Hindawi Publishing CorporationISRNMathematical AnalysisVolume 2013 Article ID 549876 5 pageshttpdxdoiorg1011552013549876

Research ArticleOn Singular Dissipative Fourth-Order DifferentialOperator in Lim-4 Case

Ekin ULurlu and Elgiz Bairamov

Department of Mathematics Ankara University Tandogan 06100 Ankara Turkey

Correspondence should be addressed to Elgiz Bairamov bairamovscienceankaraedutr

Received 26 June 2013 Accepted 25 July 2013

Academic Editors D D Hai and W Shen

Copyright copy 2013 E Ugurlu and E Bairamov This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A singular dissipative fourth-order differential operator in lim-4 case is considered To investigate the spectral analysis of thisoperator it is passed to the inverse operator with the help of Everittrsquos method Finally using Lidskiırsquos theorem it is proved that thesystem of all eigen- and associated functions of this operator (also the boundary value problem) is complete

1 Introduction

In 1910 Weyl showed that [1] the singular second-order dif-ferential operators can be fallen into two classes operators inlimit-circle case and operators in limit-point caseThe opera-tors in limit-circle case have the solutions that are all in squareintegrable space However in limit-point case only one lin-early independent solution can be in square integrable spaceThe development of this theory belongs to Titchmarsh [2]After these fundamental works second-order singular differ-ential operators have been developed by many authors (egsee [3ndash6])

Following the methods of Weyl and Titchmarsh the the-ory for higher order equations and Hamiltonian systems wasconstructed in [7ndash19]

In [7] a regular self-adjoint fourth-order boundary valueproblem was investigated Further Greenrsquos function and theresolvent operator were constructed In [8 13] such a con-struction was done for the singular self-adjoint fourth orderboundary value problem In [9 10] higher-order differentialequations whose coefficients are complex-valued were stud-ied In 1974 Walker [20] showed that an arbitrary order self-adjoint eigenvalue problem can be represented as an equiv-alent self-adjoint Hamiltonian systems Further the devel-opments in the theory of singular self-adjoint Hamiltoniansystems were given in [14ndash19]

On the other hand an important class of the nonself-adjoint operators is the class of the dissipative operators [21]

It is known that all eigenvalues of the dissipative operators liein the closed upper half-plane but this analysis is so weakThere are somemethods to complete the analysis of the dissi-pative operators Some of them are Livsicrsquos Kreinrsquos and Lid-skiırsquos theorems and functional model [21 22]These methodswere used in the literature for the second-order differentialoperators (see [22ndash26]) In this paper a singular dissipativefourth order differential operator in lim-4 case is investigatedIn particular using Lidskiırsquos theorem it is shown that thesystem of all eigen- and associated functions is complete in1198712(Λ)

2 Preliminaries

Let 119871 denote the linear nonself-adjoint operator in theHilbertspace119867with the domain119863(119871) The element 119910 isin 119863(119871) 119910 = 0is called a root function of the operator 119871 correspondingto the eigenvalue 120582

0 if all powers of 119871 are defined on this

element and (119871 minus 1205820119868)119899119910 = 0 for some 119899 gt 0

The functions 1199101 1199102 119910

119896are called the associated

functions of the eigenfunction 1199100if they belong to 119863(119871) and

the equalities 119871119910119895= 1205820119910119895+ 119910119895minus1

119895 = 1 2 119896 holdThe completeness of the system of all eigen- and associ-

ated functions of 119871 is equivalent to the completeness of thesystem of all root functions of this operator

If for the operator 119871 with dense domain 119863(119871) in 119867 theinequality I(119871119910 119910) ge 0 (119910 isin 119863(119871)) holds then 119871 is calleddissipative

2 ISRNMathematical Analysis

Theorem 1 (see [26]) Let 119871 be an invertible operator Thenminus119871 is dissipative if and only if the inverse operator 119871

minus1 of 119871 isdissipative

It is known that a kernel 119860(119905 119904) of an integral operatorAdefned by

A119891 = int

119887

119886

119860 (119905 119904) 119891 (119904) 119889119904 119891 isin 1198712(119886 119887) (1)

where minusinfin le 119886 lt 119887 le infin is called a Hilbert-Schmidt kernelif |119860(119905 119904)|

2 is integrable on (119886 119887)2

int

119887

119886

int

119887

119886

|119860 (119905 119904)|2119889119905119889119904 lt infin (2)

A kernel satisfying the property 119860(119905 119904) = 119860(119904 119905) is calleda Hermitian kernel Properties of Hermitian kernels andrelated integral operators can be found in [27] (further see[28]) Now we shall remind some results Let us consider theequation which is related with (1) as

120583A119891 = 119891 (3)

The function 119891which differs from zero is called a charac-teristic function ofAwhich corresponds to the characteristicvalue 120583 If a Hermitian Hilbert-Schmidt kernel is not nullthen it posseses at least one characteristic value and this char-acteristic value (every characteristic value) is real Furtherthere is a finite orthonormal base of characteristic functionsfor each characteristic value 120583 ofA Using the union of thesebases one obtains an orthonormal system of characteristicfunctions of the kernel 119860(119905 119904) If 120593

1 1205932 120593

119899are distinct

members of such a system belonging respectively to the char-acteristic value 120583

1 1205832 120583

119899(not necessarily all different)

then for 1 le V le 119899 120593V otimes 120593V = 120593V(119905)120593V(119904) is an orthonormalsystem of two variables For such a system the equivalence

119860 (119905 119904) ≗

infin

sum

119899=1

120593119899otimes 120593119899

120583119899

(4)

holds Moreover if 119892 = A119891 119891 isin 1198712 then the equivalence

119892 ≗

infin

sum

119899=1

(119892 120593119899) 120593119899=

infin

sum

119899=1

(119891 120593119899)

120583119899

120593119899 (5)

holds Further the series in (5) are relatively uniformly abso-lutely convergent However equivalence can be replaced byequality Before showing this let us consider the equality

119892 (119905) = 120582int119860 (119905 119904) 119891 (119904) 119889119904 (6)

Following the same idea of [27 pages 22] if 119860(119905 119904) is an 1198712-

kernel such that119860(119905 119904) is continuous and 119891 is an 1198712-function

then 119892(119905) is continuousHence if 119860(119905 119904) is a continuous Hermitian 119871

2-kernel (see[27 pages 127]) and 119892 = A119891 119891 isin 119871

2 then the equality

119892 (119905) =

infin

sum

119899=1

(119892 120593119899) 120593119899(119905) =

infin

sum

119899=1

(119891 120593119899)

120583119899

120593119899(119905) (7)

holds and the series are uniformly absolutely convergent Inthis case taking 119892(119905) = (119891 119860(119905 sdot)) the series given in (4) con-verge to the continuous kernel 119860(119905 119904) uniformly in thevariable 119905 for every 119904 Hence from (4) one can get

119860 (119905 119905) =

infin

sum

119899=1

1003816100381610038161003816120593119899 (119905)1003816100381610038161003816

2

120583119899

(8)

Integrating both side from 119886 to 119887 we obtain

int

119887

119886

119860 (119905 119905) 119889119905 =

infin

sum

119899=1

1

120583119899

(9)

Consequently if 119860(119905 119905) is integrable on (119886 119887) then the seriesconverge This implies that A is of trace-class (nuclear) Forthe definition and properties of trace-class operators see forexample [21]

Above arguments given in [27] hold for Hermitian ker-nels However from arbitrary 119871

2-kernels one can pass to theHermitian kernels For example 119870

1(119905 119904) = 119860(119905 119904) + 119860(119904 119905)

and1198702(119905 119904) = 119894(119860(119905 119904)minus119860(119904 119905)) are Hermitian kernelsThese

Hermitian kernels are continuous and 1198712-kernels if 119860(119905 119904) is

so Hence above arguments hold for 1198701(119905 119904) and 119870

2(119905 119904) if

119860(119905 119904) is continuous 1198712-kernel such that 119860(119905 119905) is integrable

on (119886 119887) Further

119860 =1

2(1198701minus 1198941198702) (10)

Hence if 1198701and 119870

2are trace-class kernels then so is 119860

However this result has been given in [29 pages 526] as adefinition

LidskiırsquosTheorem (see [21 page 231]) If the dissipative opera-tor 119871 is the trace class operator then its system of root functionsis complete in the Hilbert space 119867

3 Statement of the Problem

The fourth order differential expression is considered as

120578 (119910) = 119910119894V

+ 119902 (119909) 119910 119909 isin Λ = [119886 119887) (11)

where minusinfin lt 119886 lt 119887 le infin 119886 is the regular point and 119887 is thesingular point for 120578 and 119902 is real-valued Lebesgue measur-able and locally integrable function on Λ

Let 1198712(Λ) be the Hilbert space consisting of all functions119910 such that int119887

119886|119910(119909)|

2119889119909 lt infinwith the inner product (119910 120594) =

int119887

119886119910(119909)120594(119909)119889119909Let

Ω =119910 isin 1198712(Λ) 119910 119910

1015840 11991010158401015840 119910101584010158401015840

isin ACloc (Λ) 120578 (119910) isin 1198712(Λ)

(12)

where ACloc(Λ) denotes the set of all locally absolutely con-tinuous functions on Λ For arbitrary 119910 120594 isin Ω Greenrsquosformula is obtained as

int

1199092

1199091

120578 (119910) 120594 (119909) 119889119909 minus int

1199092

1199091

119910 (119909) 120578 (120594) 119889119909

= [119910 120594] (1199092) minus [119910 120594] (119909

1)

(13)

ISRNMathematical Analysis 3

where 119886 le 1199091

lt 1199092

le 119887 and [119910 120594](119909) = 119910101584010158401015840(119909)120594(119909) minus

