ(119898)

119896int

119904+(1119899)

119904

119885119894119873

(119897 119904 ℎ(119898)

119896) 119889119897)


119864(119868119861

119873119898

sum

119896=1

119868119891119898=ℎ(119898)

119896)119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 ℎ(119898)

119896) 119889119897


119864119868119861(119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 119910) 119889119897|119910=119891119898

)


int119861

(119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 119910) 119889119897|119910=119891119898

)119889119875

= int119861

(119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 119910) 119889119897|119910=119883119904

)119889119875

(49)

and we get that

120577119894119873

(119904 119883119904(119905 119909)) = lim inf

119899rarrinfin

119899

times [119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 119910) 119889119897|119910=119883119904(119905119909)

]


119899119864[int

119904+(1119899)

119904

119885119894119873

(119897 119905 119909) 119889119897

100381610038161003816100381610038161003816100381610038161003816

F119904]

119875-as(50)



lim119899rarrinfin

119899int

119904+(1119899)

119904

119885119894119873

(119897 119905 119909) 119889119897 = 119885119894119873

(119904 119905 119909)


(51)


120577119894119873

(119904 119883119904(119905 119909)) = 119864 [119885

119894119873

(119904 119905 119909)10038161003816100381610038161003816F

119904] = 119885

119894119873

(119904 119905 119909)


(52)


120577119894119873

(119904 119883119904(119905 119909)) = 119885

119894119873

(119904 119905 119909)


(53)


119873rarrinfin120577119894119873


119894=1(lim

119873rarrinfin120577119894119873


120577 (119905 119909) =

infin

sum

119894=1


120577119894119873

(119905 119909)) 119890119894 (119905 119909) isin 119862

120577 (119905 119909) = 0 (119905 119909) notin 119862

(54)



119885 (120596 119904 119905 119909) =

infin

sum

119894=1


119885119894119873

(120596 119904 119905 119909)) 119890119894 (55)





3(Γ2(119905 Γ

1(119905 119909))) with


119901

P(Ω 119862

120575(119877))

Γ1(119905 119909) = 119884 (sdot 119905 119909)


119901

P(Ω 119862

120575(119877)) 997888rarr 119871

119901

(Ω 119877)

Γ2(119905 119881) = 119881 (119905)

Γ3 119871

119901

(Ω 119877) 997888rarr 119877 Γ3120585 = 119864120585

(56)


|119881(119905) minus 119880(119904)|119871119901(Ω119877)

le |119881 (119905) minus 119881 (119904)|119871119901(Ω119877)

+ 119890minus120575119901119904

|119881 minus 119880||119871

119901

P(Ω119862120575(119877))

(57)


3is continuous





119905119905ge0






119889119883119906

(119904) = 119860119883119906

(119904) 119889119904 + 119865 (119904 119883119906

119904) 119889119904

+ 119866 (119904 119883119906

119904) 119877 (119904 119883

119906

119904 119906 (119904)) 119889119904 + 119866 (119904 119883

119906

119904) 119889119882 (119904)

119904 isin [119905infin)

119883119906

119905= 119909

(58)


(119904 119905 119909) or simply by 119883119906


119869 (119906) = 119864int

infin

119905

119890minus120582119904

119892 (119904 119883119906

119904 119906 (119904)) 119889119904 (59)




120595 (119905 119909 119911) = inf 119892 (119905 119909 119906) + 119911119877 (119905 119909 119906) 119906 isin 119880

(60)


Γ (119905 119909 119911) = 119906 isin 119880 119892 (119905 119909 119906) + 119911119877 (119905 119909 119906) = 120595 (119905 119909 119911)

(61)




119898119892

119862)









119892



120582 gt (minus120575 minus 120583 +1198712

119911

2) or (minus120575 +

1198712

119877

2 (119901 minus 1))

or (1198712

119877119898119892

2 (119902 minus 119898119892)minus 120578 (119902)119898

119892)

(62)


119869 (119906) = 120592 (119905 119909) + 119864int

infin

119905

119890minus120582119904

[minus120595 (119904 119883119906

119904 120577 (119904 119883

119906

119904)) + 120577 (119904 119883

119906

119904)

times 119877 (119904 119883119906

119904 119906 (119904)) + 119892 (119904 119883

119906

119904 119906 (119904))] 119889119904

(63)


119905119905ge0



119882119906

(119904) = 119882 (119904) + int

119904

119905and119904

119877 (120590119883119906

120590 119906 (120590)) 119889120590 119904 isin [0infin)

120588 (119879) = exp(int119879

119905

minus119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882 (119904)

minus1

2int

119879

119905

1003816100381610038161003816119877 (119904 119883119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)

(64)



such that

119875119906

|F119879

= 120588 (119879) 119875|F119879

(65)


119905119905ge0


119889119883119906

(119904) = 119860119883119906

(119904) 119889119904 + 119865 (119904 119883119906

119904) 119889119904

+ 119866 (119904 119883119906

119904) 119889119882

119906

(119904) 119904 isin [119905infin)

119883119906

119905= 119909

(66)


F119906

119905119905ge0 119875

119906


119883119906

(119904) = 119890(119904minus119905)119860

119909 (0) + int

119904

119905

119890(119904minus120590)119860

119865 (120590119883119906

120590) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883119906

120590) 119889119882

119906

(120590) 119904 isin [119905infin)

119883119906

119905= 119909 isin C

119884119906

(119904) minus 119884119906

(119879) + int

119879

119904

119885119906

(120590) 119889119882119906

(120590) + 120582int

119879

119904

119884119906

(120590) 119889120590

= int

119879

119904

120595 (120590119883119906

120590 119885

119906

(120590)) 119889120590 0 le 119904 le 119879 lt infin

(67)


119884119906

(119904) + int

119879

119904

119890minus120582120590

119885119906

(120590) 119889119882 (120590)

= int

119879

119904

119890minus120582120590

[120595 (120590119883119906

120590 119885

119906

(120590))

minus119885119906

(120590) 119877 (120590119883119906

120590 119906 (120590))] 119889120590

+ 119890minus120582119879

119884119906

(119879)

(68)


119864119906

[120588(119879)minus119903

] = 119864119906

[exp 119903 (int119879

119905

119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882

119906

(119904)

minus1

2int

119879

119905

1003816100381610038161003816119877 (119904 119883119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)]

= 119864119906

[exp(int119879

119905

119903119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882

119906

(119904)

minus1

2int

119879

119905

11990321003816100381610038161003816119877 (119904 119883

119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)

times exp 119903 (119903 minus 1)2

int

119879

119905

1003816100381610038161003816119877 (119904 119883119906

119904 119906 (119904))

10038161003816100381610038162

119889119904]

le 119890(12)119903(119903minus1)119879119871

2

119877119864119906

times exp(int119879

119905

2119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882

119906

(119904)

minus1

2int

119879

119905

41003816100381610038161003816119877 (119904 119883

119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)

= 119890(12)119903(119903minus1)119879119871

2

119877

(69)

It follows that

119864(int

119879

119905

|119890minus120582119904

119885119906

(119904)|2

119889119904)

12

= 119864119906

[(int

119879

119905

10038161003816100381610038161003816119890minus120582119904

119885119906

(119904)10038161003816100381610038161003816

2

119889119904)

12

120588minus1

]

le (119864119906

int

119879

119905

10038161003816100381610038161003816119890minus120582119904

119885119906

(119904)10038161003816100381610038161003816

2

119889119904)

12

times (119864119906

120588minus2

)12

lt infin

(70)


119890minus120582119879

119864119884119906

(119879) minus 119884119906

(119905)

= 119864int

119879

119905

119890minus120582120590

[minus120595 (120590119883119906

120590 119885

119906

(120590))

+119885119906

(120590) 119877 (120590119883119906

120590 119906 (120590))] 119889120590

(71)


120575(119877)) so that

119864119906

|119884(119879 119905 119909)|119901

le 119862 exp (minus119901120575119879) (72)


119864 |119884 (119879 119905 119909)| = 119864119906

(120588minus1

(119879) |119884 (119879 119905 119909)|)

le 119864(120588minus119901(119901minus1)

)(119901minus1)119901

119864(|119884 (119879 119905 119909)|119901

)1119901

le 119862119890((1198712

1198772(119901minus1))minus120575))119879

(73)


|119883119906

119904|119902


1198641003816100381610038161003816119883

119906

119879

1003816100381610038161003816119898119892

le 119862119890(1198712

119877119898119892(2119902minus2119898

119892)minus1

minus120578(119902)119898119892)119879

(74)



119864int

infin

119905

119890minus120582120590 1003816100381610038161003816119892 (120590119883

119906

120590 119906 (120590))

1003816100381610038161003816 119889120590 lt infin (75)

Since 119884119906(119905 119905 119909) = 120592(119905 119909) and 119885119906

(119904 119905 119909) = 120577(119904 119883119906

119904(119905 119909)) 119875-


119890minus120582119879

119864119884119906

(119879) minus 119907 (119905 119909)

= 119864int

119879

119905

119890minus120582120590

[minus120595 (120590119883119906

120590 120577 (120590 119883

119906

120590))

+120577 (120590119883119906

120590) 119877 (120590119883

119906

120590 119906 (120590))] 119889120590

(76)


119890minus120582120590

119892(120590119883119906

120590 119906(120590))119889120590 and


119869 (119906) = 120592 (119905 119909)

+ 119864int

infin

119905

119890minus120582119904

[minus120595 (119904 119883119906

119904 120577 (119904 119883

119906

119904)) + 120577 (119904 119883

119906

119904) 119877

times (119904 119883119906

119904 119906 (119904)) + 119892 (119904 119883

119906

119904 119906 (119904))] 119889119904

(77)






119906 (119904) = Γ0(119904 119883

119906

119904 120577 (119904 119883

119906

119904))


(78)



119889119883 (119904)=119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904+119866 (119904 119883

119904)

times(119877 (119904 119883119904 Γ

0(119904 119883

119904 120577 (119904 119883

119904))) 119889119904 + 119889119882 (119904))

119904 isin [119905infin)

119883119905= 119909

(79)


119906 (119904) = Γ0(119904 119883

119904 120577 (119904 119883

119904)) (80)







119905119905ge0


F119905119905ge0 119875119882 119906119883

119906

) where (ΩF F119905119905ge0 119875 119882) is an




119869 (119906) = 119864int

infin

119905

119890minus120582119904

119892 (119904 119883119906

119904 119906 (119904)) 119889119904 (81)



119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904 + 119866 (119904 119883

119904)

times (119877 (119904 119883119904 Γ

0(119904 119883

119904 120577 (119904 119883

119904))) 119889119904 +119889119882 (119904))

119904 isin [119905infin)

119883119905= 119909

(82)


119905119905ge0 119875119882)



119883 (119904) = 119890(119904minus119905)119860

119909 (0) + int

119904

119905

119890(119904minus120590)119860

119865 (120590119883120590) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883120590) 119877

times (120590119883120590 Γ

0(120590119883

120590 120577 (120590 119883

120590))) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883120590) 119889119882

120590 119904 isin [119905infin)

(83)

119883119905= 119909 (84)




119905119905ge0 119875119882)We define

120588 (119879) = exp(int119879

119905

minus119877lowast

(120590 119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590))) 119889119882 (120590)

minus1

2int

119879

119905

1003816100381610038161003816119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590)))10038161003816100381610038162

119889120590)

(85)


1198820

(119904) = 119882 (119904) + int

119904

119905and119904

119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590))) 119889120590


(86)


1198750

|F119879

= 120588 (119879) 119875|F119879

(87)

Let us denote by F0

119905119905ge0


[0infin) F0

119905119905ge0 119875

0


119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119905 119883119904) 119889119904

+ 119866 (119904 119883119904) 119889119882

0

(119904) 119904 isin [119905infin)

119883119905= 119909

120588 (119879) = exp(int119879

119905

minus119877lowast

(120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590)) 119889119882

0

(120590)

+ 12int

119879

119905

1003816100381610038161003816119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590)))10038161003816100381610038162

119889120590)

(88)




119905119905ge0


119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904

+ 119866 (119904 119883119904) 119889119882 (119904) 119904 isin [119905infin)

119883119905= 119909

(89)


1198821

(119904) = 119882 (119904) minus int

119904

119905and119904

119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590))) 119889120590

(90)





0



119906

) one has

119869 (119906 119905 119909) ge 120592 (119905 119909) (91)


119906 (119904) = Γ0(119904 119883

119906

119904 120577 (119904 119883

119906

119904))

119875 minus 119886119904 119891119900119903 119886119897119898119900119904119905 119890119907119890119903119910 119904 isin [119905infin)

(92)



119906 (119904) = Γ0(119904 119883

119904 120577 (119904 119883

119904)) (93)


6 Applications



119889119911119906

(119905 120585) = Δ119911119906

(119905 120585) 119889119905 + 119891 (119905 119911119906

119905(120585)) 119889119905

+

119889

sum

119894=1

119892119894(119905 119911

119906

119905(120585)) [119903

119894

(120585) 119906119894

(119905) 119889119905 + 119889119882119894

(119905)]

119911119906

0(120579 120585) = 119909 (120579 120585) 120585 isin 119861 120579 isin [minus1 0]

119911119906

(119905 120585) = 0 119905 isin [0infin) 120585 isin 120597119861

(94)

Here119882 = (1198821

1198822

119882119889



119867 = 1198712


119865 (119905 119909) (120585) = 119891 (119905 119909 (120585))

(119866 (119905 119909) 119911) (120585) =

119889

sum

119894=1

119892119894(119905 119909 (120585)) 119911

119894

(120585)


(95)


(119861) withdomain11988222

(119861)⋂11988212


Hypothesis 1 holds



119869 (119906) = 119864int

infin

0

119890minus120582119905

[int119861

120590 (120585 119911119906

119905(120585)) 119889120585 + 119906

2

(119905)] 119889119905 (96)



119861

120590(119905 119910(120585) 119906)119889120585 + 1199062 and 119877(119910 119906) = (int

119861

1199031

(120585)1199061

119889120585

int119861

1199032

(120585)1199062

119889120585 int119861

119903119889

(120585)119906119889

119889120585) for 119910 isin 119862([minus1 0]119867) 119906 =

(1199061

1199062

119906119889


0



119905119905ge0 119875119882 119906 119911

sdot(sdot)) and

setting

119906 (119904) = Γ0(119904 119911

119904(sdot) 120577 (119904 119911

119904(sdot))) (97)


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014





(119883sdot)10038161003816100381610038161003816119871119902

P(Ω119862120578(C))

le 119888120572119872|119884|

119871119902

P(Ω119871119902

120578(119867))

times (int

infin

0

119903(120572minus1)119902

1015840

119890(120596+120578)119903119902

1015840

119889119903)

11199021015840

(25)



119864|119884 (119903)|119902

le 119888119902119864(int

119903

0


119866 (120590119883120590)10038161003816100381610038161003816

2

1198712(Ξ119867)

119889120590)

1199022

le 119871119902

119888119902119864

times (int

119903

0


1198902120596(119903minus120590)

(1 +1003816100381610038161003816119883120590

10038161003816100381610038162

119862) 119889120590)

1199022

(26)

which implies that

[119864|119884 (119903)|119902

]2119902

le 1198712

1198882119902

119902int

119903

0


times 1198902120596(119903minus120590)

[119864(1 +1003816100381610038161003816119883120590

1003816100381610038161003816119862)119902

]2119902

119889120590

(27)

so that

1198902120578119903

[119864|119884 (119903)|119902

]2119902

le 1198621int

119903

0


1198902(120596+120578)(119903minus120590)

1198902120578120590

119889120590

+ 1198622int

119903

0


1198902(120596+120578)(119903minus120590)

times 1198902120578120590

[1198641003816100381610038161003816119883120590

1003816100381610038161003816119902

119862]2119902

119889120590

(28)




int

infin

0

119890119902120578119903

119864|119884 (119903)|119902

119889119904le 1198621(int

infin

0

119904minus2120572minus2120574

1198902(120596+120578)119904

119889119904)

1199022

int

infin

0

119890119902120578119904

119889119904

+ 1198622(int

infin

0

119904minus2120572minus2120574

1198902(120596+120578)119904

119889119904)

1199022

times int

infin

0

119890119902120578119904

1198641003816100381610038161003816119883119904

1003816100381610038161003816119902

119862119889119904

(29)

This shows that |119884|119871119902

P(Ω119871119902

120578(119867))


If 1198831

sdot 119883

2


120578and 1198841 1198842 are


minus 119884210038161003816100381610038161003816119871119902

P(Ω119871119902

120578(119867))

le 1198711198881119902

120572

100381610038161003816100381610038161198831

sdotminus 119883

2

sdot

10038161003816100381610038161003816119871119902

P(Ω119871119902

120578(C))

times (int

infin

0

119904minus2120572minus2120574

1198902(120596+120578)119904

119889119904)

12

(30)


1015840




119878 (119904) = 119890119904119860

for 119904 ge 0 119878 (119904) = 119868 for 119904 lt 0 (31)



+ int

119904

0

119868[119905infin)

(120590) 119878 (119904 minus 120590) 119865 (120590119883120590) 119889120590

+ int

119904

0

119868[119905infin)

(120590) 119878 (119904 minus 120590)

times 119866 (120590119883120590) 119889119882 (120590) 119904 isin [0infin)


(32)


119902





1003816100381610038161003816ℎ (1199091 119910) minus ℎ (1199092 119910)1003816100381610038161003816 le 120572

10038161003816100381610038161199091 minus 11990921003816100381610038161003816 (33)


2isin 119861 y isin 119863 Let 120601(119910)








120578times


120578as follows

Φ(119883sdot 119905 119909)

119904(119897) = 119878 (119904 + 119897 minus 119905) 119909 (0)

+ int

119904+119897

0

119868[119905infin)

(120590) 119878 (119904 + 119897 minus 120590) 119865 (120590119883120590) 119889120590

+ int

119904+119897

0

119868[119905infin)

(120590) 119878 (119904 + 119897 minus 120590)

times 119866 (120590119883120590) 119889119882 (120590)


Φ(119883sdot 119905 119909)

119904(119897) = 119909 ((119904 + 119897 minus 119905) or (minus120591))


(34)








sdot it is


toH119902




119905)

and H119902




the norm

|119909|119902

= 119864|119909|119902

119862 (35)

is finite



119883 (119904) = 119890(119904minus119905)119860

119909 (0) + int

119904

119905

119890(119904minus120590)119860

119865 (120590119883120590) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883120590) 119889119882 (120590) 119904 isin [119905infin)

119883119905= 119909 isin C

119884 (119904) minus 119884 (119879) + int

119879

119904

119885 (120590) 119889119882 (120590) + 120582int

119879

119904

119884 (120590) 119889120590

= int

119879

119904

120595 (120590119883120590 119884 (120590) 119885 (120590)) 119889120590 0 le 119904 le 119879 lt infin

(36)






119910 119871

119911gt 0

120583 isin 119877 and119898 ge 11003816100381610038161003816120595 (119904 119909 1199101 1199111) minus 120595 (119904 119909 1199102 1199112)

1003816100381610038161003816

le 119871119910

10038161003816100381610038161199101 minus 11991021003816100381610038161003816 + 119871119911

10038161003816100381610038161199111 minus 11991121003816100381610038161003816

1003816100381610038161003816120595 (119904 119909 119910 119911)1003816100381610038161003816 le 119871 (1 + |119909|

119898

119862+10038161003816100381610038161199101003816100381610038161003816 + |119911|)

⟨120595 (119904 119909 1199101 119911) minus 120595 (119904 119909 119910

2 119911) 119910

1minus 119910

2⟩119870ge 120583

10038161003816100381610038161199101 minus 119910210038161003816100381610038162

(37)


2isin 119870 119911 119911

1 and 119911

2isin




equation

119884 (119904) minus 119884 (119879) + int

119879

119904

119885 (120590) 119889119882 (120590) + 120582int

119879

119904

119884 (120590) 119889120590

= int

119879

119904

120595 (120590119883120590 119884 (120590) 119885 (120590)) 119889120590 0 le 119904 le 119879 lt infin

(38)



sdotis a



119902 ge 119898119901 120578 gt120575

119898 (39)



119902

P(Ω 119871

119902


2

1199112))


by (119884(119883sdot)(119904) 119885(119883

sdot)(119904)) 119904 ge 0

(ii) The estimate

119864sup119904ge0

(119884 (119883sdot) (119904))

119901

119890119901120575119904

+ 119864(int

infin

0

11989021205751205901003816100381610038161003816119884(119883sdot

)(120590)10038161003816100381610038162

119889120590)

1199012

+ 119864(int

infin

0

11989021205751205901003816100381610038161003816119885 (119883sdot

) (120590)10038161003816100381610038162

119889120590)

1199012

le 119888(1 +1003816100381610038161003816119883sdot

1003816100381610038161003816119898

119871119902

P(Ω119871119902

120578(C))

)119901

(40)



119871119901

P(Ω 119862

120575(119870))


sdot) 119885(119883


119871119902

P(Ω 119871

119902

120578(C)) toK119901

120575 and 119883

sdotrarr 119884(119883

sdot) is continuous

from 119871119902

P(Ω 119871

119902

120578(C)) to 119871119901

P(Ω 119862

120575(119870))


P(Ω 119871

119902


119871119902

P(Ω 119862

120578(C))






1015840

= minus(120575 + 120583 minus (1198712



120578(119902)times K

119901



119871119901

P(Ω 119862





120575(119877))


sdot(119905 119909) isin H

119902

120578(119902)



toH119902

120578(119902)


119901

120575of the


sdot) 119885(119883

sdot)) is


120578(119902)to K

119901

120575while X

sdotrarr (Y(X

sdot)) is


120578(119902)to119871119901

P(Ω 119862



120578(119902)times K

119901

120575is the unique



119902

(ΩCF119905) toK119901

120575(119905)








119883119897(119904 119883

119904(119905 119909)) = 119883

119897(119905 119909) 119897 isin [119904infin) (41)



119884 (119897 119904 119883119904(119905 119909)) = 119884 (119897 119905 119909) for 119897 isin [119904infin)

119885 (119897 119904 119883119904(119905 119909)) = 119885 (119897 119905 119909) for aa 119897 isin [119904infin)

(42)






Let now 119891 isin 119871119902


119899 119899 isin 119873 such that


Hence 119891119899



this section



= 119871(Ξ 119877) =


(119883(119905 119909) 119884(119905 119909) 119885(119905 119909)) of (36) satisfies

119884 (119904 119905 119909) = 120592 (119904 119883119904(119905 119909)) 119885 (119904 119905 119909) = 120577 (119904 119883

119904(119905 119909))


(44)



=


1lt 119905

2lt infin Δ gt

0 and 1199091 119909


100381610038161003816100381610038161003816100381610038161003816

119864 int

1199051+Δ

1199051

119885119894119873

(119904 1199051 119909

1) 119889119904 minus 119864int

1199052+Δ

1199052

119885119894119873

(119904 1199052 119909

2) 119889119904

100381610038161003816100381610038161003816100381610038161003816

le 119864int

1199052

1199051

10038161003816100381610038161003816119885119894119873

(119904 1199051 119909

1)10038161003816100381610038161003816119889119904

+ 119864int

1199051+Δ

1199052

10038161003816100381610038161003816119885119894119873

(119904 1199051 119909

1) minus 119885

119894119873

(119904 1199052 119909

2)10038161003816100381610038161003816119889119904

+ 119864int

1199052+Δ

1199051+Δ

10038161003816100381610038161003816119885119894119873

(119904 1199052 119909

2)10038161003816100381610038161003816119889119904


le 211987310038161003816100381610038161199052 minus 1199051

1003816100381610038161003816 + Δ12

119890minus120575(1199051+Δ)

times(119864(int

infin

0

119890212057511990410038161003816100381610038161003816119885119894119873

(119904 1199051 119909

1)minus119885

119894119873

(119904 1199052 119909

2)10038161003816100381610038161003816

2

119889119904)

1199012

)

1119901

le 211987310038161003816100381610038161199052 minus 1199051

1003816100381610038161003816 + Δ12

119890minus120575(1199051+Δ)

times (119864(int

infin

0

11989021205751199041003816100381610038161003816119885 (119904 1199051 1199091)minus119885 (119904 1199052 1199092)

10038161003816100381610038162

119889119904)

1199012

)

1119901

(45)


119905

119885119894119873


119864int119905+Δ

119905

119864119885119894119873


(ΩCF119905) to 119877


120577119894119873


119899119864int

119905+(1119899)

119905

119885119894119873

(119904 119905 119909) 119889119904


(46)


denote 119864[119885119894119873

(119897 119904 119910)]|119910=119883119904(119905119909)


119904(119905 119909) with the map 119910 rarr 119864[119885

119894119873


119904-


119891119898 Ω 997888rarr C 119891

119898=

119873119898

sum

119896=1

ℎ(119898)

119896119868119891119898=ℎ(119898)

119896 119873

119898isin 119873 (47)

where ℎ(119898)1 ℎ

(119898)


119873119898

119896=1119891

119898=

ℎ(119898)

119896 such that

1003816100381610038161003816119891119898 (120596) minus 119883119904(120596)



int119861

int

119904+(1119899)

119904

119885119894119873

(119897 119905 119909) 119889119897119889119875

= int119861

int

119904+(1119899)

119904

119885119894119873

(119897 119904 119883119904) 119889119897119889119875

= 119864119868119861int

119904+(1119899)

119904

119885119894119873

(119897 119904 119883119904) 119889119897


119864(119868119861int

119904+(1119899)

119904

119885119894119873

(119897 119904 119891119898) 119889119897)


119873119898

sum

119896=1

119864(119868119861119868119891119898=ℎ(119898)

119896int

119904+(1119899)

119904

119885119894119873

(119897 119904 ℎ(119898)

119896) 119889119897)


119864(119868119861

119873119898

sum

119896=1

119868119891119898=ℎ(119898)

119896)119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 ℎ(119898)

119896) 119889119897


119864119868119861(119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 119910) 119889119897|119910=119891119898

)


int119861

(119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 119910) 119889119897|119910=119891119898

)119889119875

= int119861

(119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 119910) 119889119897|119910=119883119904

)119889119875

(49)

and we get that

120577119894119873

(119904 119883119904(119905 119909)) = lim inf

119899rarrinfin

119899

times [119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 119910) 119889119897|119910=119883119904(119905119909)

]


119899119864[int

119904+(1119899)

119904

119885119894119873

(119897 119905 119909) 119889119897

100381610038161003816100381610038161003816100381610038161003816

F119904]

119875-as(50)



lim119899rarrinfin

119899int

119904+(1119899)

119904

119885119894119873

(119897 119905 119909) 119889119897 = 119885119894119873

(119904 119905 119909)


(51)


120577119894119873

(119904 119883119904(119905 119909)) = 119864 [119885

119894119873

(119904 119905 119909)10038161003816100381610038161003816F

119904] = 119885

119894119873

(119904 119905 119909)


(52)


120577119894119873

(119904 119883119904(119905 119909)) = 119885

119894119873

(119904 119905 119909)


(53)


119873rarrinfin120577119894119873


119894=1(lim

119873rarrinfin120577119894119873


120577 (119905 119909) =

infin

sum

119894=1


120577119894119873

(119905 119909)) 119890119894 (119905 119909) isin 119862

120577 (119905 119909) = 0 (119905 119909) notin 119862

(54)



119885 (120596 119904 119905 119909) =

infin

sum

119894=1


119885119894119873

(120596 119904 119905 119909)) 119890119894 (55)





3(Γ2(119905 Γ

1(119905 119909))) with


119901

P(Ω 119862

120575(119877))

Γ1(119905 119909) = 119884 (sdot 119905 119909)


119901

P(Ω 119862

120575(119877)) 997888rarr 119871

119901

(Ω 119877)

Γ2(119905 119881) = 119881 (119905)

Γ3 119871

119901

(Ω 119877) 997888rarr 119877 Γ3120585 = 119864120585

(56)


|119881(119905) minus 119880(119904)|119871119901(Ω119877)

le |119881 (119905) minus 119881 (119904)|119871119901(Ω119877)

+ 119890minus120575119901119904

|119881 minus 119880||119871

119901

P(Ω119862120575(119877))

(57)


3is continuous





119905119905ge0






119889119883119906

(119904) = 119860119883119906

(119904) 119889119904 + 119865 (119904 119883119906

119904) 119889119904

+ 119866 (119904 119883119906

119904) 119877 (119904 119883

119906

119904 119906 (119904)) 119889119904 + 119866 (119904 119883

119906

119904) 119889119882 (119904)

119904 isin [119905infin)

119883119906

119905= 119909

(58)


(119904 119905 119909) or simply by 119883119906


119869 (119906) = 119864int

infin

119905

119890minus120582119904

119892 (119904 119883119906

119904 119906 (119904)) 119889119904 (59)




120595 (119905 119909 119911) = inf 119892 (119905 119909 119906) + 119911119877 (119905 119909 119906) 119906 isin 119880

(60)


Γ (119905 119909 119911) = 119906 isin 119880 119892 (119905 119909 119906) + 119911119877 (119905 119909 119906) = 120595 (119905 119909 119911)

(61)




119898119892

119862)









119892



120582 gt (minus120575 minus 120583 +1198712

119911

2) or (minus120575 +

1198712

119877

2 (119901 minus 1))

or (1198712

119877119898119892

2 (119902 minus 119898119892)minus 120578 (119902)119898

119892)

(62)


119869 (119906) = 120592 (119905 119909) + 119864int

infin

119905

119890minus120582119904

[minus120595 (119904 119883119906

119904 120577 (119904 119883

119906

119904)) + 120577 (119904 119883

119906

119904)

times 119877 (119904 119883119906

119904 119906 (119904)) + 119892 (119904 119883

119906

119904 119906 (119904))] 119889119904

(63)


119905119905ge0



119882119906

(119904) = 119882 (119904) + int

119904

119905and119904

119877 (120590119883119906

120590 119906 (120590)) 119889120590 119904 isin [0infin)

120588 (119879) = exp(int119879

119905

minus119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882 (119904)

minus1

2int

119879

119905

1003816100381610038161003816119877 (119904 119883119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)

(64)



such that

119875119906

|F119879

= 120588 (119879) 119875|F119879

(65)


119905119905ge0


119889119883119906

(119904) = 119860119883119906

(119904) 119889119904 + 119865 (119904 119883119906

119904) 119889119904

+ 119866 (119904 119883119906

119904) 119889119882

119906

(119904) 119904 isin [119905infin)

119883119906

119905= 119909

(66)


F119906

119905119905ge0 119875

119906


119883119906

(119904) = 119890(119904minus119905)119860

119909 (0) + int

119904

119905

119890(119904minus120590)119860

119865 (120590119883119906

120590) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883119906

120590) 119889119882

119906

(120590) 119904 isin [119905infin)

119883119906

119905= 119909 isin C

119884119906

(119904) minus 119884119906

(119879) + int

119879

119904

119885119906

(120590) 119889119882119906

(120590) + 120582int

119879

119904

119884119906

(120590) 119889120590

= int

119879

119904

120595 (120590119883119906

120590 119885

119906

(120590)) 119889120590 0 le 119904 le 119879 lt infin

(67)


119884119906

(119904) + int

119879

119904

119890minus120582120590

119885119906

(120590) 119889119882 (120590)

= int

119879

119904

119890minus120582120590

[120595 (120590119883119906

120590 119885

119906

(120590))

minus119885119906

(120590) 119877 (120590119883119906

120590 119906 (120590))] 119889120590

+ 119890minus120582119879

119884119906

(119879)

(68)


119864119906

[120588(119879)minus119903

] = 119864119906

[exp 119903 (int119879

119905

119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882

119906

(119904)

minus1

2int

119879

119905

1003816100381610038161003816119877 (119904 119883119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)]

= 119864119906

[exp(int119879

119905

119903119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882

119906

(119904)

minus1

2int

119879

119905

11990321003816100381610038161003816119877 (119904 119883

119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)

times exp 119903 (119903 minus 1)2

int

119879

119905

1003816100381610038161003816119877 (119904 119883119906

119904 119906 (119904))

10038161003816100381610038162

119889119904]

le 119890(12)119903(119903minus1)119879119871

2

119877119864119906

times exp(int119879

119905

2119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882

119906

(119904)

minus1

2int

119879

119905

41003816100381610038161003816119877 (119904 119883

119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)

= 119890(12)119903(119903minus1)119879119871

2

119877

(69)

It follows that

119864(int

119879

119905

|119890minus120582119904

119885119906

(119904)|2

119889119904)

12

= 119864119906

[(int

119879

119905

10038161003816100381610038161003816119890minus120582119904

119885119906

(119904)10038161003816100381610038161003816

2

119889119904)