119910(119909)120594101584010158401015840

(119909) + 1199101015840(119909)12059410158401015840(119909) minus 119910

10158401015840(119909)1205941015840(119909) Greenrsquos formula

implies the fact that if 119910(119909 120582) and 120594(119909 120582) both satisfy theequation 120578(119910) = 120582119910 for the same value of 120582 then [119910 120594](119909)

is independent of 119909 and depends only on 120582 Further forarbitrary 119910 120594 isin Ω at singular point 119887 the limits [119910 120594](119887) =

lim119909rarr119887

[119910 120594](119909) and [119910 120594](119887) = lim119909rarr119887

[119910 120594](119909) exist andare finiteThe latter also follows fromGreenrsquos formula In factit is sufficient to get the second factors with their complexconjugates on the left-hand side in Greenrsquos formula

In this paper it is assumed that 119902(119909) satisfies the lim-4 caseconditions at 119887 (see [30] and references therein) Lim-4 caseis also known as Weylrsquos limit-circle case [8]

Let us consider the solutions 1205931(119909 120582) 120593

2(119909 120582) 120595

1(119909 120582)

and 1205952(119909 120582) of the equation

119910119894V

+ 119902 (119909) 119910 = 120582119910 119909 isin Λ (14)

where 120582 is some complex parameter satisfying the conditions

[[[

[

1205931(119886 120582) 120593

2(119886 120582) 120595

1(119886 120582) 120595

2(119886 120582)

1205931015840

1(119886 120582) 120593

1015840

2(119886 120582) 120595

1015840

1(119886 120582) 120595

1015840

2(119886 120582)

12059310158401015840

1(119886 120582) 120593

10158401015840

2(119886 120582) 120595

10158401015840

1(119886 120582) 120595

10158401015840

2(119886 120582)

120593101584010158401015840

1(119886 120582) 120593

101584010158401015840

2(119886 120582) 120595

101584010158401015840

1(119886 120582) 120595

101584010158401015840

2(119886 120582)

]]]

]

=

[[[

[

1205722

0 1205741

0

0 1205732

0 1205791

0 minus1205731

0 minus1205792

minus1205721

0 minus1205742

0

]]]

]

(15)

where all 120572119894 120573119894 120574119894 and 120579

119894(119894 = 1 2) are real numbers such

that 12057221205742minus12057211205741= 1 and 120573

11205791minus12057321205792= 1 For the existence of

these solutions see [7 8 13] Since lim-4 case holds for 120578 allsolutions 120593

119894(119909 120582) and 120595

119894(119909 120582) belong to 119871

2(Λ) It is clear that

[120593119903 120595119904](119886) = 120575

119903119904(1 le 119903 119904 le 2) where 120575

119903119904is the Kronocker

delta [120593119903 120593119904](119886) = 0 and [120595

119903 120595119904](119886) = 0

Let us set 119906119894(119909) = 120593

119894(119909 0) and 119911

119894(119909) = 120595

119894(119909 0) (119894 =

1 2) with 119909 isin Λ Then 119906119894and 119911

119894become the real solutions

of 120578(119910) = 0 (119909 isin Λ) Further they belong to Ω Hence forarbitrary 119910 isin Ω the values [119910 119906

1](119887) [119910 119906

2](119887) [119910 119911

1](119887)

and [119910 1199112](119887) exist and are finite

Let 119863(119873) be the set of all functions 119910 isin Ω satisfying theboundary conditions

1205721119910 (119886) + 120572

2119910101584010158401015840

(119886) = 0 (16)

12057311199101015840(119886) + 120573

211991010158401015840(119886) = 0 (17)

[119910 1199061] (119887) minus ℎ

1[119910 1199111] (119887) = 0 (18)

[119910 1199062] (119887) minus ℎ

2[119910 1199112] (119887) = 0 (19)

where120572119894and120573119894(119894 = 1 2) are real numbers given as previously

stated and ℎ1and ℎ

2are some complex numbers such that

ℎ1= Rℎ

1+ 119894Iℎ

1and ℎ2= Rℎ

2+ 119894Iℎ

2withIℎ

119894gt 0 119894 = 1 2

It should be noted that for any solutions 119910(119909 120582) of (14)conditions (16) and (17) respectively can be written as

[119910 1205931] (119886) = 0 (20)

[119910 1205932] (119886) = 0 (21)

The operator 119873 is defined on 119863(119873) as 119873119910 = 120578(119910) 119910 isin

119863(119873) 119909 isin Λ The main aim of this paper is to investigatethe spectral analysis of the operator 119873 (the boundary valueproblem (14)ndash(19))

4 Completeness Theorems

Let 119910119895(119909 120582) 1 le 119895 le 119903 be any 119903 (1 le 119903 le 4) solutions

of 120578(119910) = 120582119910 The notation 1198821199101 1199102 119910

119903 = 119882119910

1(119909 120582)

1199102(119909 120582) 119910

119903(119909 120582) denotes the Wronskian of order 119903 of

this set of functions [7 8 31]It is known that the following equality holds [7 8 13]

1198821199101 1199102 1199103 1199104 = minus [119910

1 1199102] (119909) [119910

3 1199104] (119909)

+ [1199101 1199103] (119909) [119910

2 1199104] (119909)

minus [1199101 1199104] (119909) [119910

2 1199103] (119909)

(22)

This equation also shows that 1198821199101 1199102 1199103 1199104 of any four

solutions of 120578(119910) = 120582119910 is independent of 119909 and depends onlyon 120582 [7 8]

Now consider the solutions 1205991(119909 120582) and 120599

2(119909 120582) of the

equation 120578(119910) = 120582119910 119909 isin Λ satisfying the conditions

[[[[

[

[1205991 1199061] (119887) [120599

2 1199061] (119887)

[1205991 1199111] (119887) [120599

2 1199111] (119887)

[1205991 1199062] (119887) [120599

2 1199062] (119887)

[1205991 1199112] (119887) [120599

2 1199112] (119887)

]]]]

]

=

[[[

[

ℎ1

0

1 0

0 ℎ2

0 1

]]]

]

(23)

For the existence of these solutions given with the previousconditions see [13] Clearly these solutions satisfy conditions(18) and (19) respectively Now let us set

119882(120582) = 1198821205931 1205932 1205991 1205992 (24)

Then119882(120582) becomes an entire function and the zeros of119882(120582)

coincide with the eigenvalues of the operator 119873 [13] Thisimplies that all zeros of 119882 (all eigenvalues of 119873) are discreteand that possible limit points of these zeros (eigenvalues of119873) can only occur at infinity

Using (22) we immediately have

1198821199061 1199062 1199111 1199112 = 1

[[[[

[

[1199061 1199061] (119909) [119906

2 1199061] (119909) [119911

1 1199061] (119909) [119911

2 1199061] (119909)

[1199061 1199062] (119909) [119906

2 1199062] (119909) [119911

1 1199062] (119909) [119911

2 1199062] (119909)

[1199061 1199111] (119909) [119906

2 1199111] (119909) [119911

1 1199111] (119909) [119911

2 1199111] (119909)

[1199061 1199112] (119909) [119906

2 1199112] (119909) [119911

1 1199112] (119909) [119911

2 1199112] (119909)

]]]]

]

=

[[[

[

0 0 minus1 0

0 0 0 minus1

1 0 0 0

0 1 0 0

]]]

]

(25)

4 ISRNMathematical Analysis

Hence the Plucker identity for the fourth-order case isobtained (see [13 p 435])

[119910 120594] (119909) = [119910 1199061] (119909) [120594 119911

1] (119909)

minus [119910 1199111] (119909) [120594 119906

1] (119909)

+ [119910 1199062] (119909) [120594 119911

2] (119909)

minus [119910 1199112] (119909) [120594 119906

2] (119909)

(26)

where 119910 120594 isin 119863(119873)

Theorem 2 The operator 119873 is dissipative in 1198712(Λ)

Proof For 119910 isin 119863(119873) we have

(119873119910 119910) minus (119910119873119910) = [119910 119910] (119887) minus [119910 119910] (119886) (27)

Since 119910 isin 119863(119873) a direct calculation shows that

[119910 119910] (119886) = 0 (28)

Further using (26) and conditions (18) and (19) one obtains

[119910 119910] (119887) = 2119894Iℎ1

1003816100381610038161003816[119910 1199111](119887)

1003816100381610038161003816

2

+ 2119894Iℎ2

1003816100381610038161003816[119910 1199112](119887)

1003816100381610038161003816

2

(29)

Substituting (28) and (29) into (27) it is obtained that

I (119873119910 119910) = Iℎ1

1003816100381610038161003816[119910 1199111] (119887)

1003816100381610038161003816

2

+ Iℎ2

1003816100381610038161003816[119910 1199112] (119887)

1003816100381610038161003816

2

(30)

and this completes the proof

Theorem 2 shows that all eigenvalues of119873 lie in the closedupper half-plane

Theorem 3 The operator 119873 has no real eigenvalue

Proof For 119910 isin 119863(119873) a direct calculation shows that

I (119873119910 119910) = I (1205821003817100381710038171003817119910

1003817100381710038171003817

2

) (31)

Now let 120582 = 1205820be a real eigenvalue of 119873 and let 120593

1(119909 1205820)

be the corresponding eigenfunctionThen (31) and (30) gives[1205931 1199111](119887) = [120593

1 1199112](119887) = 0 Using these equalities in (18) and

(19) one gets that [1205931 1199061](119887) = [120593

1 1199062](119887) = 0

Let us consider the solution 1205951(119909 1205820) Hence using (26)

it is obtained that

1 = [1205931 1205951] (119886) = [120593

1 1205951] (119887)