12

120588minus1

]

le (119864119906

int

119879

119905

10038161003816100381610038161003816119890minus120582119904

119885119906

(119904)10038161003816100381610038161003816

2

119889119904)

12

times (119864119906

120588minus2

)12

lt infin

(70)


119890minus120582119879

119864119884119906

(119879) minus 119884119906

(119905)

= 119864int

119879

119905

119890minus120582120590

[minus120595 (120590119883119906

120590 119885

119906

(120590))

+119885119906

(120590) 119877 (120590119883119906

120590 119906 (120590))] 119889120590

(71)


120575(119877)) so that

119864119906

|119884(119879 119905 119909)|119901

le 119862 exp (minus119901120575119879) (72)


119864 |119884 (119879 119905 119909)| = 119864119906

(120588minus1

(119879) |119884 (119879 119905 119909)|)

le 119864(120588minus119901(119901minus1)

)(119901minus1)119901

119864(|119884 (119879 119905 119909)|119901

)1119901

le 119862119890((1198712

1198772(119901minus1))minus120575))119879

(73)


|119883119906

119904|119902


1198641003816100381610038161003816119883

119906

119879

1003816100381610038161003816119898119892

le 119862119890(1198712

119877119898119892(2119902minus2119898

119892)minus1

minus120578(119902)119898119892)119879

(74)



119864int

infin

119905

119890minus120582120590 1003816100381610038161003816119892 (120590119883

119906

120590 119906 (120590))

1003816100381610038161003816 119889120590 lt infin (75)

Since 119884119906(119905 119905 119909) = 120592(119905 119909) and 119885119906

(119904 119905 119909) = 120577(119904 119883119906

119904(119905 119909)) 119875-


119890minus120582119879

119864119884119906

(119879) minus 119907 (119905 119909)

= 119864int

119879

119905

119890minus120582120590

[minus120595 (120590119883119906

120590 120577 (120590 119883

119906

120590))

+120577 (120590119883119906

120590) 119877 (120590119883

119906

120590 119906 (120590))] 119889120590

(76)


119890minus120582120590

119892(120590119883119906

120590 119906(120590))119889120590 and


119869 (119906) = 120592 (119905 119909)

+ 119864int

infin

119905

119890minus120582119904

[minus120595 (119904 119883119906

119904 120577 (119904 119883

119906

119904)) + 120577 (119904 119883

119906

119904) 119877

times (119904 119883119906

119904 119906 (119904)) + 119892 (119904 119883

119906

119904 119906 (119904))] 119889119904

(77)






119906 (119904) = Γ0(119904 119883

119906

119904 120577 (119904 119883

119906

119904))


(78)



119889119883 (119904)=119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904+119866 (119904 119883

119904)

times(119877 (119904 119883119904 Γ

0(119904 119883

119904 120577 (119904 119883

119904))) 119889119904 + 119889119882 (119904))

119904 isin [119905infin)

119883119905= 119909

(79)


119906 (119904) = Γ0(119904 119883

119904 120577 (119904 119883

119904)) (80)







119905119905ge0


F119905119905ge0 119875119882 119906119883

119906

) where (ΩF F119905119905ge0 119875 119882) is an




119869 (119906) = 119864int

infin

119905

119890minus120582119904

119892 (119904 119883119906

119904 119906 (119904)) 119889119904 (81)



119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904 + 119866 (119904 119883

119904)

times (119877 (119904 119883119904 Γ

0(119904 119883

119904 120577 (119904 119883

119904))) 119889119904 +119889119882 (119904))

119904 isin [119905infin)

119883119905= 119909

(82)


119905119905ge0 119875119882)



119883 (119904) = 119890(119904minus119905)119860

119909 (0) + int

119904

119905

119890(119904minus120590)119860

119865 (120590119883120590) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883120590) 119877

times (120590119883120590 Γ

0(120590119883

120590 120577 (120590 119883

120590))) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883120590) 119889119882

120590 119904 isin [119905infin)

(83)

119883119905= 119909 (84)




119905119905ge0 119875119882)We define

120588 (119879) = exp(int119879

119905

minus119877lowast

(120590 119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590))) 119889119882 (120590)

minus1

2int

119879

119905

1003816100381610038161003816119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590)))10038161003816100381610038162

119889120590)

(85)


1198820

(119904) = 119882 (119904) + int

119904

119905and119904

119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590))) 119889120590


(86)


1198750

|F119879

= 120588 (119879) 119875|F119879

(87)

Let us denote by F0

119905119905ge0


[0infin) F0

119905119905ge0 119875

0


119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119905 119883119904) 119889119904

+ 119866 (119904 119883119904) 119889119882

0

(119904) 119904 isin [119905infin)

119883119905= 119909

120588 (119879) = exp(int119879

119905

minus119877lowast

(120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590)) 119889119882

0

(120590)

+ 12int

119879

119905

1003816100381610038161003816119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590)))10038161003816100381610038162

119889120590)

(88)




119905119905ge0


119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904

+ 119866 (119904 119883119904) 119889119882 (119904) 119904 isin [119905infin)

119883119905= 119909

(89)


1198821

(119904) = 119882 (119904) minus int

119904

119905and119904

119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590))) 119889120590

(90)





0



119906

) one has

119869 (119906 119905 119909) ge 120592 (119905 119909) (91)


119906 (119904) = Γ0(119904 119883

119906

119904 120577 (119904 119883

119906

119904))

119875 minus 119886119904 119891119900119903 119886119897119898119900119904119905 119890119907119890119903119910 119904 isin [119905infin)

(92)



119906 (119904) = Γ0(119904 119883

119904 120577 (119904 119883

119904)) (93)


6 Applications



119889119911119906

(119905 120585) = Δ119911119906

(119905 120585) 119889119905 + 119891 (119905 119911119906

119905(120585)) 119889119905

+

119889

sum

119894=1

119892119894(119905 119911

119906

119905(120585)) [119903

119894

(120585) 119906119894

(119905) 119889119905 + 119889119882119894

(119905)]

119911119906

0(120579 120585) = 119909 (120579 120585) 120585 isin 119861 120579 isin [minus1 0]

119911119906

(119905 120585) = 0 119905 isin [0infin) 120585 isin 120597119861

(94)

Here119882 = (1198821

1198822

119882119889



119867 = 1198712


119865 (119905 119909) (120585) = 119891 (119905 119909 (120585))

(119866 (119905 119909) 119911) (120585) =

119889

sum

119894=1

119892119894(119905 119909 (120585)) 119911

119894

(120585)


(95)


(119861) withdomain11988222

(119861)⋂11988212


Hypothesis 1 holds



119869 (119906) = 119864int

infin

0

119890minus120582119905

[int119861

120590 (120585 119911119906

119905(120585)) 119889120585 + 119906

2

(119905)] 119889119905 (96)



119861

120590(119905 119910(120585) 119906)119889120585 + 1199062 and 119877(119910 119906) = (int

119861

1199031

(120585)1199061

119889120585

int119861

1199032

(120585)1199062

119889120585 int119861

119903119889

(120585)119906119889

119889120585) for 119910 isin 119862([minus1 0]119867) 119906 =

(1199061

1199062

119906119889


0



119905119905ge0 119875119882 119906 119911

sdot(sdot)) and

setting

119906 (119904) = Γ0(119904 119911

119904(sdot) 120577 (119904 119911

119904(sdot))) (97)


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014





120578times


120578as follows

Φ(119883sdot 119905 119909)

119904(119897) = 119878 (119904 + 119897 minus 119905) 119909 (0)

+ int

119904+119897

0

119868[119905infin)

(120590) 119878 (119904 + 119897 minus 120590) 119865 (120590119883120590) 119889120590

+ int

119904+119897

0

119868[119905infin)

(120590) 119878 (119904 + 119897 minus 120590)

times 119866 (120590119883120590) 119889119882 (120590)


Φ(119883sdot 119905 119909)

119904(119897) = 119909 ((119904 + 119897 minus 119905) or (minus120591))


(34)








sdot it is


toH119902




119905)

and H119902




the norm

|119909|119902

= 119864|119909|119902

119862 (35)

is finite



119883 (119904) = 119890(119904minus119905)119860

119909 (0) + int

119904

119905

119890(119904minus120590)119860

119865 (120590119883120590) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883120590) 119889119882 (120590) 119904 isin [119905infin)

119883119905= 119909 isin C

119884 (119904) minus 119884 (119879) + int

119879

119904

119885 (120590) 119889119882 (120590) + 120582int

119879

119904

119884 (120590) 119889120590

= int

119879

119904

120595 (120590119883120590 119884 (120590) 119885 (120590)) 119889120590 0 le 119904 le 119879 lt infin

(36)






119910 119871

119911gt 0

120583 isin 119877 and119898 ge 11003816100381610038161003816120595 (119904 119909 1199101 1199111) minus 120595 (119904 119909 1199102 1199112)

1003816100381610038161003816

le 119871119910

10038161003816100381610038161199101 minus 11991021003816100381610038161003816 + 119871119911

10038161003816100381610038161199111 minus 11991121003816100381610038161003816

1003816100381610038161003816120595 (119904 119909 119910 119911)1003816100381610038161003816 le 119871 (1 + |119909|

119898

119862+10038161003816100381610038161199101003816100381610038161003816 + |119911|)

⟨120595 (119904 119909 1199101 119911) minus 120595 (119904 119909 119910

2 119911) 119910

1minus 119910

2⟩119870ge 120583

10038161003816100381610038161199101 minus 119910210038161003816100381610038162

(37)


2isin 119870 119911 119911

1 and 119911

2isin




equation

119884 (119904) minus 119884 (119879) + int

119879

119904

119885 (120590) 119889119882 (120590) + 120582int

119879

119904

119884 (120590) 119889120590

= int

119879

119904

120595 (120590119883120590 119884 (120590) 119885 (120590)) 119889120590 0 le 119904 le 119879 lt infin

(38)



sdotis a



119902 ge 119898119901 120578 gt120575

119898 (39)



119902

P(Ω 119871

119902


2

1199112))


by (119884(119883sdot)(119904) 119885(119883

sdot)(119904)) 119904 ge 0

(ii) The estimate

119864sup119904ge0

(119884 (119883sdot) (119904))

119901

119890119901120575119904

+ 119864(int

infin

0

11989021205751205901003816100381610038161003816119884(119883sdot

)(120590)10038161003816100381610038162

119889120590)

1199012

+ 119864(int

infin

0

11989021205751205901003816100381610038161003816119885 (119883sdot

) (120590)10038161003816100381610038162

119889120590)

1199012

le 119888(1 +1003816100381610038161003816119883sdot

1003816100381610038161003816119898

119871119902

P(Ω119871119902

120578(C))

)119901

(40)



119871119901

P(Ω 119862

120575(119870))


sdot) 119885(119883


119871119902

P(Ω 119871

119902

120578(C)) toK119901

120575 and 119883

sdotrarr 119884(119883

sdot) is continuous

from 119871119902

P(Ω 119871

119902

120578(C)) to 119871119901

P(Ω 119862

120575(119870))


P(Ω 119871

119902


119871119902

P(Ω 119862

120578(C))






1015840

= minus(120575 + 120583 minus (1198712



120578(119902)times K

119901



119871119901

P(Ω 119862





120575(119877))


sdot(119905 119909) isin H

119902

120578(119902)



toH119902

120578(119902)


119901

120575of the


sdot) 119885(119883

sdot)) is


120578(119902)to K

119901

120575while X

sdotrarr (Y(X

sdot)) is


120578(119902)to119871119901

P(Ω 119862



120578(119902)times K

119901

120575is the unique



119902

(ΩCF119905) toK119901

120575(119905)








119883119897(119904 119883

119904(119905 119909)) = 119883

119897(119905 119909) 119897 isin [119904infin) (41)



119884 (119897 119904 119883119904(119905 119909)) = 119884 (119897 119905 119909) for 119897 isin [119904infin)

119885 (119897 119904 119883119904(119905 119909)) = 119885 (119897 119905 119909) for aa 119897 isin [119904infin)

(42)






Let now 119891 isin 119871119902


119899 119899 isin 119873 such that


Hence 119891119899



this section



= 119871(Ξ 119877) =


(119883(119905 119909) 119884(119905 119909) 119885(119905 119909)) of (36) satisfies

119884 (119904 119905 119909) = 120592 (119904 119883119904(119905 119909)) 119885 (119904 119905 119909) = 120577 (119904 119883

119904(119905 119909))


(44)



=


1lt 119905

2lt infin Δ gt

0 and 1199091 119909


100381610038161003816100381610038161003816100381610038161003816

119864 int

1199051+Δ

1199051

119885119894119873

(119904 1199051 119909

1) 119889119904 minus 119864int

1199052+Δ

1199052

119885119894119873

(119904 1199052 119909

2) 119889119904

100381610038161003816100381610038161003816100381610038161003816

le 119864int

1199052

1199051

10038161003816100381610038161003816119885119894119873

(119904 1199051 119909

1)10038161003816100381610038161003816119889119904

+ 119864int

1199051+Δ

1199052

10038161003816100381610038161003816119885119894119873

(119904 1199051 119909

1) minus 119885

119894119873

(119904 1199052 119909

2)10038161003816100381610038161003816119889119904

+ 119864int

1199052+Δ

1199051+Δ

10038161003816100381610038161003816119885119894119873

(119904 1199052 119909

2)10038161003816100381610038161003816119889119904


le 211987310038161003816100381610038161199052 minus 1199051

1003816100381610038161003816 + Δ12

119890minus120575(1199051+Δ)

times(119864(int

infin

0

119890212057511990410038161003816100381610038161003816119885119894119873

(119904 1199051 119909

1)minus119885

119894119873

(119904 1199052 119909

2)10038161003816100381610038161003816

2

119889119904)

1199012

)

1119901

le 211987310038161003816100381610038161199052 minus 1199051

1003816100381610038161003816 + Δ12

119890minus120575(1199051+Δ)

times (119864(int

infin

0

11989021205751199041003816100381610038161003816119885 (119904 1199051 1199091)minus119885 (119904 1199052 1199092)

10038161003816100381610038162

119889119904)

1199012

)

1119901

(45)


119905

119885119894119873


119864int119905+Δ

119905

119864119885119894119873


(ΩCF119905) to 119877


120577119894119873


119899119864int

119905+(1119899)

119905

119885119894119873

(119904 119905 119909) 119889119904


(46)


denote 119864[119885119894119873

(119897 119904 119910)]|119910=119883119904(119905119909)


119904(119905 119909) with the map 119910 rarr 119864[119885

119894119873


119904-


119891119898 Ω 997888rarr C 119891

119898=

119873119898

sum

119896=1

ℎ(119898)

119896119868119891119898=ℎ(119898)

119896 119873

119898isin 119873 (47)

where ℎ(119898)1 ℎ

(119898)


119873119898

119896=1119891

119898=

ℎ(119898)

119896 such that

1003816100381610038161003816119891119898 (120596) minus 119883119904(120596)



int119861

int

119904+(1119899)

119904

119885119894119873

(119897 119905 119909) 119889119897119889119875

= int119861

int

119904+(1119899)

119904

119885119894119873

(119897 119904 119883119904) 119889119897119889119875

= 119864119868119861int

119904+(1119899)

119904

119885119894119873

(119897 119904 119883119904) 119889119897


119864(119868119861int

119904+(1119899)

119904

119885119894119873

(119897 119904 119891119898) 119889119897)


119873119898

sum

119896=1

119864(119868119861119868119891119898=ℎ(119898)

119896int

119904+(1119899)

119904

119885119894119873

(119897 119904 ℎ(119898)

119896) 119889119897)


119864(119868119861

119873119898

sum

119896=1

119868119891119898=ℎ(119898)

119896)119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 ℎ(119898)

119896) 119889119897


119864119868119861(119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 119910) 119889119897|119910=119891119898

)


int119861

(119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 119910) 119889119897|119910=119891119898

)119889119875

= int119861

(119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 119910) 119889119897|119910=119883119904

)119889119875

(49)

and we get that

120577119894119873

(119904 119883119904(119905 119909)) = lim inf

119899rarrinfin

119899

times [119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 119910) 119889119897|119910=119883119904(119905119909)

]


119899119864[int

119904+(1119899)

119904

119885119894119873

(119897 119905 119909) 119889119897

100381610038161003816100381610038161003816100381610038161003816

F119904]

119875-as(50)



lim119899rarrinfin

119899int

119904+(1119899)

119904

119885119894119873

(119897 119905 119909) 119889119897 = 119885119894119873

(119904 119905 119909)


(51)


120577119894119873

(119904 119883119904(119905 119909)) = 119864 [119885

119894119873

(119904 119905 119909)10038161003816100381610038161003816F

119904] = 119885

119894119873

(119904 119905 119909)


(52)


120577119894119873

(119904 119883119904(119905 119909)) = 119885

119894119873

(119904 119905 119909)


(53)


119873rarrinfin120577119894119873


119894=1(lim

119873rarrinfin120577119894119873


120577 (119905 119909) =

infin

sum

119894=1


120577119894119873

(119905 119909)) 119890119894 (119905 119909) isin 119862

120577 (119905 119909) = 0 (119905 119909) notin 119862

(54)



119885 (120596 119904 119905 119909) =

infin

sum

119894=1


119885119894119873

(120596 119904 119905 119909)) 119890119894 (55)





3(Γ2(119905 Γ

1(119905 119909))) with


119901

P(Ω 119862

120575(119877))

Γ1(119905 119909) = 119884 (sdot 119905 119909)


119901

P(Ω 119862

120575(119877)) 997888rarr 119871

119901

(Ω 119877)

Γ2(119905 119881) = 119881 (119905)

Γ3 119871

119901

(Ω 119877) 997888rarr 119877 Γ3120585 = 119864120585

(56)


|119881(119905) minus 119880(119904)|119871119901(Ω119877)

le |119881 (119905) minus 119881 (119904)|119871119901(Ω119877)

+ 119890minus120575119901119904

|119881 minus 119880||119871

119901

P(Ω119862120575(119877))

(57)


3is continuous





119905119905ge0






119889119883119906

(119904) = 119860119883119906

(119904) 119889119904 + 119865 (119904 119883119906

119904) 119889119904

+ 119866 (119904 119883119906

119904) 119877 (119904 119883

119906

119904 119906 (119904)) 119889119904 + 119866 (119904 119883

119906

119904) 119889119882 (119904)

119904 isin [119905infin)

119883119906

119905= 119909

(58)


(119904 119905 119909) or simply by 119883119906


119869 (119906) = 119864int

infin

119905

119890minus120582119904

119892 (119904 119883119906

119904 119906 (119904)) 119889119904 (59)




120595 (119905 119909 119911) = inf 119892 (119905 119909 119906) + 119911119877 (119905 119909 119906) 119906 isin 119880

(60)


Γ (119905 119909 119911) = 119906 isin 119880 119892 (119905 119909 119906) + 119911119877 (119905 119909 119906) = 120595 (119905 119909 119911)

(61)




119898119892

119862)









119892



120582 gt (minus120575 minus 120583 +1198712

119911

2) or (minus120575 +

1198712

119877

2 (119901 minus 1))

or (1198712

119877119898119892

2 (119902 minus 119898119892)minus 120578 (119902)119898

119892)

(62)


119869 (119906) = 120592 (119905 119909) + 119864int

infin

119905

119890minus120582119904

[minus120595 (119904 119883119906

119904 120577 (119904 119883

119906

119904)) + 120577 (119904 119883

119906

119904)

times 119877 (119904 119883119906

119904 119906 (119904)) + 119892 (119904 119883

119906

119904 119906 (119904))] 119889119904

(63)


119905119905ge0



119882119906

(119904) = 119882 (119904) + int

119904

119905and119904

119877 (120590119883119906

120590 119906 (120590)) 119889120590 119904 isin [0infin)

120588 (119879) = exp(int119879

119905

minus119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882 (119904)

minus1

2int

119879

119905

1003816100381610038161003816119877 (119904 119883119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)

(64)



such that

119875119906

|F119879

= 120588 (119879) 119875|F119879

(65)


119905119905ge0


119889119883119906

(119904) = 119860119883119906

(119904) 119889119904 + 119865 (119904 119883119906

119904) 119889119904

+ 119866 (119904 119883119906

119904) 119889119882

119906

(119904) 119904 isin [119905infin)

119883119906

119905= 119909

(66)


F119906

119905119905ge0 119875

119906


119883119906

(119904) = 119890(119904minus119905)119860

119909 (0) + int

119904

119905

119890(119904minus120590)119860

119865 (120590119883119906

120590) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883119906

120590) 119889119882

119906

(120590) 119904 isin [119905infin)

119883119906

119905= 119909 isin C

119884119906

(119904) minus 119884119906

(119879) + int

119879

119904

119885119906

(120590) 119889119882119906

(120590) + 120582int

119879

119904

119884119906

(120590) 119889120590

= int

119879

119904

120595 (120590119883119906

120590 119885

119906

(120590)) 119889120590 0 le 119904 le 119879 lt infin

(67)


119884119906

(119904) + int

119879

119904

119890minus120582120590

119885119906

(120590) 119889119882 (120590)

= int

119879

119904

119890minus120582120590

[120595 (120590119883119906

120590 119885

119906

(120590))

minus119885119906

(120590) 119877 (120590119883119906

120590 119906 (120590))] 119889120590

+ 119890minus120582119879

119884119906

(119879)

(68)


119864119906

[120588(119879)minus119903

] = 119864119906

[exp 119903 (int119879

119905

119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882

119906

(119904)

minus1

2int

119879

119905

1003816100381610038161003816119877 (119904 119883119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)]

= 119864119906

[exp(int119879

119905

119903119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882

119906

(119904)

minus1

2int

119879

119905

11990321003816100381610038161003816119877 (119904 119883

119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)

times exp 119903 (119903 minus 1)2

int

119879

119905

1003816100381610038161003816119877 (119904 119883119906

119904 119906 (119904))

10038161003816100381610038162

119889119904]

le 119890(12)119903(119903minus1)119879119871

2

119877119864119906

times exp(int119879

119905

2119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882

119906

(119904)

minus1

2int

119879

119905

41003816100381610038161003816119877 (119904 119883

119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)

= 119890(12)119903(119903minus1)119879119871

2

119877

(69)

It follows that

119864(int

119879

119905

|119890minus120582119904

119885119906

(119904)|2

119889119904)

12

= 119864119906

[(int

119879

119905

10038161003816100381610038161003816119890minus120582119904

119885119906

(119904)10038161003816100381610038161003816

2

119889119904)

12

120588minus1

]

le (119864119906

int

119879

119905

10038161003816100381610038161003816119890minus120582119904

119885119906

(119904)10038161003816100381610038161003816

2

119889119904)

12

times (119864119906

120588minus2

)12

lt infin

(70)


119890minus120582119879

119864119884119906

(119879) minus 119884119906

(119905)

= 119864int

119879

119905

119890minus120582120590

[minus120595 (120590119883119906

120590 119885

119906

(120590))

+119885119906

(120590) 119877 (120590119883119906

120590 119906 (120590))] 119889120590

(71)


120575(119877)) so that

119864119906

|119884(119879 119905 119909)|119901

le 119862 exp (minus119901120575119879) (72)


119864 |119884 (119879 119905 119909)| = 119864119906

(120588minus1

(119879) |119884 (119879 119905 119909)|)

le 119864(120588minus119901(119901minus1)

)(119901minus1)119901

119864(|119884 (119879 119905 119909)|119901

)1119901

le 119862119890((1198712

1198772(119901minus1))minus120575))119879

(73)


|119883119906

119904|119902


1198641003816100381610038161003816119883

119906

119879

1003816100381610038161003816119898119892

le 119862119890(1198712

119877119898119892(2119902minus2119898

119892)minus1

minus120578(119902)119898119892)119879

(74)



119864int

infin

119905

119890minus120582120590 1003816100381610038161003816119892 (120590119883

119906

120590 119906 (120590))

1003816100381610038161003816 119889120590 lt infin (75)

Since 119884119906(119905 119905 119909) = 120592(119905 119909) and 119885119906

(119904 119905 119909) = 120577(119904 119883119906

119904(119905 119909)) 119875-


119890minus120582119879

119864119884119906

(119879) minus 119907 (119905 119909)

= 119864int

119879

119905

119890minus120582120590

[minus120595 (120590119883119906

120590 120577 (120590 119883

119906

120590))

+120577 (120590119883119906

120590) 119877 (120590119883

119906

120590 119906 (120590))] 119889120590

(76)


119890minus120582120590

119892(120590119883119906

120590 119906(120590))119889120590 and


119869 (119906) = 120592 (119905 119909)

+ 119864int

infin

119905

119890minus120582119904

[minus120595 (119904 119883119906

119904 120577 (119904 119883

119906

119904)) + 120577 (119904 119883

119906

119904) 119877

times (119904 119883119906

119904 119906 (119904)) + 119892 (119904 119883

119906

119904 119906 (119904))] 119889119904

(77)






119906 (119904) = Γ0(119904 119883

119906

119904 120577 (119904 119883

119906

119904))


(78)



119889119883 (119904)=119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904+119866 (119904 119883

119904)

times(119877 (119904 119883119904 Γ

0(119904 119883

119904 120577 (119904 119883

119904))) 119889119904 + 119889119882 (119904))

119904 isin [119905infin)

119883119905= 119909

(79)


119906 (119904) = Γ0(119904 119883

119904 120577 (119904 119883

119904)) (80)







119905119905ge0


F119905119905ge0 119875119882 119906119883

119906

) where (ΩF F119905119905ge0 119875 119882) is an




119869 (119906) = 119864int

infin

119905

119890minus120582119904

119892 (119904 119883119906

119904 119906 (119904)) 119889119904 (81)



119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904 + 119866 (119904 119883

119904)

times (119877 (119904 119883119904 Γ

0(119904 119883

119904 120577 (119904 119883

119904))) 119889119904 +119889119882 (119904))

119904 isin [119905infin)

119883119905= 119909

(82)


119905119905ge0 119875119882)



119883 (119904) = 119890(119904minus119905)119860

119909 (0) + int

119904

119905

119890(119904minus120590)119860

119865 (120590119883120590) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883120590) 119877

times (120590119883120590 Γ

0(120590119883

120590 120577 (120590 119883

120590))) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883120590) 119889119882

120590 119904 isin [119905infin)

(83)

119883119905= 119909 (84)




119905119905ge0 119875119882)We define

120588 (119879) = exp(int119879

119905

minus119877lowast

(120590 119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590))) 119889119882 (120590)

minus1

2int

119879

119905

1003816100381610038161003816119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590)))10038161003816100381610038162

119889120590)

(85)


1198820

(119904) = 119882 (119904) + int

119904

119905and119904

119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590))) 119889120590


(86)


1198750

|F119879

= 120588 (119879) 119875|F119879

(87)

Let us denote by F0

119905119905ge0


[0infin) F0

119905119905ge0 119875

0


119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119905 119883119904) 119889119904

+ 119866 (119904 119883119904) 119889119882

0

(119904) 119904 isin [119905infin)

119883119905= 119909

120588 (119879) = exp(int119879

119905

minus119877lowast

(120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590)) 119889119882

0

(120590)

+ 12int

119879

119905

1003816100381610038161003816119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590)))10038161003816100381610038162

119889120590)

(88)




119905119905ge0


119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904

+ 119866 (119904 119883119904) 119889119882 (119904) 119904 isin [119905infin)

119883119905= 119909

(89)


1198821

(119904) = 119882 (119904) minus int

119904

119905and119904

119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590))) 119889120590

(90)





0



119906

) one has

119869 (119906 119905 119909) ge 120592 (119905 119909) (91)


119906 (119904) = Γ0(119904 119883

119906

119904 120577 (119904 119883

119906

119904))

119875 minus 119886119904 119891119900119903 119886119897119898119900119904119905 119890119907119890119903119910 119904 isin [119905infin)

(92)



119906 (119904) = Γ0(119904 119883

119904 120577 (119904 119883

119904)) (93)


6 Applications



119889119911119906

(119905 120585) = Δ119911119906

(119905 120585) 119889119905 + 119891 (119905 119911119906

119905(120585)) 119889119905

+

119889

sum

119894=1

119892119894(119905 119911

119906

119905(120585)) [119903

119894

(120585) 119906119894

(119905) 119889119905 + 119889119882119894

(119905)]

119911119906

0(120579 120585) = 119909 (120579 120585) 120585 isin 119861 120579 isin [minus1 0]

119911119906

(119905 120585) = 0 119905 isin [0infin) 120585 isin 120597119861

(94)

Here119882 = (1198821

1198822

119882119889



119867 = 1198712


119865 (119905 119909) (120585) = 119891 (119905 119909 (120585))

(119866 (119905 119909) 119911) (120585) =

119889

sum

119894=1

119892119894(119905 119909 (120585)) 119911

119894

(120585)


(95)


(119861) withdomain11988222

(119861)⋂11988212


Hypothesis 1 holds



119869 (119906) = 119864int

infin

0

119890minus120582119905

[int119861

120590 (120585 119911119906

119905(120585)) 119889120585 + 119906

2

(119905)] 119889119905 (96)



119861

120590(119905 119910(120585) 119906)119889120585 + 1199062 and 119877(119910 119906) = (int

119861

1199031

(120585)1199061

119889120585

int119861

1199032

(120585)1199062

119889120585 int119861

119903119889

(120585)119906119889

119889120585) for 119910 isin 119862([minus1 0]119867) 119906 =

(1199061

1199062

119906119889


0



119905119905ge0 119875119882 119906 119911

sdot(sdot)) and

setting

119906 (119904) = Γ0(119904 119911

119904(sdot) 120577 (119904 119911

119904(sdot))) (97)


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014





119871119901

P(Ω 119862

120575(119870))


sdot) 119885(119883


119871119902

P(Ω 119871

119902

120578(C)) toK119901

120575 and 119883

sdotrarr 119884(119883

sdot) is continuous

from 119871119902

P(Ω 119871

119902

120578(C)) to 119871119901

P(Ω 119862

120575(119870))


P(Ω 119871

119902


119871119902

P(Ω 119862

120578(C))






1015840

= minus(120575 + 120583 minus (1198712



120578(119902)times K

119901



119871119901

P(Ω 119862





120575(119877))


sdot(119905 119909) isin H

119902

120578(119902)



toH119902

120578(119902)


119901

120575of the


sdot) 119885(119883

sdot)) is


120578(119902)to K

119901

120575while X

sdotrarr (Y(X

sdot)) is


120578(119902)to119871119901

P(Ω 119862



120578(119902)times K

119901

120575is the unique



119902

(ΩCF119905) toK119901

120575(119905)








119883119897(119904 119883

119904(119905 119909)) = 119883

119897(119905 119909) 119897 isin [119904infin) (41)



119884 (119897 119904 119883119904(119905 119909)) = 119884 (119897 119905 119909) for 119897 isin [119904infin)

119885 (119897 119904 119883119904(119905 119909)) = 119885 (119897 119905 119909) for aa 119897 isin [119904infin)

(42)






Let now 119891 isin 119871119902


119899 119899 isin 119873 such that


Hence 119891119899



this section



= 119871(Ξ 119877) =


(119883(119905 119909) 119884(119905 119909) 119885(119905 119909)) of (36) satisfies

119884 (119904 119905 119909) = 120592 (119904 119883119904(119905 119909)) 119885 (119904 119905 119909) = 120577 (119904 119883

119904(119905 119909))


(44)



=


1lt 119905

2lt infin Δ gt

0 and 1199091 119909


100381610038161003816100381610038161003816100381610038161003816

119864 int

1199051+Δ

1199051

119885119894119873

(119904 1199051 119909

1) 119889119904 minus 119864int

1199052+Δ

1199052

119885119894119873

(119904 1199052 119909

2) 119889119904

100381610038161003816100381610038161003816100381610038161003816

le 119864int

1199052

1199051

10038161003816100381610038161003816119885119894119873

(119904 1199051 119909

1)10038161003816100381610038161003816119889119904

+ 119864int

1199051+Δ

1199052

10038161003816100381610038161003816119885119894119873

(119904 1199051 119909

1) minus 119885

119894119873

(119904 1199052 119909

2)10038161003816100381610038161003816119889119904

+ 119864int

1199052+Δ

1199051+Δ

10038161003816100381610038161003816119885119894119873

(119904 1199052 119909

2)10038161003816100381610038161003816119889119904


le 211987310038161003816100381610038161199052 minus 1199051

1003816100381610038161003816 + Δ12

119890minus120575(1199051+Δ)

times(119864(int

infin

0

119890212057511990410038161003816100381610038161003816119885119894119873

(119904 1199051 119909

1)minus119885

119894119873

(119904 1199052 119909

2)10038161003816100381610038161003816

2

119889119904)