= [1205931 1199061] (119887) [120595

1 1199111] (119887)

minus [1205931 1199111] (119887) [120595

1 1199061] (119887)

+ [1205931 1199062] (119887) [120595

1 1199112] (119887)

minus [1205931 1199112] (119887) [120595

1 1199062] (119887) = 0

(32)

This contradiction completes the proof

From Theorem 3 it is obtained that all eigenvalues of 119873lie in the open upper half-plane In particular zero is not aneigenvalue of 119873 Hence the operator 119873minus1 exists

Consider the solutions 1199061(119909) 119906

2(119909) V

1(119909) and V

2(119909)

where V1(119909) = 119906

1(119909) minus ℎ

11199111(119909) and V

2(119909) = 119906

2(119909) minus

ℎ21199112(119909) 119906

1(119909) and 119906

2(119909) satisfy conditions (16) and (17)

respectively and V1(119909) and V

2(119909) satisfy conditions (18) and

(19) respectivelyThe equation119873119910 = 119891(119909) 119910 isin 119863(119873) 119909 isin Λ is equivalent

to the nonhomogeneous differential equation

119910119894V

+ 119902 (119909) 119910 = 119891 (119909) 119910 isin 119863 (119873) 119909 isin Λ (33)

subject to the boundary conditions

[119910 1199061] (119886) = 0

[119910 1199062] (119886) = 0

[119910 1199061] (119887) minus ℎ

1[119910 1199111] (119887) = 0

[119910 1199062] (119887) minus ℎ

2[119910 1199112] (119887) = 0

(34)

(compare the boundary conditions at 119886 with (20) and (21))Using Everittrsquos method [7] (further see [8 13]) the generalsolution is obtained as

119910 (119909) = int

119887

119886

119866 (119909 120577) 119891 (120577) 119889120577 (35)

where

119866 (119909 120577) = V119879 (120577) 119906 (119909) 119886 le 119909 lt 120577

V119879 (119909) 119906 (120577) 120577 lt 119909 le 119887(36)

V(119909) = 119906(119909) minus ℎ119911(119909) and

119906 (119909) = (1199061(119909)

1199062(119909)

) 119911 (119909) = (1199111(119909)

1199112(119909)

) ℎ = (ℎ1

0

0 ℎ2

)

(37)

The operator 119870 defined by

119870119891 = int

119887

119886

119866 (119909 120577) 119891 (120577) 119889120577 (38)

where 119891 isin 1198712(Λ) is the inverse operator of 119873 Hence the

completeness of the system of all eigen- and associated func-tions of 119873 is equivalent to the completeness of those of 119870 in1198712(Λ)Since 119866(119909 120577) is a continuous Hilbert-Schmidt kernel and

119866(119909 119909) is integrable on [119886 119887) the operator119870 is of trace classLet us consider the operator minus119870 Since119873 is dissipative in

1198712(Λ) minus119870 is also dissipative in 119871

2(Λ) Thus all conditions are

satisfied for Lidskiırsquos theorem So we have the following

Theorem 4 The system of all root functions of minus119870 (also 119870) iscomplete in 119871

2(Λ)

Since the completeness of the system of root functions(eigen- and associated functions) of 119870 is equivalent to thecompleteness of those of 119873 is obtained that the following

Theorem 5 All eigenvalues of the problem (14)ndash(19) lie in theopen upper half-plane and they are purely discrete The systemof all eigen- and associated functions of the problem (14)ndash(19)is complete in 119871

2(Λ)

ISRNMathematical Analysis 5

References

[1] H Weyl ldquoUber gewohnliche Differentialgleichungen mit Sin-gularitaten und die zugehorigen Entwicklungen willkurlicherFunktionenrdquo Mathematische Annalen vol 68 no 2 pp 220ndash269 1910

[2] E C TitchmarshEigenfunction Expansions Associatedwith Sec-ond Order Differential Equations Part 1 Oxford UniversityPress 2nd edition 1962

[3] E A Coddington and N Levinson Theory of Ordinary Differ-ential Equations McGraw-Hill New York NY USA 1955

[4] M A Naimark Linear Differential Operators Nauka MoscowRussia 2nd edition 1969 English translation Ungar New YorkNY USA 1st edition parts 1 1967 1st edition parts 2 1968

[5] I M Glazman Direct Methods of Qualitative Spectral Analysisof Singular Differential Operators Israel Program for ScientificTranslations Jerusalem Israel 1965

[6] F V Atkinson Discrete and Continuous Boundary ProblemsAcademic Press New York NY USA 1964

[7] W N Everitt ldquoThe Sturm-Liouville problem for fourth-orderdifferential equationsrdquo The Quarterly Journal of Mathematicsvol 8 pp 146ndash160 1957

[8] W N Everitt ldquoFourth order singular differential equationsrdquoMathematische Annalen vol 149 pp 320ndash340 1963

[9] W N Everitt ldquoSingular differential equations I The even ordercaserdquoMathematische Annalen vol 156 pp 9ndash24 1964

[10] W N Everitt ldquoSingular differential equations II Some self-adjoint even order casesrdquoTheQuarterly Journal of Mathematicsvol 18 pp 13ndash32 1967

[11] W N Everitt D B Hinton and J S W Wong ldquoOn the stronglimit-119899 classification of linear ordinary differential expressionsof order 2119899rdquo Proceedings of the London Mathematical Societyvol 29 pp 351ndash367 1974

[12] V I Kogan and F S Rofe-Beketov ldquoOn square-integrable solu-tions of symmetric systems of differential equations of arbitraryorderrdquo Proceedings of the Royal Society of Edinburgh A vol 74pp 5ndash40 1974

[13] C T Fulton ldquoThe Bessel-squared equation in the lim-2 lim-3and lim-4 casesrdquoThe Quarterly Journal of Mathematics vol 40no 160 pp 423ndash456 1989

[14] A M Krall ldquo119872(120582) theory for singular Hamiltonian systemswith one singular pointrdquo SIAM Journal on Mathematical Anal-ysis vol 20 no 3 pp 664ndash700 1989

[15] A M Krall ldquo119872(120582) theory for singular Hamiltonian systemswith two singular pointsrdquo SIAM Journal on MathematicalAnalysis vol 20 no 3 pp 700ndash715 1989

[16] D B Hinton and J K Shaw ldquoOn Titchmarsh-Weyl119872(120582)-func-tions for linear Hamiltonian systemsrdquo Journal of DifferentialEquations vol 40 no 3 pp 316ndash342 1981

[17] D BHinton and J K Shaw ldquoHamiltonian systems of limit pointor limit circle type with both endpoints singularrdquo Journal ofDifferential Equations vol 50 no 3 pp 444ndash464 1983

[18] D B Hinton and J K Shaw ldquoParameterization of the 119872(120582)

function for a Hamiltonian system of limit circle typerdquo Proceed-ings of the Royal Society of Edinburgh A vol 93 no 3-4 pp 349ndash360 1983

[19] D B Hinton and J K Shaw ldquoOn boundary value problems forHamiltonian systems with two singular pointsrdquo SIAM Journalon Mathematical Analysis vol 15 no 2 pp 272ndash286 1984

[20] P W Walker ldquoA vector-matrix formulation for formally sym-metric ordinary differential equations with applications to solu-tions of integrable squarerdquo Journal of the London MathematicalSociety vol 9 no 2 pp 151ndash159 197475

[21] I C Gohberg and M G Kreın Introduction to the Theoryof Linear Nonselfadjoint Operators American MathematicalSociety Providence RI USA 1969

[22] B S Pavlov ldquoSpectral analysis of a dissipative singularSchrodinger operator in terms of a functional modelrdquo in PartialDifferential Equations vol 65 of Itogi Nauki i Tekhniki SeriyaSovremennye Problemy Matematiki Fundamentalrsquonye Naprav-leniya pp 95ndash163 1991 English translation in Partial Differen-tial Equations VIII vol 65 of Encyclopaedia of MathematicalSciences pp 87ndash163 1996

[23] G Guseinov ldquoCompleteness theorem for the dissipative Sturm-Liouville operatorrdquo Doga Turkish Journal of Mathematics vol17 no 1 pp 48ndash54 1993

[24] E Bairamov and A M Krall ldquoDissipative operators generatedby the Sturm-Liouville differential expression in the Weyl limitcircle caserdquo Journal of Mathematical Analysis and Applicationsvol 254 no 1 pp 178ndash190 2001

[25] E Ugurlu and E Bairamov ldquoDissipative operators with impul-sive conditionsrdquo Journal of Mathematical Chemistry vol 51 pp1670ndash1680 2013

[26] Z Wang and H Wu ldquoDissipative non-self-adjoint Sturm-Liouville operators and completeness of their eigenfunctionsrdquoJournal of Mathematical Analysis and Applications vol 394 no1 pp 1ndash12 2012

[27] F Smithies Integral Equations Cambridge University PressNew York NY USA 1958

[28] F Riesz and B Sz-Nagy Functional Analysis Frederick UngarNew York NY USA 6th edition 1972

[29] E Prugovecki Quantum Mechanics in Hilbert Space vol 92Academic Press New York NY USA 2nd edition 1981

[30] M S P Eastham ldquoThe limit-2119899 case of symmetric differentialoperators of order 2119899rdquo Proceedings of the London MathematicalSociety vol 38 no 2 pp 272ndash294 1979

[31] E L Ince Ordinary Differential Equations London UK 1927

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Decision SciencesAdvances in

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Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 ISRNMathematical Analysis

Theorem 1 (see [26]) Let 119871 be an invertible operator Thenminus119871 is dissipative if and only if the inverse operator 119871

minus1 of 119871 isdissipative

It is known that a kernel 119860(119905 119904) of an integral operatorAdefned by