1199012

)

1119901

le 211987310038161003816100381610038161199052 minus 1199051

1003816100381610038161003816 + Δ12

119890minus120575(1199051+Δ)

times (119864(int

infin

0

11989021205751199041003816100381610038161003816119885 (119904 1199051 1199091)minus119885 (119904 1199052 1199092)

10038161003816100381610038162

119889119904)

1199012

)

1119901

(45)


119905

119885119894119873


119864int119905+Δ

119905

119864119885119894119873


(ΩCF119905) to 119877


120577119894119873


119899119864int

119905+(1119899)

119905

119885119894119873

(119904 119905 119909) 119889119904


(46)


denote 119864[119885119894119873

(119897 119904 119910)]|119910=119883119904(119905119909)


119904(119905 119909) with the map 119910 rarr 119864[119885

119894119873


119904-


119891119898 Ω 997888rarr C 119891

119898=

119873119898

sum

119896=1

ℎ(119898)

119896119868119891119898=ℎ(119898)

119896 119873

119898isin 119873 (47)

where ℎ(119898)1 ℎ

(119898)


119873119898

119896=1119891

119898=

ℎ(119898)

119896 such that

1003816100381610038161003816119891119898 (120596) minus 119883119904(120596)



int119861

int

119904+(1119899)

119904

119885119894119873

(119897 119905 119909) 119889119897119889119875

= int119861

int

119904+(1119899)

119904

119885119894119873

(119897 119904 119883119904) 119889119897119889119875

= 119864119868119861int

119904+(1119899)

119904

119885119894119873

(119897 119904 119883119904) 119889119897


119864(119868119861int

119904+(1119899)

119904

119885119894119873

(119897 119904 119891119898) 119889119897)


119873119898

sum

119896=1

119864(119868119861119868119891119898=ℎ(119898)

119896int

119904+(1119899)

119904

119885119894119873

(119897 119904 ℎ(119898)

119896) 119889119897)


119864(119868119861

119873119898

sum

119896=1

119868119891119898=ℎ(119898)

119896)119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 ℎ(119898)

119896) 119889119897


119864119868119861(119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 119910) 119889119897|119910=119891119898

)


int119861

(119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 119910) 119889119897|119910=119891119898

)119889119875

= int119861

(119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 119910) 119889119897|119910=119883119904

)119889119875

(49)

and we get that

120577119894119873

(119904 119883119904(119905 119909)) = lim inf

119899rarrinfin

119899

times [119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 119910) 119889119897|119910=119883119904(119905119909)

]


119899119864[int

119904+(1119899)

119904

119885119894119873

(119897 119905 119909) 119889119897

100381610038161003816100381610038161003816100381610038161003816

F119904]

119875-as(50)



lim119899rarrinfin

119899int

119904+(1119899)

119904

119885119894119873

(119897 119905 119909) 119889119897 = 119885119894119873

(119904 119905 119909)


(51)


120577119894119873

(119904 119883119904(119905 119909)) = 119864 [119885

119894119873

(119904 119905 119909)10038161003816100381610038161003816F

119904] = 119885

119894119873

(119904 119905 119909)


(52)


120577119894119873

(119904 119883119904(119905 119909)) = 119885

119894119873

(119904 119905 119909)


(53)


119873rarrinfin120577119894119873


119894=1(lim

119873rarrinfin120577119894119873


120577 (119905 119909) =

infin

sum

119894=1


120577119894119873

(119905 119909)) 119890119894 (119905 119909) isin 119862

120577 (119905 119909) = 0 (119905 119909) notin 119862

(54)



119885 (120596 119904 119905 119909) =

infin

sum

119894=1


119885119894119873

(120596 119904 119905 119909)) 119890119894 (55)





3(Γ2(119905 Γ

1(119905 119909))) with


119901

P(Ω 119862

120575(119877))

Γ1(119905 119909) = 119884 (sdot 119905 119909)


119901

P(Ω 119862

120575(119877)) 997888rarr 119871

119901

(Ω 119877)

Γ2(119905 119881) = 119881 (119905)

Γ3 119871

119901

(Ω 119877) 997888rarr 119877 Γ3120585 = 119864120585

(56)


|119881(119905) minus 119880(119904)|119871119901(Ω119877)

le |119881 (119905) minus 119881 (119904)|119871119901(Ω119877)

+ 119890minus120575119901119904

|119881 minus 119880||119871

119901

P(Ω119862120575(119877))

(57)


3is continuous





119905119905ge0






119889119883119906

(119904) = 119860119883119906

(119904) 119889119904 + 119865 (119904 119883119906

119904) 119889119904

+ 119866 (119904 119883119906

119904) 119877 (119904 119883

119906

119904 119906 (119904)) 119889119904 + 119866 (119904 119883

119906

119904) 119889119882 (119904)

119904 isin [119905infin)

119883119906

119905= 119909

(58)


(119904 119905 119909) or simply by 119883119906


119869 (119906) = 119864int

infin

119905

119890minus120582119904

119892 (119904 119883119906

119904 119906 (119904)) 119889119904 (59)




120595 (119905 119909 119911) = inf 119892 (119905 119909 119906) + 119911119877 (119905 119909 119906) 119906 isin 119880

(60)


Γ (119905 119909 119911) = 119906 isin 119880 119892 (119905 119909 119906) + 119911119877 (119905 119909 119906) = 120595 (119905 119909 119911)

(61)




119898119892

119862)









119892



120582 gt (minus120575 minus 120583 +1198712

119911

2) or (minus120575 +

1198712

119877

2 (119901 minus 1))

or (1198712

119877119898119892

2 (119902 minus 119898119892)minus 120578 (119902)119898

119892)

(62)


119869 (119906) = 120592 (119905 119909) + 119864int

infin

119905

119890minus120582119904

[minus120595 (119904 119883119906

119904 120577 (119904 119883

119906

119904)) + 120577 (119904 119883

119906

119904)

times 119877 (119904 119883119906

119904 119906 (119904)) + 119892 (119904 119883

119906

119904 119906 (119904))] 119889119904

(63)


119905119905ge0



119882119906

(119904) = 119882 (119904) + int

119904

119905and119904

119877 (120590119883119906

120590 119906 (120590)) 119889120590 119904 isin [0infin)

120588 (119879) = exp(int119879

119905

minus119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882 (119904)

minus1

2int

119879

119905

1003816100381610038161003816119877 (119904 119883119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)

(64)



such that

119875119906

|F119879

= 120588 (119879) 119875|F119879

(65)


119905119905ge0


119889119883119906

(119904) = 119860119883119906

(119904) 119889119904 + 119865 (119904 119883119906

119904) 119889119904

+ 119866 (119904 119883119906

119904) 119889119882

119906

(119904) 119904 isin [119905infin)

119883119906

119905= 119909

(66)


F119906

119905119905ge0 119875

119906


119883119906

(119904) = 119890(119904minus119905)119860

119909 (0) + int

119904

119905

119890(119904minus120590)119860

119865 (120590119883119906

120590) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883119906

120590) 119889119882

119906

(120590) 119904 isin [119905infin)

119883119906

119905= 119909 isin C

119884119906

(119904) minus 119884119906

(119879) + int

119879

119904

119885119906

(120590) 119889119882119906

(120590) + 120582int

119879

119904

119884119906

(120590) 119889120590

= int

119879

119904

120595 (120590119883119906

120590 119885

119906

(120590)) 119889120590 0 le 119904 le 119879 lt infin

(67)


119884119906

(119904) + int

119879

119904

119890minus120582120590

119885119906

(120590) 119889119882 (120590)

= int

119879

119904

119890minus120582120590

[120595 (120590119883119906

120590 119885

119906

(120590))

minus119885119906

(120590) 119877 (120590119883119906

120590 119906 (120590))] 119889120590

+ 119890minus120582119879

119884119906

(119879)

(68)


119864119906

[120588(119879)minus119903

] = 119864119906

[exp 119903 (int119879

119905

119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882

119906

(119904)

minus1

2int

119879

119905

1003816100381610038161003816119877 (119904 119883119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)]

= 119864119906

[exp(int119879

119905

119903119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882

119906

(119904)

minus1

2int

119879

119905

11990321003816100381610038161003816119877 (119904 119883

119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)

times exp 119903 (119903 minus 1)2

int

119879

119905

1003816100381610038161003816119877 (119904 119883119906

119904 119906 (119904))

10038161003816100381610038162

119889119904]

le 119890(12)119903(119903minus1)119879119871

2

119877119864119906

times exp(int119879

119905

2119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882

119906

(119904)

minus1

2int

119879

119905

41003816100381610038161003816119877 (119904 119883

119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)

= 119890(12)119903(119903minus1)119879119871

2

119877

(69)

It follows that

119864(int

119879

119905

|119890minus120582119904

119885119906

(119904)|2

119889119904)

12

= 119864119906

[(int

119879

119905

10038161003816100381610038161003816119890minus120582119904

119885119906

(119904)10038161003816100381610038161003816

2

119889119904)

12

120588minus1

]

le (119864119906

int

119879

119905

10038161003816100381610038161003816119890minus120582119904

119885119906

(119904)10038161003816100381610038161003816

2

119889119904)

12

times (119864119906

120588minus2

)12

lt infin

(70)


119890minus120582119879

119864119884119906

(119879) minus 119884119906

(119905)

= 119864int

119879

119905

119890minus120582120590

[minus120595 (120590119883119906

120590 119885

119906

(120590))

+119885119906

(120590) 119877 (120590119883119906

120590 119906 (120590))] 119889120590

(71)


120575(119877)) so that

119864119906

|119884(119879 119905 119909)|119901

le 119862 exp (minus119901120575119879) (72)


119864 |119884 (119879 119905 119909)| = 119864119906

(120588minus1

(119879) |119884 (119879 119905 119909)|)

le 119864(120588minus119901(119901minus1)

)(119901minus1)119901

119864(|119884 (119879 119905 119909)|119901

)1119901

le 119862119890((1198712

1198772(119901minus1))minus120575))119879

(73)


|119883119906

119904|119902


1198641003816100381610038161003816119883

119906

119879

1003816100381610038161003816119898119892

le 119862119890(1198712

119877119898119892(2119902minus2119898

119892)minus1

minus120578(119902)119898119892)119879

(74)



119864int

infin

119905

119890minus120582120590 1003816100381610038161003816119892 (120590119883

119906

120590 119906 (120590))

1003816100381610038161003816 119889120590 lt infin (75)

Since 119884119906(119905 119905 119909) = 120592(119905 119909) and 119885119906

(119904 119905 119909) = 120577(119904 119883119906

119904(119905 119909)) 119875-


119890minus120582119879

119864119884119906

(119879) minus 119907 (119905 119909)

= 119864int

119879

119905

119890minus120582120590

[minus120595 (120590119883119906

120590 120577 (120590 119883

119906

120590))

+120577 (120590119883119906

120590) 119877 (120590119883

119906

120590 119906 (120590))] 119889120590

(76)


119890minus120582120590

119892(120590119883119906

120590 119906(120590))119889120590 and


119869 (119906) = 120592 (119905 119909)

+ 119864int

infin

119905

119890minus120582119904

[minus120595 (119904 119883119906

119904 120577 (119904 119883

119906

119904)) + 120577 (119904 119883

119906

119904) 119877

times (119904 119883119906

119904 119906 (119904)) + 119892 (119904 119883

119906

119904 119906 (119904))] 119889119904

(77)






119906 (119904) = Γ0(119904 119883

119906

119904 120577 (119904 119883

119906

119904))


(78)



119889119883 (119904)=119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904+119866 (119904 119883

119904)

times(119877 (119904 119883119904 Γ

0(119904 119883

119904 120577 (119904 119883

119904))) 119889119904 + 119889119882 (119904))

119904 isin [119905infin)

119883119905= 119909

(79)


119906 (119904) = Γ0(119904 119883

119904 120577 (119904 119883

119904)) (80)







119905119905ge0


F119905119905ge0 119875119882 119906119883

119906

) where (ΩF F119905119905ge0 119875 119882) is an




119869 (119906) = 119864int

infin

119905

119890minus120582119904

119892 (119904 119883119906

119904 119906 (119904)) 119889119904 (81)



119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904 + 119866 (119904 119883

119904)

times (119877 (119904 119883119904 Γ

0(119904 119883

119904 120577 (119904 119883

119904))) 119889119904 +119889119882 (119904))

119904 isin [119905infin)

119883119905= 119909

(82)


119905119905ge0 119875119882)



119883 (119904) = 119890(119904minus119905)119860

119909 (0) + int

119904

119905

119890(119904minus120590)119860

119865 (120590119883120590) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883120590) 119877

times (120590119883120590 Γ

0(120590119883

120590 120577 (120590 119883

120590))) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883120590) 119889119882

120590 119904 isin [119905infin)

(83)

119883119905= 119909 (84)




119905119905ge0 119875119882)We define

120588 (119879) = exp(int119879

119905

minus119877lowast

(120590 119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590))) 119889119882 (120590)

minus1

2int

119879

119905

1003816100381610038161003816119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590)))10038161003816100381610038162

119889120590)

(85)


1198820

(119904) = 119882 (119904) + int

119904

119905and119904

119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590))) 119889120590


(86)


1198750

|F119879

= 120588 (119879) 119875|F119879

(87)

Let us denote by F0

119905119905ge0


[0infin) F0

119905119905ge0 119875

0


119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119905 119883119904) 119889119904

+ 119866 (119904 119883119904) 119889119882

0

(119904) 119904 isin [119905infin)

119883119905= 119909

120588 (119879) = exp(int119879

119905

minus119877lowast

(120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590)) 119889119882

0

(120590)

+ 12int

119879

119905

1003816100381610038161003816119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590)))10038161003816100381610038162

119889120590)

(88)




119905119905ge0


119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904

+ 119866 (119904 119883119904) 119889119882 (119904) 119904 isin [119905infin)

119883119905= 119909

(89)


1198821

(119904) = 119882 (119904) minus int

119904

119905and119904

119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590))) 119889120590

(90)





0



119906

) one has

119869 (119906 119905 119909) ge 120592 (119905 119909) (91)


119906 (119904) = Γ0(119904 119883

119906

119904 120577 (119904 119883

119906

119904))

119875 minus 119886119904 119891119900119903 119886119897119898119900119904119905 119890119907119890119903119910 119904 isin [119905infin)

(92)



119906 (119904) = Γ0(119904 119883

119904 120577 (119904 119883

119904)) (93)


6 Applications



119889119911119906

(119905 120585) = Δ119911119906

(119905 120585) 119889119905 + 119891 (119905 119911119906

119905(120585)) 119889119905

+

119889

sum

119894=1

119892119894(119905 119911

119906

119905(120585)) [119903

119894

(120585) 119906119894

(119905) 119889119905 + 119889119882119894

(119905)]

119911119906

0(120579 120585) = 119909 (120579 120585) 120585 isin 119861 120579 isin [minus1 0]

119911119906

(119905 120585) = 0 119905 isin [0infin) 120585 isin 120597119861

(94)

Here119882 = (1198821

1198822

119882119889



119867 = 1198712


119865 (119905 119909) (120585) = 119891 (119905 119909 (120585))

(119866 (119905 119909) 119911) (120585) =

119889

sum

119894=1

119892119894(119905 119909 (120585)) 119911

119894

(120585)


(95)


(119861) withdomain11988222

(119861)⋂11988212


Hypothesis 1 holds



119869 (119906) = 119864int

infin

0

119890minus120582119905

[int119861

120590 (120585 119911119906

119905(120585)) 119889120585 + 119906

2

(119905)] 119889119905 (96)



119861

120590(119905 119910(120585) 119906)119889120585 + 1199062 and 119877(119910 119906) = (int

119861

1199031

(120585)1199061

119889120585

int119861

1199032

(120585)1199062

119889120585 int119861

119903119889

(120585)119906119889

119889120585) for 119910 isin 119862([minus1 0]119867) 119906 =

(1199061

1199062

119906119889


0



119905119905ge0 119875119882 119906 119911

sdot(sdot)) and

setting

119906 (119904) = Γ0(119904 119911

119904(sdot) 120577 (119904 119911

119904(sdot))) (97)


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014




le 211987310038161003816100381610038161199052 minus 1199051

1003816100381610038161003816 + Δ12

119890minus120575(1199051+Δ)

times(119864(int

infin

0

119890212057511990410038161003816100381610038161003816119885119894119873

(119904 1199051 119909

1)minus119885

119894119873

(119904 1199052 119909

2)10038161003816100381610038161003816

2

119889119904)