A119891 = int

119887

119886

119860 (119905 119904) 119891 (119904) 119889119904 119891 isin 1198712(119886 119887) (1)

where minusinfin le 119886 lt 119887 le infin is called a Hilbert-Schmidt kernelif |119860(119905 119904)|

2 is integrable on (119886 119887)2

int

119887

119886

int

119887

119886

|119860 (119905 119904)|2119889119905119889119904 lt infin (2)

A kernel satisfying the property 119860(119905 119904) = 119860(119904 119905) is calleda Hermitian kernel Properties of Hermitian kernels andrelated integral operators can be found in [27] (further see[28]) Now we shall remind some results Let us consider theequation which is related with (1) as

120583A119891 = 119891 (3)

The function 119891which differs from zero is called a charac-teristic function ofAwhich corresponds to the characteristicvalue 120583 If a Hermitian Hilbert-Schmidt kernel is not nullthen it posseses at least one characteristic value and this char-acteristic value (every characteristic value) is real Furtherthere is a finite orthonormal base of characteristic functionsfor each characteristic value 120583 ofA Using the union of thesebases one obtains an orthonormal system of characteristicfunctions of the kernel 119860(119905 119904) If 120593

1 1205932 120593

119899are distinct

members of such a system belonging respectively to the char-acteristic value 120583

1 1205832 120583

119899(not necessarily all different)

then for 1 le V le 119899 120593V otimes 120593V = 120593V(119905)120593V(119904) is an orthonormalsystem of two variables For such a system the equivalence

119860 (119905 119904) ≗

infin

sum

119899=1

120593119899otimes 120593119899

120583119899

(4)

holds Moreover if 119892 = A119891 119891 isin 1198712 then the equivalence

119892 ≗

infin

sum

119899=1

(119892 120593119899) 120593119899=

infin

sum

119899=1

(119891 120593119899)

120583119899

120593119899 (5)

holds Further the series in (5) are relatively uniformly abso-lutely convergent However equivalence can be replaced byequality Before showing this let us consider the equality

119892 (119905) = 120582int119860 (119905 119904) 119891 (119904) 119889119904 (6)

Following the same idea of [27 pages 22] if 119860(119905 119904) is an 1198712-

kernel such that119860(119905 119904) is continuous and 119891 is an 1198712-function

then 119892(119905) is continuousHence if 119860(119905 119904) is a continuous Hermitian 119871

2-kernel (see[27 pages 127]) and 119892 = A119891 119891 isin 119871

2 then the equality

119892 (119905) =

infin

sum

119899=1

(119892 120593119899) 120593119899(119905) =

infin

sum

119899=1

(119891 120593119899)

120583119899

120593119899(119905) (7)

holds and the series are uniformly absolutely convergent Inthis case taking 119892(119905) = (119891 119860(119905 sdot)) the series given in (4) con-verge to the continuous kernel 119860(119905 119904) uniformly in thevariable 119905 for every 119904 Hence from (4) one can get

119860 (119905 119905) =

infin

sum

119899=1

1003816100381610038161003816120593119899 (119905)1003816100381610038161003816

2

120583119899

(8)

Integrating both side from 119886 to 119887 we obtain

int

119887

119886

119860 (119905 119905) 119889119905 =

infin

sum

119899=1

1

120583119899

(9)

Consequently if 119860(119905 119905) is integrable on (119886 119887) then the seriesconverge This implies that A is of trace-class (nuclear) Forthe definition and properties of trace-class operators see forexample [21]

Above arguments given in [27] hold for Hermitian ker-nels However from arbitrary 119871

2-kernels one can pass to theHermitian kernels For example 119870

1(119905 119904) = 119860(119905 119904) + 119860(119904 119905)

and1198702(119905 119904) = 119894(119860(119905 119904)minus119860(119904 119905)) are Hermitian kernelsThese

Hermitian kernels are continuous and 1198712-kernels if 119860(119905 119904) is

so Hence above arguments hold for 1198701(119905 119904) and 119870

2(119905 119904) if

119860(119905 119904) is continuous 1198712-kernel such that 119860(119905 119905) is integrable

on (119886 119887) Further

119860 =1

2(1198701minus 1198941198702) (10)

Hence if 1198701and 119870

2are trace-class kernels then so is 119860

However this result has been given in [29 pages 526] as adefinition

LidskiırsquosTheorem (see [21 page 231]) If the dissipative opera-tor 119871 is the trace class operator then its system of root functionsis complete in the Hilbert space 119867

3 Statement of the Problem

The fourth order differential expression is considered as

120578 (119910) = 119910119894V

+ 119902 (119909) 119910 119909 isin Λ = [119886 119887) (11)

where minusinfin lt 119886 lt 119887 le infin 119886 is the regular point and 119887 is thesingular point for 120578 and 119902 is real-valued Lebesgue measur-able and locally integrable function on Λ

Let 1198712(Λ) be the Hilbert space consisting of all functions119910 such that int119887

119886|119910(119909)|

2119889119909 lt infinwith the inner product (119910 120594) =

int119887

119886119910(119909)120594(119909)119889119909Let

Ω =119910 isin 1198712(Λ) 119910 119910

1015840 11991010158401015840 119910101584010158401015840

isin ACloc (Λ) 120578 (119910) isin 1198712(Λ)

(12)

where ACloc(Λ) denotes the set of all locally absolutely con-tinuous functions on Λ For arbitrary 119910 120594 isin Ω Greenrsquosformula is obtained as

int

1199092

1199091

120578 (119910) 120594 (119909) 119889119909 minus int

1199092

1199091

119910 (119909) 120578 (120594) 119889119909

= [119910 120594] (1199092) minus [119910 120594] (119909

1)

(13)

ISRNMathematical Analysis 3

where 119886 le 1199091

lt 1199092

le 119887 and [119910 120594](119909) = 119910101584010158401015840(119909)120594(119909) minus

119910(119909)120594101584010158401015840

(119909) + 1199101015840(119909)12059410158401015840(119909) minus 119910

10158401015840(119909)1205941015840(119909) Greenrsquos formula

implies the fact that if 119910(119909 120582) and 120594(119909 120582) both satisfy theequation 120578(119910) = 120582119910 for the same value of 120582 then [119910 120594](119909)

is independent of 119909 and depends only on 120582 Further forarbitrary 119910 120594 isin Ω at singular point 119887 the limits [119910 120594](119887) =

lim119909rarr119887

[119910 120594](119909) and [119910 120594](119887) = lim119909rarr119887

[119910 120594](119909) exist andare finiteThe latter also follows fromGreenrsquos formula In factit is sufficient to get the second factors with their complexconjugates on the left-hand side in Greenrsquos formula

In this paper it is assumed that 119902(119909) satisfies the lim-4 caseconditions at 119887 (see [30] and references therein) Lim-4 caseis also known as Weylrsquos limit-circle case [8]

Let us consider the solutions 1205931(119909 120582) 120593

2(119909 120582) 120595

1(119909 120582)

and 1205952(119909 120582) of the equation

119910119894V

+ 119902 (119909) 119910 = 120582119910 119909 isin Λ (14)

where 120582 is some complex parameter satisfying the conditions

[[[

[

1205931(119886 120582) 120593

2(119886 120582) 120595

1(119886 120582) 120595

2(119886 120582)

1205931015840

1(119886 120582) 120593

1015840

2(119886 120582) 120595

1015840

1(119886 120582) 120595

1015840

2(119886 120582)

12059310158401015840

1(119886 120582) 120593

10158401015840

2(119886 120582) 120595

10158401015840

1(119886 120582) 120595

10158401015840

2(119886 120582)

120593101584010158401015840

1(119886 120582) 120593

101584010158401015840

2(119886 120582) 120595

101584010158401015840

1(119886 120582) 120595

101584010158401015840

2(119886 120582)

]]]

]

=

[[[

[

1205722

0 1205741

0

0 1205732

0 1205791

0 minus1205731

0 minus1205792

minus1205721

0 minus1205742

0

]]]

]

(15)

where all 120572119894 120573119894 120574119894 and 120579

119894(119894 = 1 2) are real numbers such

that 12057221205742minus12057211205741= 1 and 120573

11205791minus12057321205792= 1 For the existence of

these solutions see [7 8 13] Since lim-4 case holds for 120578 allsolutions 120593

119894(119909 120582) and 120595

119894(119909 120582) belong to 119871

2(Λ) It is clear that

[120593119903 120595119904](119886) = 120575

119903119904(1 le 119903 119904 le 2) where 120575

119903119904is the Kronocker

delta [120593119903 120593119904](119886) = 0 and [120595

119903 120595119904](119886) = 0

Let us set 119906119894(119909) = 120593

119894(119909 0) and 119911

119894(119909) = 120595

119894(119909 0) (119894 =

1 2) with 119909 isin Λ Then 119906119894and 119911

119894become the real solutions

of 120578(119910) = 0 (119909 isin Λ) Further they belong to Ω Hence forarbitrary 119910 isin Ω the values [119910 119906

1](119887) [119910 119906

2](119887) [119910 119911

1](119887)

and [119910 1199112](119887) exist and are finite

Let 119863(119873) be the set of all functions 119910 isin Ω satisfying theboundary conditions

1205721119910 (119886) + 120572

2119910101584010158401015840

(119886) = 0 (16)

12057311199101015840(119886) + 120573

211991010158401015840(119886) = 0 (17)

[119910 1199061] (119887) minus ℎ

1[119910 1199111] (119887) = 0 (18)

[119910 1199062] (119887) minus ℎ

2[119910 1199112] (119887) = 0 (19)

where120572119894and120573119894(119894 = 1 2) are real numbers given as previously

stated and ℎ1and ℎ

2are some complex numbers such that

ℎ1= Rℎ

1+ 119894Iℎ

1and ℎ2= Rℎ

2+ 119894Iℎ

2withIℎ

119894gt 0 119894 = 1 2

It should be noted that for any solutions 119910(119909 120582) of (14)conditions (16) and (17) respectively can be written as

[119910 1205931] (119886) = 0 (20)

[119910 1205932] (119886) = 0 (21)

The operator 119873 is defined on 119863(119873) as 119873119910 = 120578(119910) 119910 isin

119863(119873) 119909 isin Λ The main aim of this paper is to investigatethe spectral analysis of the operator 119873 (the boundary valueproblem (14)ndash(19))

4 Completeness Theorems

Let 119910119895(119909 120582) 1 le 119895 le 119903 be any 119903 (1 le 119903 le 4) solutions

of 120578(119910) = 120582119910 The notation 1198821199101 1199102 119910

119903 = 119882119910

1(119909 120582)

1199102(119909 120582) 119910

119903(119909 120582) denotes the Wronskian of order 119903 of

this set of functions [7 8 31]It is known that the following equality holds [7 8 13]

1198821199101 1199102 1199103 1199104 = minus [119910

1 1199102] (119909) [119910

3 1199104] (119909)

+ [1199101 1199103] (119909) [119910

2 1199104] (119909)

minus [1199101 1199104] (119909) [119910

2 1199103] (119909)

(22)

This equation also shows that 1198821199101 1199102 1199103 1199104 of any four

solutions of 120578(119910) = 120582119910 is independent of 119909 and depends onlyon 120582 [7 8]

Now consider the solutions 1205991(119909 120582) and 120599

2(119909 120582) of the

equation 120578(119910) = 120582119910 119909 isin Λ satisfying the conditions

[[[[

[

[1205991 1199061] (119887) [120599

2 1199061] (119887)

[1205991 1199111] (119887) [120599

2 1199111] (119887)

[1205991 1199062] (119887) [120599

2 1199062] (119887)

[1205991 1199112] (119887) [120599

2 1199112] (119887)

]]]]

]

=

[[[

[

ℎ1

0

1 0

0 ℎ2

0 1

]]]

]

(23)

For the existence of these solutions given with the previousconditions see [13] Clearly these solutions satisfy conditions(18) and (19) respectively Now let us set

119882(120582) = 1198821205931 1205932 1205991 1205992 (24)

Then119882(120582) becomes an entire function and the zeros of119882(120582)

coincide with the eigenvalues of the operator 119873 [13] Thisimplies that all zeros of 119882 (all eigenvalues of 119873) are discreteand that possible limit points of these zeros (eigenvalues of119873) can only occur at infinity

Using (22) we immediately have

1198821199061 1199062 1199111 1199112 = 1

[[[[

[

[1199061 1199061] (119909) [119906

2 1199061] (119909) [119911

1 1199061] (119909) [119911

2 1199061] (119909)

[1199061 1199062] (119909) [119906

2 1199062] (119909) [119911

1 1199062] (119909) [119911

2 1199062] (119909)

[1199061 1199111] (119909) [119906

2 1199111] (119909) [119911

1 1199111] (119909) [119911

2 1199111] (119909)

[1199061 1199112] (119909) [119906

2 1199112] (119909) [119911

1 1199112] (119909) [119911

2 1199112] (119909)

]]]]

]

=

[[[

[

0 0 minus1 0

0 0 0 minus1

1 0 0 0

0 1 0 0

]]]

]

(25)

4 ISRNMathematical Analysis

Hence the Plucker identity for the fourth-order case isobtained (see [13 p 435])

[119910 120594] (119909) = [119910 1199061] (119909) [120594 119911

1] (119909)

minus [119910 1199111] (119909) [120594 119906

1] (119909)

+ [119910 1199062] (119909) [120594 119911

2] (119909)

minus [119910 1199112] (119909) [120594 119906

2] (119909)

(26)

where 119910 120594 isin 119863(119873)

Theorem 2 The operator 119873 is dissipative in 1198712(Λ)

Proof For 119910 isin 119863(119873) we have

(119873119910 119910) minus (119910119873119910) = [119910 119910] (119887) minus [119910 119910] (119886) (27)

Since 119910 isin 119863(119873) a direct calculation shows that

[119910 119910] (119886) = 0 (28)

Further using (26) and conditions (18) and (19) one obtains

[119910 119910] (119887) = 2119894Iℎ1

1003816100381610038161003816[119910 1199111](119887)

1003816100381610038161003816

2

+ 2119894Iℎ2

1003816100381610038161003816[119910 1199112](119887)

1003816100381610038161003816

2

(29)

Substituting (28) and (29) into (27) it is obtained that

I (119873119910 119910) = Iℎ1

1003816100381610038161003816[119910 1199111] (119887)

1003816100381610038161003816

2

+ Iℎ2

1003816100381610038161003816[119910 1199112] (119887)

1003816100381610038161003816

2

(30)

and this completes the proof

Theorem 2 shows that all eigenvalues of119873 lie in the closedupper half-plane

Theorem 3 The operator 119873 has no real eigenvalue

Proof For 119910 isin 119863(119873) a direct calculation shows that

I (119873119910 119910) = I (1205821003817100381710038171003817119910

1003817100381710038171003817

2

) (31)

Now let 120582 = 1205820be a real eigenvalue of 119873 and let 120593

1(119909 1205820)

be the corresponding eigenfunctionThen (31) and (30) gives[1205931 1199111](119887) = [120593

1 1199112](119887) = 0 Using these equalities in (18) and

(19) one gets that [1205931 1199061](119887) = [120593

1 1199062](119887) = 0

Let us consider the solution 1205951(119909 1205820) Hence using (26)

it is obtained that

1 = [1205931 1205951] (119886) = [120593

1 1205951] (119887)

= [1205931 1199061] (119887) [120595

1 1199111] (119887)

minus [1205931 1199111] (119887) [120595

1 1199061] (119887)

+ [1205931 1199062] (119887) [120595

1 1199112] (119887)

minus [1205931 1199112] (119887) [120595

1 1199062] (119887) = 0

(32)

This contradiction completes the proof

From Theorem 3 it is obtained that all eigenvalues of 119873lie in the open upper half-plane In particular zero is not aneigenvalue of 119873 Hence the operator 119873minus1 exists

Consider the solutions 1199061(119909) 119906

2(119909) V

1(119909) and V

2(119909)

where V1(119909) = 119906

1(119909) minus ℎ

11199111(119909) and V

2(119909) = 119906

2(119909) minus

ℎ21199112(119909) 119906

1(119909) and 119906

2(119909) satisfy conditions (16) and (17)

respectively and V1(119909) and V

2(119909) satisfy conditions (18) and

(19) respectivelyThe equation119873119910 = 119891(119909) 119910 isin 119863(119873) 119909 isin Λ is equivalent

to the nonhomogeneous differential equation

119910119894V

+ 119902 (119909) 119910 = 119891 (119909) 119910 isin 119863 (119873) 119909 isin Λ (33)

subject to the boundary conditions

[119910 1199061] (119886) = 0

[119910 1199062] (119886) = 0

[119910 1199061] (119887) minus ℎ

1[119910 1199111] (119887) = 0

[119910 1199062] (119887) minus ℎ

2[119910 1199112] (119887) = 0

(34)

(compare the boundary conditions at 119886 with (20) and (21))Using Everittrsquos method [7] (further see [8 13]) the generalsolution is obtained as

119910 (119909) = int

119887

119886

119866 (119909 120577) 119891 (120577) 119889120577 (35)

where

119866 (119909 120577) = V119879 (120577) 119906 (119909) 119886 le 119909 lt 120577

V119879 (119909) 119906 (120577) 120577 lt 119909 le 119887(36)

V(119909) = 119906(119909) minus ℎ119911(119909) and

119906 (119909) = (1199061(119909)

1199062(119909)

) 119911 (119909) = (1199111(119909)

1199112(119909)

) ℎ = (ℎ1

0

0 ℎ2

)

(37)

The operator 119870 defined by

119870119891 = int

119887

119886

119866 (119909 120577) 119891 (120577) 119889120577 (38)

where 119891 isin 1198712(Λ) is the inverse operator of 119873 Hence the

completeness of the system of all eigen- and associated func-tions of 119873 is equivalent to the completeness of those of 119870 in1198712(Λ)Since 119866(119909 120577) is a continuous Hilbert-Schmidt kernel and

119866(119909 119909) is integrable on [119886 119887) the operator119870 is of trace classLet us consider the operator minus119870 Since119873 is dissipative in

1198712(Λ) minus119870 is also dissipative in 119871

2(Λ) Thus all conditions are

satisfied for Lidskiırsquos theorem So we have the following

Theorem 4 The system of all root functions of minus119870 (also 119870) iscomplete in 119871

2(Λ)

Since the completeness of the system of root functions(eigen- and associated functions) of 119870 is equivalent to thecompleteness of those of 119873 is obtained that the following

Theorem 5 All eigenvalues of the problem (14)ndash(19) lie in theopen upper half-plane and they are purely discrete The systemof all eigen- and associated functions of the problem (14)ndash(19)is complete in 119871

2(Λ)

ISRNMathematical Analysis 5

References

[1] H Weyl ldquoUber gewohnliche Differentialgleichungen mit Sin-gularitaten und die zugehorigen Entwicklungen willkurlicherFunktionenrdquo Mathematische Annalen vol 68 no 2 pp 220ndash269 1910

[2] E C TitchmarshEigenfunction Expansions Associatedwith Sec-ond Order Differential Equations Part 1 Oxford UniversityPress 2nd edition 1962

[3] E A Coddington and N Levinson Theory of Ordinary Differ-ential Equations McGraw-Hill New York NY USA 1955