1199012

)

1119901

le 211987310038161003816100381610038161199052 minus 1199051

1003816100381610038161003816 + Δ12

119890minus120575(1199051+Δ)

times (119864(int

infin

0

11989021205751199041003816100381610038161003816119885 (119904 1199051 1199091)minus119885 (119904 1199052 1199092)

10038161003816100381610038162

119889119904)

1199012

)

1119901

(45)


119905

119885119894119873


119864int119905+Δ

119905

119864119885119894119873


(ΩCF119905) to 119877


120577119894119873


119899119864int

119905+(1119899)

119905

119885119894119873

(119904 119905 119909) 119889119904


(46)


denote 119864[119885119894119873

(119897 119904 119910)]|119910=119883119904(119905119909)


119904(119905 119909) with the map 119910 rarr 119864[119885

119894119873


119904-


119891119898 Ω 997888rarr C 119891

119898=

119873119898

sum

119896=1

ℎ(119898)

119896119868119891119898=ℎ(119898)

119896 119873

119898isin 119873 (47)

where ℎ(119898)1 ℎ

(119898)


119873119898

119896=1119891

119898=

ℎ(119898)

119896 such that

1003816100381610038161003816119891119898 (120596) minus 119883119904(120596)



int119861

int

119904+(1119899)

119904

119885119894119873

(119897 119905 119909) 119889119897119889119875

= int119861

int

119904+(1119899)

119904

119885119894119873

(119897 119904 119883119904) 119889119897119889119875

= 119864119868119861int

119904+(1119899)

119904

119885119894119873

(119897 119904 119883119904) 119889119897


119864(119868119861int

119904+(1119899)

119904

119885119894119873

(119897 119904 119891119898) 119889119897)


119873119898

sum

119896=1

119864(119868119861119868119891119898=ℎ(119898)

119896int

119904+(1119899)

119904

119885119894119873

(119897 119904 ℎ(119898)

119896) 119889119897)


119864(119868119861

119873119898

sum

119896=1

119868119891119898=ℎ(119898)

119896)119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 ℎ(119898)

119896) 119889119897


119864119868119861(119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 119910) 119889119897|119910=119891119898

)


int119861

(119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 119910) 119889119897|119910=119891119898

)119889119875

= int119861

(119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 119910) 119889119897|119910=119883119904

)119889119875

(49)

and we get that

120577119894119873

(119904 119883119904(119905 119909)) = lim inf

119899rarrinfin

119899

times [119864int

119904+(1119899)

119904

119885119894119873

(119897 119904 119910) 119889119897|119910=119883119904(119905119909)

]


119899119864[int

119904+(1119899)

119904

119885119894119873

(119897 119905 119909) 119889119897

100381610038161003816100381610038161003816100381610038161003816

F119904]

119875-as(50)



lim119899rarrinfin

119899int

119904+(1119899)

119904

119885119894119873

(119897 119905 119909) 119889119897 = 119885119894119873

(119904 119905 119909)


(51)


120577119894119873

(119904 119883119904(119905 119909)) = 119864 [119885

119894119873

(119904 119905 119909)10038161003816100381610038161003816F

119904] = 119885

119894119873

(119904 119905 119909)


(52)


120577119894119873

(119904 119883119904(119905 119909)) = 119885

119894119873

(119904 119905 119909)


(53)


119873rarrinfin120577119894119873


119894=1(lim

119873rarrinfin120577119894119873


120577 (119905 119909) =

infin

sum

119894=1


120577119894119873

(119905 119909)) 119890119894 (119905 119909) isin 119862

120577 (119905 119909) = 0 (119905 119909) notin 119862

(54)



119885 (120596 119904 119905 119909) =

infin

sum

119894=1


119885119894119873

(120596 119904 119905 119909)) 119890119894 (55)





3(Γ2(119905 Γ

1(119905 119909))) with


119901

P(Ω 119862

120575(119877))

Γ1(119905 119909) = 119884 (sdot 119905 119909)


119901

P(Ω 119862

120575(119877)) 997888rarr 119871

119901

(Ω 119877)

Γ2(119905 119881) = 119881 (119905)

Γ3 119871

119901

(Ω 119877) 997888rarr 119877 Γ3120585 = 119864120585

(56)


|119881(119905) minus 119880(119904)|119871119901(Ω119877)

le |119881 (119905) minus 119881 (119904)|119871119901(Ω119877)

+ 119890minus120575119901119904

|119881 minus 119880||119871

119901

P(Ω119862120575(119877))

(57)


3is continuous





119905119905ge0






119889119883119906

(119904) = 119860119883119906

(119904) 119889119904 + 119865 (119904 119883119906

119904) 119889119904

+ 119866 (119904 119883119906

119904) 119877 (119904 119883

119906

119904 119906 (119904)) 119889119904 + 119866 (119904 119883

119906

119904) 119889119882 (119904)

119904 isin [119905infin)

119883119906

119905= 119909

(58)


(119904 119905 119909) or simply by 119883119906


119869 (119906) = 119864int

infin

119905

119890minus120582119904

119892 (119904 119883119906

119904 119906 (119904)) 119889119904 (59)




120595 (119905 119909 119911) = inf 119892 (119905 119909 119906) + 119911119877 (119905 119909 119906) 119906 isin 119880

(60)


Γ (119905 119909 119911) = 119906 isin 119880 119892 (119905 119909 119906) + 119911119877 (119905 119909 119906) = 120595 (119905 119909 119911)

(61)




119898119892

119862)









119892



120582 gt (minus120575 minus 120583 +1198712

119911

2) or (minus120575 +

1198712

119877

2 (119901 minus 1))

or (1198712

119877119898119892

2 (119902 minus 119898119892)minus 120578 (119902)119898

119892)

(62)


119869 (119906) = 120592 (119905 119909) + 119864int

infin

119905

119890minus120582119904

[minus120595 (119904 119883119906

119904 120577 (119904 119883

119906

119904)) + 120577 (119904 119883

119906

119904)

times 119877 (119904 119883119906

119904 119906 (119904)) + 119892 (119904 119883

119906

119904 119906 (119904))] 119889119904

(63)


119905119905ge0



119882119906

(119904) = 119882 (119904) + int

119904

119905and119904

119877 (120590119883119906

120590 119906 (120590)) 119889120590 119904 isin [0infin)

120588 (119879) = exp(int119879

119905

minus119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882 (119904)

minus1

2int

119879

119905

1003816100381610038161003816119877 (119904 119883119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)

(64)



such that

119875119906

|F119879

= 120588 (119879) 119875|F119879

(65)


119905119905ge0


119889119883119906

(119904) = 119860119883119906

(119904) 119889119904 + 119865 (119904 119883119906

119904) 119889119904

+ 119866 (119904 119883119906

119904) 119889119882

119906

(119904) 119904 isin [119905infin)

119883119906

119905= 119909

(66)


F119906

119905119905ge0 119875

119906


119883119906

(119904) = 119890(119904minus119905)119860

119909 (0) + int

119904

119905

119890(119904minus120590)119860

119865 (120590119883119906

120590) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883119906

120590) 119889119882

119906

(120590) 119904 isin [119905infin)

119883119906

119905= 119909 isin C

119884119906

(119904) minus 119884119906

(119879) + int

119879

119904

119885119906

(120590) 119889119882119906

(120590) + 120582int

119879

119904

119884119906

(120590) 119889120590

= int

119879

119904

120595 (120590119883119906

120590 119885

119906

(120590)) 119889120590 0 le 119904 le 119879 lt infin

(67)


119884119906

(119904) + int

119879

119904

119890minus120582120590

119885119906

(120590) 119889119882 (120590)

= int

119879

119904

119890minus120582120590

[120595 (120590119883119906

120590 119885

119906

(120590))

minus119885119906

(120590) 119877 (120590119883119906

120590 119906 (120590))] 119889120590

+ 119890minus120582119879

119884119906

(119879)

(68)


119864119906

[120588(119879)minus119903

] = 119864119906

[exp 119903 (int119879

119905

119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882

119906

(119904)

minus1

2int

119879

119905

1003816100381610038161003816119877 (119904 119883119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)]

= 119864119906

[exp(int119879

119905

119903119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882

119906

(119904)

minus1

2int

119879

119905

11990321003816100381610038161003816119877 (119904 119883

119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)

times exp 119903 (119903 minus 1)2

int

119879

119905

1003816100381610038161003816119877 (119904 119883119906

119904 119906 (119904))

10038161003816100381610038162

119889119904]

le 119890(12)119903(119903minus1)119879119871

2

119877119864119906

times exp(int119879

119905

2119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882

119906

(119904)

minus1

2int

119879

119905

41003816100381610038161003816119877 (119904 119883

119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)

= 119890(12)119903(119903minus1)119879119871

2

119877

(69)

It follows that

119864(int

119879

119905

|119890minus120582119904

119885119906

(119904)|2

119889119904)

12

= 119864119906

[(int

119879

119905

10038161003816100381610038161003816119890minus120582119904

119885119906

(119904)10038161003816100381610038161003816

2

119889119904)

12

120588minus1

]

le (119864119906

int

119879

119905

10038161003816100381610038161003816119890minus120582119904

119885119906

(119904)10038161003816100381610038161003816

2

119889119904)

12

times (119864119906

120588minus2

)12

lt infin

(70)


119890minus120582119879

119864119884119906

(119879) minus 119884119906

(119905)

= 119864int

119879

119905

119890minus120582120590

[minus120595 (120590119883119906

120590 119885

119906

(120590))

+119885119906

(120590) 119877 (120590119883119906

120590 119906 (120590))] 119889120590

(71)


120575(119877)) so that

119864119906

|119884(119879 119905 119909)|119901

le 119862 exp (minus119901120575119879) (72)


119864 |119884 (119879 119905 119909)| = 119864119906

(120588minus1

(119879) |119884 (119879 119905 119909)|)

le 119864(120588minus119901(119901minus1)

)(119901minus1)119901

119864(|119884 (119879 119905 119909)|119901

)1119901

le 119862119890((1198712

1198772(119901minus1))minus120575))119879

(73)


|119883119906

119904|119902


1198641003816100381610038161003816119883

119906

119879

1003816100381610038161003816119898119892

le 119862119890(1198712

119877119898119892(2119902minus2119898

119892)minus1

minus120578(119902)119898119892)119879

(74)



119864int

infin

119905

119890minus120582120590 1003816100381610038161003816119892 (120590119883

119906

120590 119906 (120590))

1003816100381610038161003816 119889120590 lt infin (75)

Since 119884119906(119905 119905 119909) = 120592(119905 119909) and 119885119906

(119904 119905 119909) = 120577(119904 119883119906

119904(119905 119909)) 119875-


119890minus120582119879

119864119884119906

(119879) minus 119907 (119905 119909)

= 119864int

119879

119905

119890minus120582120590

[minus120595 (120590119883119906

120590 120577 (120590 119883

119906

120590))

+120577 (120590119883119906

120590) 119877 (120590119883

119906

120590 119906 (120590))] 119889120590

(76)


119890minus120582120590

119892(120590119883119906

120590 119906(120590))119889120590 and


119869 (119906) = 120592 (119905 119909)

+ 119864int

infin

119905

119890minus120582119904

[minus120595 (119904 119883119906

119904 120577 (119904 119883

119906

119904)) + 120577 (119904 119883

119906

119904) 119877

times (119904 119883119906

119904 119906 (119904)) + 119892 (119904 119883

119906

119904 119906 (119904))] 119889119904

(77)






119906 (119904) = Γ0(119904 119883

119906

119904 120577 (119904 119883

119906

119904))


(78)



119889119883 (119904)=119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904+119866 (119904 119883

119904)

times(119877 (119904 119883119904 Γ

0(119904 119883

119904 120577 (119904 119883

119904))) 119889119904 + 119889119882 (119904))

119904 isin [119905infin)

119883119905= 119909

(79)


119906 (119904) = Γ0(119904 119883

119904 120577 (119904 119883

119904)) (80)







119905119905ge0


F119905119905ge0 119875119882 119906119883

119906

) where (ΩF F119905119905ge0 119875 119882) is an




119869 (119906) = 119864int

infin

119905

119890minus120582119904

119892 (119904 119883119906

119904 119906 (119904)) 119889119904 (81)



119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904 + 119866 (119904 119883

119904)

times (119877 (119904 119883119904 Γ

0(119904 119883

119904 120577 (119904 119883

119904))) 119889119904 +119889119882 (119904))

119904 isin [119905infin)

119883119905= 119909

(82)


119905119905ge0 119875119882)



119883 (119904) = 119890(119904minus119905)119860

119909 (0) + int

119904

119905

119890(119904minus120590)119860

119865 (120590119883120590) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883120590) 119877

times (120590119883120590 Γ

0(120590119883

120590 120577 (120590 119883

120590))) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883120590) 119889119882

120590 119904 isin [119905infin)

(83)

119883119905= 119909 (84)




119905119905ge0 119875119882)We define

120588 (119879) = exp(int119879

119905

minus119877lowast

(120590 119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590))) 119889119882 (120590)

minus1

2int

119879

119905

1003816100381610038161003816119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590)))10038161003816100381610038162

119889120590)

(85)


1198820

(119904) = 119882 (119904) + int

119904

119905and119904

119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590))) 119889120590


(86)


1198750

|F119879

= 120588 (119879) 119875|F119879

(87)

Let us denote by F0

119905119905ge0


[0infin) F0

119905119905ge0 119875

0


119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119905 119883119904) 119889119904

+ 119866 (119904 119883119904) 119889119882

0

(119904) 119904 isin [119905infin)

119883119905= 119909

120588 (119879) = exp(int119879

119905

minus119877lowast

(120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590)) 119889119882

0

(120590)

+ 12int

119879

119905

1003816100381610038161003816119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590)))10038161003816100381610038162

119889120590)

(88)




119905119905ge0


119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904

+ 119866 (119904 119883119904) 119889119882 (119904) 119904 isin [119905infin)

119883119905= 119909

(89)


1198821

(119904) = 119882 (119904) minus int

119904

119905and119904

119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590))) 119889120590

(90)





0



119906

) one has

119869 (119906 119905 119909) ge 120592 (119905 119909) (91)


119906 (119904) = Γ0(119904 119883

119906

119904 120577 (119904 119883

119906

119904))

119875 minus 119886119904 119891119900119903 119886119897119898119900119904119905 119890119907119890119903119910 119904 isin [119905infin)

(92)



119906 (119904) = Γ0(119904 119883

119904 120577 (119904 119883

119904)) (93)


6 Applications



119889119911119906

(119905 120585) = Δ119911119906

(119905 120585) 119889119905 + 119891 (119905 119911119906

119905(120585)) 119889119905

+

119889

sum

119894=1

119892119894(119905 119911

119906

119905(120585)) [119903

119894

(120585) 119906119894

(119905) 119889119905 + 119889119882119894

(119905)]

119911119906

0(120579 120585) = 119909 (120579 120585) 120585 isin 119861 120579 isin [minus1 0]

119911119906

(119905 120585) = 0 119905 isin [0infin) 120585 isin 120597119861

(94)

Here119882 = (1198821

1198822

119882119889



119867 = 1198712


119865 (119905 119909) (120585) = 119891 (119905 119909 (120585))

(119866 (119905 119909) 119911) (120585) =

119889

sum

119894=1

119892119894(119905 119909 (120585)) 119911

119894

(120585)


(95)


(119861) withdomain11988222

(119861)⋂11988212


Hypothesis 1 holds



119869 (119906) = 119864int

infin

0

119890minus120582119905

[int119861

120590 (120585 119911119906

119905(120585)) 119889120585 + 119906

2

(119905)] 119889119905 (96)



119861

120590(119905 119910(120585) 119906)119889120585 + 1199062 and 119877(119910 119906) = (int

119861

1199031

(120585)1199061

119889120585

int119861

1199032

(120585)1199062

119889120585 int119861

119903119889

(120585)119906119889

119889120585) for 119910 isin 119862([minus1 0]119867) 119906 =

(1199061

1199062

119906119889


0



119905119905ge0 119875119882 119906 119911

sdot(sdot)) and

setting

119906 (119904) = Γ0(119904 119911

119904(sdot) 120577 (119904 119911

119904(sdot))) (97)


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014





119885 (120596 119904 119905 119909) =

infin

sum

119894=1


119885119894119873

(120596 119904 119905 119909)) 119890119894 (55)