[4] M A Naimark Linear Differential Operators Nauka MoscowRussia 2nd edition 1969 English translation Ungar New YorkNY USA 1st edition parts 1 1967 1st edition parts 2 1968

[5] I M Glazman Direct Methods of Qualitative Spectral Analysisof Singular Differential Operators Israel Program for ScientificTranslations Jerusalem Israel 1965

[6] F V Atkinson Discrete and Continuous Boundary ProblemsAcademic Press New York NY USA 1964

[7] W N Everitt ldquoThe Sturm-Liouville problem for fourth-orderdifferential equationsrdquo The Quarterly Journal of Mathematicsvol 8 pp 146ndash160 1957

[8] W N Everitt ldquoFourth order singular differential equationsrdquoMathematische Annalen vol 149 pp 320ndash340 1963

[9] W N Everitt ldquoSingular differential equations I The even ordercaserdquoMathematische Annalen vol 156 pp 9ndash24 1964

[10] W N Everitt ldquoSingular differential equations II Some self-adjoint even order casesrdquoTheQuarterly Journal of Mathematicsvol 18 pp 13ndash32 1967

[11] W N Everitt D B Hinton and J S W Wong ldquoOn the stronglimit-119899 classification of linear ordinary differential expressionsof order 2119899rdquo Proceedings of the London Mathematical Societyvol 29 pp 351ndash367 1974

[12] V I Kogan and F S Rofe-Beketov ldquoOn square-integrable solu-tions of symmetric systems of differential equations of arbitraryorderrdquo Proceedings of the Royal Society of Edinburgh A vol 74pp 5ndash40 1974

[13] C T Fulton ldquoThe Bessel-squared equation in the lim-2 lim-3and lim-4 casesrdquoThe Quarterly Journal of Mathematics vol 40no 160 pp 423ndash456 1989

[14] A M Krall ldquo119872(120582) theory for singular Hamiltonian systemswith one singular pointrdquo SIAM Journal on Mathematical Anal-ysis vol 20 no 3 pp 664ndash700 1989

[15] A M Krall ldquo119872(120582) theory for singular Hamiltonian systemswith two singular pointsrdquo SIAM Journal on MathematicalAnalysis vol 20 no 3 pp 700ndash715 1989

[16] D B Hinton and J K Shaw ldquoOn Titchmarsh-Weyl119872(120582)-func-tions for linear Hamiltonian systemsrdquo Journal of DifferentialEquations vol 40 no 3 pp 316ndash342 1981

[17] D BHinton and J K Shaw ldquoHamiltonian systems of limit pointor limit circle type with both endpoints singularrdquo Journal ofDifferential Equations vol 50 no 3 pp 444ndash464 1983

[18] D B Hinton and J K Shaw ldquoParameterization of the 119872(120582)

function for a Hamiltonian system of limit circle typerdquo Proceed-ings of the Royal Society of Edinburgh A vol 93 no 3-4 pp 349ndash360 1983

[19] D B Hinton and J K Shaw ldquoOn boundary value problems forHamiltonian systems with two singular pointsrdquo SIAM Journalon Mathematical Analysis vol 15 no 2 pp 272ndash286 1984

[20] P W Walker ldquoA vector-matrix formulation for formally sym-metric ordinary differential equations with applications to solu-tions of integrable squarerdquo Journal of the London MathematicalSociety vol 9 no 2 pp 151ndash159 197475

[21] I C Gohberg and M G Kreın Introduction to the Theoryof Linear Nonselfadjoint Operators American MathematicalSociety Providence RI USA 1969

[22] B S Pavlov ldquoSpectral analysis of a dissipative singularSchrodinger operator in terms of a functional modelrdquo in PartialDifferential Equations vol 65 of Itogi Nauki i Tekhniki SeriyaSovremennye Problemy Matematiki Fundamentalrsquonye Naprav-leniya pp 95ndash163 1991 English translation in Partial Differen-tial Equations VIII vol 65 of Encyclopaedia of MathematicalSciences pp 87ndash163 1996

[23] G Guseinov ldquoCompleteness theorem for the dissipative Sturm-Liouville operatorrdquo Doga Turkish Journal of Mathematics vol17 no 1 pp 48ndash54 1993

[24] E Bairamov and A M Krall ldquoDissipative operators generatedby the Sturm-Liouville differential expression in the Weyl limitcircle caserdquo Journal of Mathematical Analysis and Applicationsvol 254 no 1 pp 178ndash190 2001

[25] E Ugurlu and E Bairamov ldquoDissipative operators with impul-sive conditionsrdquo Journal of Mathematical Chemistry vol 51 pp1670ndash1680 2013

[26] Z Wang and H Wu ldquoDissipative non-self-adjoint Sturm-Liouville operators and completeness of their eigenfunctionsrdquoJournal of Mathematical Analysis and Applications vol 394 no1 pp 1ndash12 2012

[27] F Smithies Integral Equations Cambridge University PressNew York NY USA 1958

[28] F Riesz and B Sz-Nagy Functional Analysis Frederick UngarNew York NY USA 6th edition 1972

[29] E Prugovecki Quantum Mechanics in Hilbert Space vol 92Academic Press New York NY USA 2nd edition 1981

[30] M S P Eastham ldquoThe limit-2119899 case of symmetric differentialoperators of order 2119899rdquo Proceedings of the London MathematicalSociety vol 38 no 2 pp 272ndash294 1979

[31] E L Ince Ordinary Differential Equations London UK 1927

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRNMathematical Analysis 3

where 119886 le 1199091

lt 1199092

le 119887 and [119910 120594](119909) = 119910101584010158401015840(119909)120594(119909) minus

119910(119909)120594101584010158401015840

(119909) + 1199101015840(119909)12059410158401015840(119909) minus 119910

10158401015840(119909)1205941015840(119909) Greenrsquos formula

implies the fact that if 119910(119909 120582) and 120594(119909 120582) both satisfy theequation 120578(119910) = 120582119910 for the same value of 120582 then [119910 120594](119909)

is independent of 119909 and depends only on 120582 Further forarbitrary 119910 120594 isin Ω at singular point 119887 the limits [119910 120594](119887) =

lim119909rarr119887

[119910 120594](119909) and [119910 120594](119887) = lim119909rarr119887

[119910 120594](119909) exist andare finiteThe latter also follows fromGreenrsquos formula In factit is sufficient to get the second factors with their complexconjugates on the left-hand side in Greenrsquos formula

In this paper it is assumed that 119902(119909) satisfies the lim-4 caseconditions at 119887 (see [30] and references therein) Lim-4 caseis also known as Weylrsquos limit-circle case [8]

Let us consider the solutions 1205931(119909 120582) 120593

2(119909 120582) 120595

1(119909 120582)

and 1205952(119909 120582) of the equation

119910119894V

+ 119902 (119909) 119910 = 120582119910 119909 isin Λ (14)

where 120582 is some complex parameter satisfying the conditions

[[[

[

1205931(119886 120582) 120593

2(119886 120582) 120595

1(119886 120582) 120595

2(119886 120582)

1205931015840

1(119886 120582) 120593

1015840

2(119886 120582) 120595

1015840

1(119886 120582) 120595

1015840

2(119886 120582)

12059310158401015840

1(119886 120582) 120593

10158401015840

2(119886 120582) 120595

10158401015840

1(119886 120582) 120595

10158401015840

2(119886 120582)

120593101584010158401015840

1(119886 120582) 120593

101584010158401015840

2(119886 120582) 120595

101584010158401015840

1(119886 120582) 120595

101584010158401015840

2(119886 120582)

]]]

]

=

[[[

[

1205722

0 1205741

0

0 1205732

0 1205791

0 minus1205731

0 minus1205792

minus1205721

0 minus1205742

0

]]]

]

(15)

where all 120572119894 120573119894 120574119894 and 120579

119894(119894 = 1 2) are real numbers such

that 12057221205742minus12057211205741= 1 and 120573

11205791minus12057321205792= 1 For the existence of

these solutions see [7 8 13] Since lim-4 case holds for 120578 allsolutions 120593

119894(119909 120582) and 120595

119894(119909 120582) belong to 119871

2(Λ) It is clear that

[120593119903 120595119904](119886) = 120575

119903119904(1 le 119903 119904 le 2) where 120575

119903119904is the Kronocker

delta [120593119903 120593119904](119886) = 0 and [120595

119903 120595119904](119886) = 0

Let us set 119906119894(119909) = 120593

119894(119909 0) and 119911

119894(119909) = 120595

119894(119909 0) (119894 =

1 2) with 119909 isin Λ Then 119906119894and 119911

119894become the real solutions

of 120578(119910) = 0 (119909 isin Λ) Further they belong to Ω Hence forarbitrary 119910 isin Ω the values [119910 119906

1](119887) [119910 119906

2](119887) [119910 119911

1](119887)

and [119910 1199112](119887) exist and are finite

Let 119863(119873) be the set of all functions 119910 isin Ω satisfying theboundary conditions

1205721119910 (119886) + 120572

2119910101584010158401015840

(119886) = 0 (16)

12057311199101015840(119886) + 120573

211991010158401015840(119886) = 0 (17)

[119910 1199061] (119887) minus ℎ

1[119910 1199111] (119887) = 0 (18)

[119910 1199062] (119887) minus ℎ

2[119910 1199112] (119887) = 0 (19)

where120572119894and120573119894(119894 = 1 2) are real numbers given as previously

stated and ℎ1and ℎ

2are some complex numbers such that

ℎ1= Rℎ

1+ 119894Iℎ

1and ℎ2= Rℎ

2+ 119894Iℎ

2withIℎ

119894gt 0 119894 = 1 2

It should be noted that for any solutions 119910(119909 120582) of (14)conditions (16) and (17) respectively can be written as