3(Γ2(119905 Γ

1(119905 119909))) with


119901

P(Ω 119862

120575(119877))

Γ1(119905 119909) = 119884 (sdot 119905 119909)


119901

P(Ω 119862

120575(119877)) 997888rarr 119871

119901

(Ω 119877)

Γ2(119905 119881) = 119881 (119905)

Γ3 119871

119901

(Ω 119877) 997888rarr 119877 Γ3120585 = 119864120585

(56)


|119881(119905) minus 119880(119904)|119871119901(Ω119877)

le |119881 (119905) minus 119881 (119904)|119871119901(Ω119877)

+ 119890minus120575119901119904

|119881 minus 119880||119871

119901

P(Ω119862120575(119877))

(57)


3is continuous





119905119905ge0






119889119883119906

(119904) = 119860119883119906

(119904) 119889119904 + 119865 (119904 119883119906

119904) 119889119904

+ 119866 (119904 119883119906

119904) 119877 (119904 119883

119906

119904 119906 (119904)) 119889119904 + 119866 (119904 119883

119906

119904) 119889119882 (119904)

119904 isin [119905infin)

119883119906

119905= 119909

(58)


(119904 119905 119909) or simply by 119883119906


119869 (119906) = 119864int

infin

119905

119890minus120582119904

119892 (119904 119883119906

119904 119906 (119904)) 119889119904 (59)




120595 (119905 119909 119911) = inf 119892 (119905 119909 119906) + 119911119877 (119905 119909 119906) 119906 isin 119880

(60)


Γ (119905 119909 119911) = 119906 isin 119880 119892 (119905 119909 119906) + 119911119877 (119905 119909 119906) = 120595 (119905 119909 119911)

(61)




119898119892

119862)









119892



120582 gt (minus120575 minus 120583 +1198712

119911

2) or (minus120575 +

1198712

119877

2 (119901 minus 1))

or (1198712

119877119898119892

2 (119902 minus 119898119892)minus 120578 (119902)119898

119892)

(62)


119869 (119906) = 120592 (119905 119909) + 119864int

infin

119905

119890minus120582119904

[minus120595 (119904 119883119906

119904 120577 (119904 119883

119906

119904)) + 120577 (119904 119883

119906

119904)

times 119877 (119904 119883119906

119904 119906 (119904)) + 119892 (119904 119883

119906

119904 119906 (119904))] 119889119904

(63)


119905119905ge0



119882119906

(119904) = 119882 (119904) + int

119904

119905and119904

119877 (120590119883119906

120590 119906 (120590)) 119889120590 119904 isin [0infin)

120588 (119879) = exp(int119879

119905

minus119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882 (119904)

minus1

2int

119879

119905

1003816100381610038161003816119877 (119904 119883119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)

(64)



such that

119875119906

|F119879

= 120588 (119879) 119875|F119879

(65)


119905119905ge0


119889119883119906

(119904) = 119860119883119906

(119904) 119889119904 + 119865 (119904 119883119906

119904) 119889119904

+ 119866 (119904 119883119906

119904) 119889119882

119906

(119904) 119904 isin [119905infin)

119883119906

119905= 119909

(66)


F119906

119905119905ge0 119875

119906


119883119906

(119904) = 119890(119904minus119905)119860

119909 (0) + int

119904

119905

119890(119904minus120590)119860

119865 (120590119883119906

120590) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883119906

120590) 119889119882

119906

(120590) 119904 isin [119905infin)

119883119906

119905= 119909 isin C

119884119906

(119904) minus 119884119906

(119879) + int

119879

119904

119885119906

(120590) 119889119882119906

(120590) + 120582int

119879

119904

119884119906

(120590) 119889120590

= int

119879

119904

120595 (120590119883119906

120590 119885

119906

(120590)) 119889120590 0 le 119904 le 119879 lt infin

(67)


119884119906

(119904) + int

119879

119904

119890minus120582120590

119885119906

(120590) 119889119882 (120590)

= int

119879

119904

119890minus120582120590

[120595 (120590119883119906

120590 119885

119906

(120590))

minus119885119906

(120590) 119877 (120590119883119906

120590 119906 (120590))] 119889120590

+ 119890minus120582119879

119884119906

(119879)

(68)


119864119906

[120588(119879)minus119903

] = 119864119906

[exp 119903 (int119879

119905

119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882

119906

(119904)

minus1

2int

119879

119905

1003816100381610038161003816119877 (119904 119883119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)]

= 119864119906

[exp(int119879

119905

119903119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882

119906

(119904)

minus1

2int

119879

119905

11990321003816100381610038161003816119877 (119904 119883

119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)

times exp 119903 (119903 minus 1)2

int

119879

119905

1003816100381610038161003816119877 (119904 119883119906

119904 119906 (119904))

10038161003816100381610038162

119889119904]

le 119890(12)119903(119903minus1)119879119871

2

119877119864119906

times exp(int119879

119905

2119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882

119906

(119904)

minus1

2int

119879

119905

41003816100381610038161003816119877 (119904 119883

119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)

= 119890(12)119903(119903minus1)119879119871

2

119877

(69)

It follows that

119864(int

119879

119905

|119890minus120582119904

119885119906

(119904)|2

119889119904)

12

= 119864119906

[(int

119879

119905

10038161003816100381610038161003816119890minus120582119904

119885119906

(119904)10038161003816100381610038161003816

2

119889119904)

12

120588minus1

]

le (119864119906

int

119879

119905

10038161003816100381610038161003816119890minus120582119904

119885119906

(119904)10038161003816100381610038161003816

2

119889119904)

12

times (119864119906

120588minus2

)12

lt infin

(70)


119890minus120582119879

119864119884119906

(119879) minus 119884119906

(119905)

= 119864int

119879

119905

119890minus120582120590

[minus120595 (120590119883119906

120590 119885

119906

(120590))

+119885119906

(120590) 119877 (120590119883119906

120590 119906 (120590))] 119889120590

(71)


120575(119877)) so that

119864119906

|119884(119879 119905 119909)|119901

le 119862 exp (minus119901120575119879) (72)


119864 |119884 (119879 119905 119909)| = 119864119906

(120588minus1

(119879) |119884 (119879 119905 119909)|)

le 119864(120588minus119901(119901minus1)

)(119901minus1)119901

119864(|119884 (119879 119905 119909)|119901

)1119901

le 119862119890((1198712

1198772(119901minus1))minus120575))119879

(73)


|119883119906

119904|119902


1198641003816100381610038161003816119883

119906

119879

1003816100381610038161003816119898119892

le 119862119890(1198712

119877119898119892(2119902minus2119898

119892)minus1

minus120578(119902)119898119892)119879

(74)



119864int

infin

119905

119890minus120582120590 1003816100381610038161003816119892 (120590119883

119906

120590 119906 (120590))

1003816100381610038161003816 119889120590 lt infin (75)

Since 119884119906(119905 119905 119909) = 120592(119905 119909) and 119885119906

(119904 119905 119909) = 120577(119904 119883119906

119904(119905 119909)) 119875-


119890minus120582119879

119864119884119906

(119879) minus 119907 (119905 119909)

= 119864int

119879

119905

119890minus120582120590

[minus120595 (120590119883119906

120590 120577 (120590 119883

119906

120590))

+120577 (120590119883119906

120590) 119877 (120590119883

119906

120590 119906 (120590))] 119889120590

(76)


119890minus120582120590

119892(120590119883119906

120590 119906(120590))119889120590 and


119869 (119906) = 120592 (119905 119909)

+ 119864int

infin

119905

119890minus120582119904

[minus120595 (119904 119883119906

119904 120577 (119904 119883

119906

119904)) + 120577 (119904 119883

119906

119904) 119877

times (119904 119883119906

119904 119906 (119904)) + 119892 (119904 119883

119906

119904 119906 (119904))] 119889119904

(77)






119906 (119904) = Γ0(119904 119883

119906

119904 120577 (119904 119883

119906

119904))


(78)



119889119883 (119904)=119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904+119866 (119904 119883

119904)

times(119877 (119904 119883119904 Γ

0(119904 119883

119904 120577 (119904 119883

119904))) 119889119904 + 119889119882 (119904))

119904 isin [119905infin)

119883119905= 119909

(79)


119906 (119904) = Γ0(119904 119883

119904 120577 (119904 119883

119904)) (80)







119905119905ge0


F119905119905ge0 119875119882 119906119883

119906

) where (ΩF F119905119905ge0 119875 119882) is an




119869 (119906) = 119864int

infin

119905

119890minus120582119904

119892 (119904 119883119906

119904 119906 (119904)) 119889119904 (81)



119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904 + 119866 (119904 119883

119904)

times (119877 (119904 119883119904 Γ

0(119904 119883

119904 120577 (119904 119883

119904))) 119889119904 +119889119882 (119904))

119904 isin [119905infin)

119883119905= 119909

(82)


119905119905ge0 119875119882)



119883 (119904) = 119890(119904minus119905)119860

119909 (0) + int

119904

119905

119890(119904minus120590)119860

119865 (120590119883120590) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883120590) 119877

times (120590119883120590 Γ

0(120590119883

120590 120577 (120590 119883

120590))) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883120590) 119889119882

120590 119904 isin [119905infin)

(83)

119883119905= 119909 (84)




119905119905ge0 119875119882)We define

120588 (119879) = exp(int119879

119905

minus119877lowast

(120590 119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590))) 119889119882 (120590)

minus1

2int

119879

119905

1003816100381610038161003816119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590)))10038161003816100381610038162

119889120590)

(85)


1198820

(119904) = 119882 (119904) + int

119904

119905and119904

119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590))) 119889120590


(86)


1198750

|F119879

= 120588 (119879) 119875|F119879

(87)

Let us denote by F0

119905119905ge0


[0infin) F0

119905119905ge0 119875

0


119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119905 119883119904) 119889119904

+ 119866 (119904 119883119904) 119889119882

0

(119904) 119904 isin [119905infin)

119883119905= 119909

120588 (119879) = exp(int119879

119905

minus119877lowast

(120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590)) 119889119882

0

(120590)

+ 12int

119879

119905

1003816100381610038161003816119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590)))10038161003816100381610038162

119889120590)

(88)




119905119905ge0


119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904

+ 119866 (119904 119883119904) 119889119882 (119904) 119904 isin [119905infin)

119883119905= 119909

(89)


1198821

(119904) = 119882 (119904) minus int

119904

119905and119904

119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590))) 119889120590

(90)





0



119906

) one has

119869 (119906 119905 119909) ge 120592 (119905 119909) (91)


119906 (119904) = Γ0(119904 119883

119906

119904 120577 (119904 119883

119906

119904))

119875 minus 119886119904 119891119900119903 119886119897119898119900119904119905 119890119907119890119903119910 119904 isin [119905infin)

(92)



119906 (119904) = Γ0(119904 119883

119904 120577 (119904 119883

119904)) (93)


6 Applications



119889119911119906

(119905 120585) = Δ119911119906

(119905 120585) 119889119905 + 119891 (119905 119911119906

119905(120585)) 119889119905

+

119889

sum

119894=1

119892119894(119905 119911

119906

119905(120585)) [119903

119894

(120585) 119906119894

(119905) 119889119905 + 119889119882119894

(119905)]

119911119906

0(120579 120585) = 119909 (120579 120585) 120585 isin 119861 120579 isin [minus1 0]

119911119906

(119905 120585) = 0 119905 isin [0infin) 120585 isin 120597119861

(94)

Here119882 = (1198821

1198822

119882119889



119867 = 1198712


119865 (119905 119909) (120585) = 119891 (119905 119909 (120585))

(119866 (119905 119909) 119911) (120585) =

119889

sum

119894=1

119892119894(119905 119909 (120585)) 119911

119894

(120585)


(95)


(119861) withdomain11988222

(119861)⋂11988212


Hypothesis 1 holds



119869 (119906) = 119864int

infin

0

119890minus120582119905

[int119861

120590 (120585 119911119906

119905(120585)) 119889120585 + 119906

2

(119905)] 119889119905 (96)



119861

120590(119905 119910(120585) 119906)119889120585 + 1199062 and 119877(119910 119906) = (int

119861

1199031

(120585)1199061

119889120585

int119861

1199032

(120585)1199062

119889120585 int119861

119903119889

(120585)119906119889

119889120585) for 119910 isin 119862([minus1 0]119867) 119906 =

(1199061

1199062

119906119889


0



119905119905ge0 119875119882 119906 119911

sdot(sdot)) and

setting

119906 (119904) = Γ0(119904 119911

119904(sdot) 120577 (119904 119911

119904(sdot))) (97)


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014





such that

119875119906

|F119879

= 120588 (119879) 119875|F119879

(65)


119905119905ge0


119889119883119906

(119904) = 119860119883119906

(119904) 119889119904 + 119865 (119904 119883119906

119904) 119889119904

+ 119866 (119904 119883119906

119904) 119889119882

119906

(119904) 119904 isin [119905infin)

119883119906

119905= 119909

(66)


F119906

119905119905ge0 119875

119906


119883119906

(119904) = 119890(119904minus119905)119860

119909 (0) + int

119904

119905

119890(119904minus120590)119860

119865 (120590119883119906

120590) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883119906

120590) 119889119882

119906

(120590) 119904 isin [119905infin)

119883119906

119905= 119909 isin C

119884119906

(119904) minus 119884119906

(119879) + int

119879

119904

119885119906

(120590) 119889119882119906

(120590) + 120582int

119879

119904

119884119906

(120590) 119889120590

= int

119879

119904

120595 (120590119883119906

120590 119885

119906

(120590)) 119889120590 0 le 119904 le 119879 lt infin

(67)


119884119906

(119904) + int

119879

119904

119890minus120582120590

119885119906

(120590) 119889119882 (120590)

= int

119879

119904

119890minus120582120590

[120595 (120590119883119906

120590 119885

119906

(120590))

minus119885119906

(120590) 119877 (120590119883119906

120590 119906 (120590))] 119889120590

+ 119890minus120582119879

119884119906

(119879)

(68)


119864119906

[120588(119879)minus119903

] = 119864119906

[exp 119903 (int119879

119905

119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882

119906

(119904)

minus1

2int

119879

119905

1003816100381610038161003816119877 (119904 119883119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)]

= 119864119906

[exp(int119879

119905

119903119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882

119906

(119904)

minus1

2int

119879

119905

11990321003816100381610038161003816119877 (119904 119883

119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)

times exp 119903 (119903 minus 1)2

int

119879

119905

1003816100381610038161003816119877 (119904 119883119906

119904 119906 (119904))

10038161003816100381610038162

119889119904]

le 119890(12)119903(119903minus1)119879119871

2

119877119864119906

times exp(int119879

119905

2119877lowast

(119904 119883119906

119904 119906 (119904)) 119889119882

119906

(119904)

minus1

2int

119879

119905

41003816100381610038161003816119877 (119904 119883

119906

119904 119906 (119904))

10038161003816100381610038162

119889119904)

= 119890(12)119903(119903minus1)119879119871

2

119877

(69)

It follows that

119864(int

119879

119905

|119890minus120582119904

119885119906

(119904)|2

119889119904)

12

= 119864119906

[(int

119879

119905

10038161003816100381610038161003816119890minus120582119904

119885119906

(119904)10038161003816100381610038161003816

2

119889119904)

12

120588minus1

]

le (119864119906

int

119879

119905

10038161003816100381610038161003816119890minus120582119904

119885119906

(119904)10038161003816100381610038161003816

2

119889119904)

12

times (119864119906

120588minus2

)12

lt infin

(70)


119890minus120582119879

119864119884119906

(119879) minus 119884119906

(119905)

= 119864int

119879

119905

119890minus120582120590

[minus120595 (120590119883119906

120590 119885

119906

(120590))

+119885119906

(120590) 119877 (120590119883119906

120590 119906 (120590))] 119889120590

(71)


120575(119877)) so that

119864119906

|119884(119879 119905 119909)|119901

le 119862 exp (minus119901120575119879) (72)


119864 |119884 (119879 119905 119909)| = 119864119906

(120588minus1

(119879) |119884 (119879 119905 119909)|)

le 119864(120588minus119901(119901minus1)

)(119901minus1)119901

119864(|119884 (119879 119905 119909)|119901

)1119901

le 119862119890((1198712

1198772(119901minus1))minus120575))119879

(73)


|119883119906

119904|119902


1198641003816100381610038161003816119883

119906

119879

1003816100381610038161003816119898119892

le 119862119890(1198712

119877119898119892(2119902minus2119898

119892)minus1

minus120578(119902)119898119892)119879

(74)