[119910 1205931] (119886) = 0 (20)

[119910 1205932] (119886) = 0 (21)

The operator 119873 is defined on 119863(119873) as 119873119910 = 120578(119910) 119910 isin

119863(119873) 119909 isin Λ The main aim of this paper is to investigatethe spectral analysis of the operator 119873 (the boundary valueproblem (14)ndash(19))

4 Completeness Theorems

Let 119910119895(119909 120582) 1 le 119895 le 119903 be any 119903 (1 le 119903 le 4) solutions

of 120578(119910) = 120582119910 The notation 1198821199101 1199102 119910

119903 = 119882119910

1(119909 120582)

1199102(119909 120582) 119910

119903(119909 120582) denotes the Wronskian of order 119903 of

this set of functions [7 8 31]It is known that the following equality holds [7 8 13]

1198821199101 1199102 1199103 1199104 = minus [119910

1 1199102] (119909) [119910

3 1199104] (119909)

+ [1199101 1199103] (119909) [119910

2 1199104] (119909)

minus [1199101 1199104] (119909) [119910

2 1199103] (119909)

(22)

This equation also shows that 1198821199101 1199102 1199103 1199104 of any four

solutions of 120578(119910) = 120582119910 is independent of 119909 and depends onlyon 120582 [7 8]

Now consider the solutions 1205991(119909 120582) and 120599

2(119909 120582) of the

equation 120578(119910) = 120582119910 119909 isin Λ satisfying the conditions

[[[[

[

[1205991 1199061] (119887) [120599

2 1199061] (119887)

[1205991 1199111] (119887) [120599

2 1199111] (119887)

[1205991 1199062] (119887) [120599

2 1199062] (119887)

[1205991 1199112] (119887) [120599

2 1199112] (119887)

]]]]

]

=

[[[

[

ℎ1

0

1 0

0 ℎ2

0 1

]]]

]

(23)

For the existence of these solutions given with the previousconditions see [13] Clearly these solutions satisfy conditions(18) and (19) respectively Now let us set

119882(120582) = 1198821205931 1205932 1205991 1205992 (24)

Then119882(120582) becomes an entire function and the zeros of119882(120582)

coincide with the eigenvalues of the operator 119873 [13] Thisimplies that all zeros of 119882 (all eigenvalues of 119873) are discreteand that possible limit points of these zeros (eigenvalues of119873) can only occur at infinity

Using (22) we immediately have

1198821199061 1199062 1199111 1199112 = 1

[[[[

[

[1199061 1199061] (119909) [119906

2 1199061] (119909) [119911

1 1199061] (119909) [119911

2 1199061] (119909)

[1199061 1199062] (119909) [119906

2 1199062] (119909) [119911

1 1199062] (119909) [119911

2 1199062] (119909)

[1199061 1199111] (119909) [119906

2 1199111] (119909) [119911

1 1199111] (119909) [119911

2 1199111] (119909)

[1199061 1199112] (119909) [119906

2 1199112] (119909) [119911

1 1199112] (119909) [119911

2 1199112] (119909)

]]]]

]

=

[[[

[

0 0 minus1 0

0 0 0 minus1

1 0 0 0

0 1 0 0

]]]

]

(25)

4 ISRNMathematical Analysis

Hence the Plucker identity for the fourth-order case isobtained (see [13 p 435])

[119910 120594] (119909) = [119910 1199061] (119909) [120594 119911

1] (119909)

minus [119910 1199111] (119909) [120594 119906

1] (119909)

+ [119910 1199062] (119909) [120594 119911

2] (119909)

minus [119910 1199112] (119909) [120594 119906

2] (119909)

(26)

where 119910 120594 isin 119863(119873)

Theorem 2 The operator 119873 is dissipative in 1198712(Λ)

Proof For 119910 isin 119863(119873) we have

(119873119910 119910) minus (119910119873119910) = [119910 119910] (119887) minus [119910 119910] (119886) (27)

Since 119910 isin 119863(119873) a direct calculation shows that

[119910 119910] (119886) = 0 (28)

Further using (26) and conditions (18) and (19) one obtains

[119910 119910] (119887) = 2119894Iℎ1

1003816100381610038161003816[119910 1199111](119887)

1003816100381610038161003816

2

+ 2119894Iℎ2

1003816100381610038161003816[119910 1199112](119887)

1003816100381610038161003816

2

(29)

Substituting (28) and (29) into (27) it is obtained that

I (119873119910 119910) = Iℎ1

1003816100381610038161003816[119910 1199111] (119887)

1003816100381610038161003816

2

+ Iℎ2

1003816100381610038161003816[119910 1199112] (119887)

1003816100381610038161003816

2

(30)

and this completes the proof

Theorem 2 shows that all eigenvalues of119873 lie in the closedupper half-plane

Theorem 3 The operator 119873 has no real eigenvalue

Proof For 119910 isin 119863(119873) a direct calculation shows that

I (119873119910 119910) = I (1205821003817100381710038171003817119910

1003817100381710038171003817

2

) (31)

Now let 120582 = 1205820be a real eigenvalue of 119873 and let 120593

1(119909 1205820)

be the corresponding eigenfunctionThen (31) and (30) gives[1205931 1199111](119887) = [120593

1 1199112](119887) = 0 Using these equalities in (18) and

(19) one gets that [1205931 1199061](119887) = [120593

1 1199062](119887) = 0

Let us consider the solution 1205951(119909 1205820) Hence using (26)

it is obtained that

1 = [1205931 1205951] (119886) = [120593

1 1205951] (119887)

= [1205931 1199061] (119887) [120595

1 1199111] (119887)

minus [1205931 1199111] (119887) [120595

1 1199061] (119887)

+ [1205931 1199062] (119887) [120595

1 1199112] (119887)

minus [1205931 1199112] (119887) [120595

1 1199062] (119887) = 0

(32)

This contradiction completes the proof

From Theorem 3 it is obtained that all eigenvalues of 119873lie in the open upper half-plane In particular zero is not aneigenvalue of 119873 Hence the operator 119873minus1 exists

Consider the solutions 1199061(119909) 119906

2(119909) V

1(119909) and V

2(119909)

where V1(119909) = 119906

1(119909) minus ℎ

11199111(119909) and V

2(119909) = 119906

2(119909) minus

ℎ21199112(119909) 119906

1(119909) and 119906

2(119909) satisfy conditions (16) and (17)

respectively and V1(119909) and V

2(119909) satisfy conditions (18) and

(19) respectivelyThe equation119873119910 = 119891(119909) 119910 isin 119863(119873) 119909 isin Λ is equivalent

to the nonhomogeneous differential equation

119910119894V

+ 119902 (119909) 119910 = 119891 (119909) 119910 isin 119863 (119873) 119909 isin Λ (33)

subject to the boundary conditions

[119910 1199061] (119886) = 0

[119910 1199062] (119886) = 0

[119910 1199061] (119887) minus ℎ

1[119910 1199111] (119887) = 0

[119910 1199062] (119887) minus ℎ

2[119910 1199112] (119887) = 0

(34)

(compare the boundary conditions at 119886 with (20) and (21))Using Everittrsquos method [7] (further see [8 13]) the generalsolution is obtained as

119910 (119909) = int

119887

119886

119866 (119909 120577) 119891 (120577) 119889120577 (35)

where

119866 (119909 120577) = V119879 (120577) 119906 (119909) 119886 le 119909 lt 120577

V119879 (119909) 119906 (120577) 120577 lt 119909 le 119887(36)

V(119909) = 119906(119909) minus ℎ119911(119909) and

119906 (119909) = (1199061(119909)

1199062(119909)

) 119911 (119909) = (1199111(119909)

1199112(119909)

) ℎ = (ℎ1

0

0 ℎ2

)

(37)

The operator 119870 defined by

119870119891 = int

119887

119886

119866 (119909 120577) 119891 (120577) 119889120577 (38)

where 119891 isin 1198712(Λ) is the inverse operator of 119873 Hence the

completeness of the system of all eigen- and associated func-tions of 119873 is equivalent to the completeness of those of 119870 in1198712(Λ)Since 119866(119909 120577) is a continuous Hilbert-Schmidt kernel and

119866(119909 119909) is integrable on [119886 119887) the operator119870 is of trace classLet us consider the operator minus119870 Since119873 is dissipative in

1198712(Λ) minus119870 is also dissipative in 119871

2(Λ) Thus all conditions are

satisfied for Lidskiırsquos theorem So we have the following

Theorem 4 The system of all root functions of minus119870 (also 119870) iscomplete in 119871

2(Λ)

Since the completeness of the system of root functions(eigen- and associated functions) of 119870 is equivalent to thecompleteness of those of 119873 is obtained that the following

Theorem 5 All eigenvalues of the problem (14)ndash(19) lie in theopen upper half-plane and they are purely discrete The systemof all eigen- and associated functions of the problem (14)ndash(19)is complete in 119871

2(Λ)

ISRNMathematical Analysis 5

References

[1] H Weyl ldquoUber gewohnliche Differentialgleichungen mit Sin-gularitaten und die zugehorigen Entwicklungen willkurlicherFunktionenrdquo Mathematische Annalen vol 68 no 2 pp 220ndash269 1910

[2] E C TitchmarshEigenfunction Expansions Associatedwith Sec-ond Order Differential Equations Part 1 Oxford UniversityPress 2nd edition 1962

[3] E A Coddington and N Levinson Theory of Ordinary Differ-ential Equations McGraw-Hill New York NY USA 1955

[4] M A Naimark Linear Differential Operators Nauka MoscowRussia 2nd edition 1969 English translation Ungar New YorkNY USA 1st edition parts 1 1967 1st edition parts 2 1968