119864int

infin

119905

119890minus120582120590 1003816100381610038161003816119892 (120590119883

119906

120590 119906 (120590))

1003816100381610038161003816 119889120590 lt infin (75)

Since 119884119906(119905 119905 119909) = 120592(119905 119909) and 119885119906

(119904 119905 119909) = 120577(119904 119883119906

119904(119905 119909)) 119875-


119890minus120582119879

119864119884119906

(119879) minus 119907 (119905 119909)

= 119864int

119879

119905

119890minus120582120590

[minus120595 (120590119883119906

120590 120577 (120590 119883

119906

120590))

+120577 (120590119883119906

120590) 119877 (120590119883

119906

120590 119906 (120590))] 119889120590

(76)


119890minus120582120590

119892(120590119883119906

120590 119906(120590))119889120590 and


119869 (119906) = 120592 (119905 119909)

+ 119864int

infin

119905

119890minus120582119904

[minus120595 (119904 119883119906

119904 120577 (119904 119883

119906

119904)) + 120577 (119904 119883

119906

119904) 119877

times (119904 119883119906

119904 119906 (119904)) + 119892 (119904 119883

119906

119904 119906 (119904))] 119889119904

(77)






119906 (119904) = Γ0(119904 119883

119906

119904 120577 (119904 119883

119906

119904))


(78)



119889119883 (119904)=119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904+119866 (119904 119883

119904)

times(119877 (119904 119883119904 Γ

0(119904 119883

119904 120577 (119904 119883

119904))) 119889119904 + 119889119882 (119904))

119904 isin [119905infin)

119883119905= 119909

(79)


119906 (119904) = Γ0(119904 119883

119904 120577 (119904 119883

119904)) (80)







119905119905ge0


F119905119905ge0 119875119882 119906119883

119906

) where (ΩF F119905119905ge0 119875 119882) is an




119869 (119906) = 119864int

infin

119905

119890minus120582119904

119892 (119904 119883119906

119904 119906 (119904)) 119889119904 (81)



119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904 + 119866 (119904 119883

119904)

times (119877 (119904 119883119904 Γ

0(119904 119883

119904 120577 (119904 119883

119904))) 119889119904 +119889119882 (119904))

119904 isin [119905infin)

119883119905= 119909

(82)


119905119905ge0 119875119882)



119883 (119904) = 119890(119904minus119905)119860

119909 (0) + int

119904

119905

119890(119904minus120590)119860

119865 (120590119883120590) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883120590) 119877

times (120590119883120590 Γ

0(120590119883

120590 120577 (120590 119883

120590))) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883120590) 119889119882

120590 119904 isin [119905infin)

(83)

119883119905= 119909 (84)




119905119905ge0 119875119882)We define

120588 (119879) = exp(int119879

119905

minus119877lowast

(120590 119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590))) 119889119882 (120590)

minus1

2int

119879

119905

1003816100381610038161003816119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590)))10038161003816100381610038162

119889120590)

(85)


1198820

(119904) = 119882 (119904) + int

119904

119905and119904

119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590))) 119889120590


(86)


1198750

|F119879

= 120588 (119879) 119875|F119879

(87)

Let us denote by F0

119905119905ge0


[0infin) F0

119905119905ge0 119875

0


119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119905 119883119904) 119889119904

+ 119866 (119904 119883119904) 119889119882

0

(119904) 119904 isin [119905infin)

119883119905= 119909

120588 (119879) = exp(int119879

119905

minus119877lowast

(120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590)) 119889119882

0

(120590)

+ 12int

119879

119905

1003816100381610038161003816119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590)))10038161003816100381610038162

119889120590)

(88)




119905119905ge0


119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904

+ 119866 (119904 119883119904) 119889119882 (119904) 119904 isin [119905infin)

119883119905= 119909

(89)


1198821

(119904) = 119882 (119904) minus int

119904

119905and119904

119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590))) 119889120590

(90)





0



119906

) one has

119869 (119906 119905 119909) ge 120592 (119905 119909) (91)


119906 (119904) = Γ0(119904 119883

119906

119904 120577 (119904 119883

119906

119904))

119875 minus 119886119904 119891119900119903 119886119897119898119900119904119905 119890119907119890119903119910 119904 isin [119905infin)

(92)



119906 (119904) = Γ0(119904 119883

119904 120577 (119904 119883

119904)) (93)


6 Applications



119889119911119906

(119905 120585) = Δ119911119906

(119905 120585) 119889119905 + 119891 (119905 119911119906

119905(120585)) 119889119905

+

119889

sum

119894=1

119892119894(119905 119911

119906

119905(120585)) [119903

119894

(120585) 119906119894

(119905) 119889119905 + 119889119882119894

(119905)]

119911119906

0(120579 120585) = 119909 (120579 120585) 120585 isin 119861 120579 isin [minus1 0]

119911119906

(119905 120585) = 0 119905 isin [0infin) 120585 isin 120597119861

(94)

Here119882 = (1198821

1198822

119882119889



119867 = 1198712


119865 (119905 119909) (120585) = 119891 (119905 119909 (120585))

(119866 (119905 119909) 119911) (120585) =

119889

sum

119894=1

119892119894(119905 119909 (120585)) 119911

119894

(120585)


(95)


(119861) withdomain11988222

(119861)⋂11988212


Hypothesis 1 holds



119869 (119906) = 119864int

infin

0

119890minus120582119905

[int119861

120590 (120585 119911119906

119905(120585)) 119889120585 + 119906

2

(119905)] 119889119905 (96)



119861

120590(119905 119910(120585) 119906)119889120585 + 1199062 and 119877(119910 119906) = (int

119861

1199031

(120585)1199061

119889120585

int119861

1199032

(120585)1199062

119889120585 int119861

119903119889

(120585)119906119889

119889120585) for 119910 isin 119862([minus1 0]119867) 119906 =

(1199061

1199062

119906119889


0



119905119905ge0 119875119882 119906 119911

sdot(sdot)) and

setting

119906 (119904) = Γ0(119904 119911

119904(sdot) 120577 (119904 119911

119904(sdot))) (97)


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014





119864int

infin

119905

119890minus120582120590 1003816100381610038161003816119892 (120590119883

119906

120590 119906 (120590))

1003816100381610038161003816 119889120590 lt infin (75)

Since 119884119906(119905 119905 119909) = 120592(119905 119909) and 119885119906

(119904 119905 119909) = 120577(119904 119883119906

119904(119905 119909)) 119875-


119890minus120582119879

119864119884119906

(119879) minus 119907 (119905 119909)

= 119864int

119879

119905

119890minus120582120590

[minus120595 (120590119883119906

120590 120577 (120590 119883

119906

120590))

+120577 (120590119883119906

120590) 119877 (120590119883

119906

120590 119906 (120590))] 119889120590

(76)


119890minus120582120590

119892(120590119883119906

120590 119906(120590))119889120590 and


119869 (119906) = 120592 (119905 119909)

+ 119864int

infin

119905

119890minus120582119904

[minus120595 (119904 119883119906

119904 120577 (119904 119883

119906

119904)) + 120577 (119904 119883

119906

119904) 119877

times (119904 119883119906

119904 119906 (119904)) + 119892 (119904 119883

119906

119904 119906 (119904))] 119889119904

(77)






119906 (119904) = Γ0(119904 119883

119906

119904 120577 (119904 119883

119906

119904))


(78)



119889119883 (119904)=119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904+119866 (119904 119883

119904)

times(119877 (119904 119883119904 Γ

0(119904 119883

119904 120577 (119904 119883

119904))) 119889119904 + 119889119882 (119904))

119904 isin [119905infin)

119883119905= 119909

(79)


119906 (119904) = Γ0(119904 119883

119904 120577 (119904 119883

119904)) (80)







119905119905ge0


F119905119905ge0 119875119882 119906119883

119906

) where (ΩF F119905119905ge0 119875 119882) is an




119869 (119906) = 119864int

infin

119905

119890minus120582119904

119892 (119904 119883119906

119904 119906 (119904)) 119889119904 (81)



119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904 + 119866 (119904 119883

119904)

times (119877 (119904 119883119904 Γ

0(119904 119883

119904 120577 (119904 119883

119904))) 119889119904 +119889119882 (119904))

119904 isin [119905infin)

119883119905= 119909

(82)


119905119905ge0 119875119882)



119883 (119904) = 119890(119904minus119905)119860

119909 (0) + int

119904

119905

119890(119904minus120590)119860

119865 (120590119883120590) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883120590) 119877

times (120590119883120590 Γ

0(120590119883

120590 120577 (120590 119883

120590))) 119889120590

+ int

119904

119905

119890(119904minus120590)119860

119866 (120590119883120590) 119889119882

120590 119904 isin [119905infin)

(83)

119883119905= 119909 (84)




119905119905ge0 119875119882)We define

120588 (119879) = exp(int119879

119905

minus119877lowast

(120590 119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590))) 119889119882 (120590)

minus1

2int

119879

119905

1003816100381610038161003816119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590)))10038161003816100381610038162

119889120590)

(85)


1198820

(119904) = 119882 (119904) + int

119904

119905and119904

119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590))) 119889120590


(86)


1198750

|F119879

= 120588 (119879) 119875|F119879

(87)

Let us denote by F0

119905119905ge0


[0infin) F0

119905119905ge0 119875

0


119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119905 119883119904) 119889119904

+ 119866 (119904 119883119904) 119889119882

0

(119904) 119904 isin [119905infin)

119883119905= 119909

120588 (119879) = exp(int119879

119905

minus119877lowast

(120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590)) 119889119882

0

(120590)

+ 12int

119879

119905

1003816100381610038161003816119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590)))10038161003816100381610038162

119889120590)

(88)




119905119905ge0


119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904

+ 119866 (119904 119883119904) 119889119882 (119904) 119904 isin [119905infin)

119883119905= 119909

(89)


1198821

(119904) = 119882 (119904) minus int

119904

119905and119904

119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590))) 119889120590

(90)





0



119906

) one has

119869 (119906 119905 119909) ge 120592 (119905 119909) (91)


119906 (119904) = Γ0(119904 119883

119906

119904 120577 (119904 119883

119906

119904))

119875 minus 119886119904 119891119900119903 119886119897119898119900119904119905 119890119907119890119903119910 119904 isin [119905infin)

(92)



119906 (119904) = Γ0(119904 119883

119904 120577 (119904 119883

119904)) (93)


6 Applications



119889119911119906

(119905 120585) = Δ119911119906

(119905 120585) 119889119905 + 119891 (119905 119911119906

119905(120585)) 119889119905

+

119889

sum

119894=1

119892119894(119905 119911

119906

119905(120585)) [119903

119894

(120585) 119906119894

(119905) 119889119905 + 119889119882119894

(119905)]

119911119906

0(120579 120585) = 119909 (120579 120585) 120585 isin 119861 120579 isin [minus1 0]

119911119906

(119905 120585) = 0 119905 isin [0infin) 120585 isin 120597119861

(94)

Here119882 = (1198821

1198822

119882119889



119867 = 1198712


119865 (119905 119909) (120585) = 119891 (119905 119909 (120585))

(119866 (119905 119909) 119911) (120585) =

119889

sum

119894=1

119892119894(119905 119909 (120585)) 119911

119894

(120585)


(95)


(119861) withdomain11988222

(119861)⋂11988212


Hypothesis 1 holds



119869 (119906) = 119864int

infin

0

119890minus120582119905

[int119861

120590 (120585 119911119906

119905(120585)) 119889120585 + 119906

2

(119905)] 119889119905 (96)



119861

120590(119905 119910(120585) 119906)119889120585 + 1199062 and 119877(119910 119906) = (int

119861

1199031

(120585)1199061

119889120585

int119861

1199032

(120585)1199062

119889120585 int119861

119903119889

(120585)119906119889

119889120585) for 119910 isin 119862([minus1 0]119867) 119906 =

(1199061

1199062

119906119889


0



119905119905ge0 119875119882 119906 119911

sdot(sdot)) and

setting

119906 (119904) = Γ0(119904 119911

119904(sdot) 120577 (119904 119911

119904(sdot))) (97)


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014





119905119905ge0 119875119882)We define

120588 (119879) = exp(int119879

119905

minus119877lowast

(120590 119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590))) 119889119882 (120590)

minus1

2int

119879

119905

1003816100381610038161003816119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590)))10038161003816100381610038162

119889120590)

(85)


1198820

(119904) = 119882 (119904) + int

119904

119905and119904

119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590))) 119889120590


(86)


1198750

|F119879

= 120588 (119879) 119875|F119879

(87)

Let us denote by F0

119905119905ge0


[0infin) F0

119905119905ge0 119875

0


119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119905 119883119904) 119889119904

+ 119866 (119904 119883119904) 119889119882

0

(119904) 119904 isin [119905infin)

119883119905= 119909

120588 (119879) = exp(int119879

119905

minus119877lowast

(120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590)) 119889119882

0

(120590)

+ 12int

119879

119905

1003816100381610038161003816119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590)))10038161003816100381610038162

119889120590)

(88)




119905119905ge0


119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904

+ 119866 (119904 119883119904) 119889119882 (119904) 119904 isin [119905infin)

119883119905= 119909

(89)


1198821

(119904) = 119882 (119904) minus int

119904

119905and119904

119877 (120590119883120590 Γ

0(120590 119883

120590 120577 (120590 119883

120590))) 119889120590

(90)





0



119906

) one has

119869 (119906 119905 119909) ge 120592 (119905 119909) (91)


119906 (119904) = Γ0(119904 119883

119906

119904 120577 (119904 119883

119906

119904))

119875 minus 119886119904 119891119900119903 119886119897119898119900119904119905 119890119907119890119903119910 119904 isin [119905infin)

(92)



119906 (119904) = Γ0(119904 119883

119904 120577 (119904 119883

119904)) (93)


6 Applications



119889119911119906

(119905 120585) = Δ119911119906

(119905 120585) 119889119905 + 119891 (119905 119911119906

119905(120585)) 119889119905

+

119889

sum

119894=1

119892119894(119905 119911

119906

119905(120585)) [119903

119894

(120585) 119906119894

(119905) 119889119905 + 119889119882119894

(119905)]

119911119906

0(120579 120585) = 119909 (120579 120585) 120585 isin 119861 120579 isin [minus1 0]

119911119906

(119905 120585) = 0 119905 isin [0infin) 120585 isin 120597119861

(94)

Here119882 = (1198821

1198822

119882119889



119867 = 1198712


119865 (119905 119909) (120585) = 119891 (119905 119909 (120585))

(119866 (119905 119909) 119911) (120585) =

119889

sum

119894=1

119892119894(119905 119909 (120585)) 119911

119894

(120585)


(95)


(119861) withdomain11988222

(119861)⋂11988212


Hypothesis 1 holds



119869 (119906) = 119864int

infin

0

119890minus120582119905

[int119861

120590 (120585 119911119906

119905(120585)) 119889120585 + 119906

2

(119905)] 119889119905 (96)



119861

120590(119905 119910(120585) 119906)119889120585 + 1199062 and 119877(119910 119906) = (int

119861

1199031

(120585)1199061

119889120585

int119861

1199032

(120585)1199062

119889120585 int119861

119903119889

(120585)119906119889

119889120585) for 119910 isin 119862([minus1 0]119867) 119906 =

(1199061

1199062

119906119889


0



119905119905ge0 119875119882 119906 119911

sdot(sdot)) and

setting

119906 (119904) = Γ0(119904 119911

119904(sdot) 120577 (119904 119911

119904(sdot))) (97)


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014





119869 (119906) = 119864int

infin

0

119890minus120582119905

[int119861

120590 (120585 119911119906

119905(120585)) 119889120585 + 119906

2

(119905)] 119889119905 (96)



119861

120590(119905 119910(120585) 119906)119889120585 + 1199062 and 119877(119910 119906) = (int

119861

1199031

(120585)1199061

119889120585

int119861

1199032

(120585)1199062

119889120585 int119861

119903119889

(120585)119906119889

119889120585) for 119910 isin 119862([minus1 0]119867) 119906 =

(1199061

1199062

119906119889


0



119905119905ge0 119875119882 119906 119911

sdot(sdot)) and

setting

119906 (119904) = Γ0(119904 119911

119904(sdot) 120577 (119904 119911

119904(sdot))) (97)


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014














Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014



infinite horizon optimal control of stochastic delay...

Documents