[5] I M Glazman Direct Methods of Qualitative Spectral Analysisof Singular Differential Operators Israel Program for ScientificTranslations Jerusalem Israel 1965

[6] F V Atkinson Discrete and Continuous Boundary ProblemsAcademic Press New York NY USA 1964

[7] W N Everitt ldquoThe Sturm-Liouville problem for fourth-orderdifferential equationsrdquo The Quarterly Journal of Mathematicsvol 8 pp 146ndash160 1957

[8] W N Everitt ldquoFourth order singular differential equationsrdquoMathematische Annalen vol 149 pp 320ndash340 1963

[9] W N Everitt ldquoSingular differential equations I The even ordercaserdquoMathematische Annalen vol 156 pp 9ndash24 1964

[10] W N Everitt ldquoSingular differential equations II Some self-adjoint even order casesrdquoTheQuarterly Journal of Mathematicsvol 18 pp 13ndash32 1967

[11] W N Everitt D B Hinton and J S W Wong ldquoOn the stronglimit-119899 classification of linear ordinary differential expressionsof order 2119899rdquo Proceedings of the London Mathematical Societyvol 29 pp 351ndash367 1974

[12] V I Kogan and F S Rofe-Beketov ldquoOn square-integrable solu-tions of symmetric systems of differential equations of arbitraryorderrdquo Proceedings of the Royal Society of Edinburgh A vol 74pp 5ndash40 1974

[13] C T Fulton ldquoThe Bessel-squared equation in the lim-2 lim-3and lim-4 casesrdquoThe Quarterly Journal of Mathematics vol 40no 160 pp 423ndash456 1989

[14] A M Krall ldquo119872(120582) theory for singular Hamiltonian systemswith one singular pointrdquo SIAM Journal on Mathematical Anal-ysis vol 20 no 3 pp 664ndash700 1989

[15] A M Krall ldquo119872(120582) theory for singular Hamiltonian systemswith two singular pointsrdquo SIAM Journal on MathematicalAnalysis vol 20 no 3 pp 700ndash715 1989

[16] D B Hinton and J K Shaw ldquoOn Titchmarsh-Weyl119872(120582)-func-tions for linear Hamiltonian systemsrdquo Journal of DifferentialEquations vol 40 no 3 pp 316ndash342 1981

[17] D BHinton and J K Shaw ldquoHamiltonian systems of limit pointor limit circle type with both endpoints singularrdquo Journal ofDifferential Equations vol 50 no 3 pp 444ndash464 1983

[18] D B Hinton and J K Shaw ldquoParameterization of the 119872(120582)

function for a Hamiltonian system of limit circle typerdquo Proceed-ings of the Royal Society of Edinburgh A vol 93 no 3-4 pp 349ndash360 1983

[19] D B Hinton and J K Shaw ldquoOn boundary value problems forHamiltonian systems with two singular pointsrdquo SIAM Journalon Mathematical Analysis vol 15 no 2 pp 272ndash286 1984

[20] P W Walker ldquoA vector-matrix formulation for formally sym-metric ordinary differential equations with applications to solu-tions of integrable squarerdquo Journal of the London MathematicalSociety vol 9 no 2 pp 151ndash159 197475

[21] I C Gohberg and M G Kreın Introduction to the Theoryof Linear Nonselfadjoint Operators American MathematicalSociety Providence RI USA 1969

[22] B S Pavlov ldquoSpectral analysis of a dissipative singularSchrodinger operator in terms of a functional modelrdquo in PartialDifferential Equations vol 65 of Itogi Nauki i Tekhniki SeriyaSovremennye Problemy Matematiki Fundamentalrsquonye Naprav-leniya pp 95ndash163 1991 English translation in Partial Differen-tial Equations VIII vol 65 of Encyclopaedia of MathematicalSciences pp 87ndash163 1996

[23] G Guseinov ldquoCompleteness theorem for the dissipative Sturm-Liouville operatorrdquo Doga Turkish Journal of Mathematics vol17 no 1 pp 48ndash54 1993

[24] E Bairamov and A M Krall ldquoDissipative operators generatedby the Sturm-Liouville differential expression in the Weyl limitcircle caserdquo Journal of Mathematical Analysis and Applicationsvol 254 no 1 pp 178ndash190 2001

[25] E Ugurlu and E Bairamov ldquoDissipative operators with impul-sive conditionsrdquo Journal of Mathematical Chemistry vol 51 pp1670ndash1680 2013

[26] Z Wang and H Wu ldquoDissipative non-self-adjoint Sturm-Liouville operators and completeness of their eigenfunctionsrdquoJournal of Mathematical Analysis and Applications vol 394 no1 pp 1ndash12 2012

[27] F Smithies Integral Equations Cambridge University PressNew York NY USA 1958

[28] F Riesz and B Sz-Nagy Functional Analysis Frederick UngarNew York NY USA 6th edition 1972

[29] E Prugovecki Quantum Mechanics in Hilbert Space vol 92Academic Press New York NY USA 2nd edition 1981

[30] M S P Eastham ldquoThe limit-2119899 case of symmetric differentialoperators of order 2119899rdquo Proceedings of the London MathematicalSociety vol 38 no 2 pp 272ndash294 1979

[31] E L Ince Ordinary Differential Equations London UK 1927

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 ISRNMathematical Analysis

Hence the Plucker identity for the fourth-order case isobtained (see [13 p 435])

[119910 120594] (119909) = [119910 1199061] (119909) [120594 119911

1] (119909)

minus [119910 1199111] (119909) [120594 119906

1] (119909)

+ [119910 1199062] (119909) [120594 119911

2] (119909)

minus [119910 1199112] (119909) [120594 119906

2] (119909)

(26)

where 119910 120594 isin 119863(119873)

Theorem 2 The operator 119873 is dissipative in 1198712(Λ)

Proof For 119910 isin 119863(119873) we have

(119873119910 119910) minus (119910119873119910) = [119910 119910] (119887) minus [119910 119910] (119886) (27)

Since 119910 isin 119863(119873) a direct calculation shows that

[119910 119910] (119886) = 0 (28)

Further using (26) and conditions (18) and (19) one obtains

[119910 119910] (119887) = 2119894Iℎ1

1003816100381610038161003816[119910 1199111](119887)

1003816100381610038161003816

2

+ 2119894Iℎ2

1003816100381610038161003816[119910 1199112](119887)

1003816100381610038161003816

2

(29)

Substituting (28) and (29) into (27) it is obtained that

I (119873119910 119910) = Iℎ1

1003816100381610038161003816[119910 1199111] (119887)

1003816100381610038161003816

2

+ Iℎ2

1003816100381610038161003816[119910 1199112] (119887)

1003816100381610038161003816

2

(30)

and this completes the proof

Theorem 2 shows that all eigenvalues of119873 lie in the closedupper half-plane

Theorem 3 The operator 119873 has no real eigenvalue

Proof For 119910 isin 119863(119873) a direct calculation shows that

I (119873119910 119910) = I (1205821003817100381710038171003817119910

1003817100381710038171003817

2

) (31)

Now let 120582 = 1205820be a real eigenvalue of 119873 and let 120593

1(119909 1205820)

be the corresponding eigenfunctionThen (31) and (30) gives[1205931 1199111](119887) = [120593

1 1199112](119887) = 0 Using these equalities in (18) and

(19) one gets that [1205931 1199061](119887) = [120593

1 1199062](119887) = 0

Let us consider the solution 1205951(119909 1205820) Hence using (26)

it is obtained that

1 = [1205931 1205951] (119886) = [120593

1 1205951] (119887)

= [1205931 1199061] (119887) [120595

1 1199111] (119887)

minus [1205931 1199111] (119887) [120595

1 1199061] (119887)

+ [1205931 1199062] (119887) [120595

1 1199112] (119887)

minus [1205931 1199112] (119887) [120595

1 1199062] (119887) = 0

(32)

This contradiction completes the proof

From Theorem 3 it is obtained that all eigenvalues of 119873lie in the open upper half-plane In particular zero is not aneigenvalue of 119873 Hence the operator 119873minus1 exists

Consider the solutions 1199061(119909) 119906

2(119909) V

1(119909) and V

2(119909)

where V1(119909) = 119906

1(119909) minus ℎ

11199111(119909) and V

2(119909) = 119906

2(119909) minus

ℎ21199112(119909) 119906

1(119909) and 119906

2(119909) satisfy conditions (16) and (17)

respectively and V1(119909) and V

2(119909) satisfy conditions (18) and

(19) respectivelyThe equation119873119910 = 119891(119909) 119910 isin 119863(119873) 119909 isin Λ is equivalent

to the nonhomogeneous differential equation

119910119894V

+ 119902 (119909) 119910 = 119891 (119909) 119910 isin 119863 (119873) 119909 isin Λ (33)

subject to the boundary conditions

[119910 1199061] (119886) = 0

[119910 1199062] (119886) = 0

[119910 1199061] (119887) minus ℎ

1[119910 1199111] (119887) = 0

[119910 1199062] (119887) minus ℎ

2[119910 1199112] (119887) = 0

(34)

(compare the boundary conditions at 119886 with (20) and (21))Using Everittrsquos method [7] (further see [8 13]) the generalsolution is obtained as

119910 (119909) = int

119887

119886

119866 (119909 120577) 119891 (120577) 119889120577 (35)

where

119866 (119909 120577) = V119879 (120577) 119906 (119909) 119886 le 119909 lt 120577

V119879 (119909) 119906 (120577) 120577 lt 119909 le 119887(36)

V(119909) = 119906(119909) minus ℎ119911(119909) and

119906 (119909) = (1199061(119909)

1199062(119909)

) 119911 (119909) = (1199111(119909)

1199112(119909)

) ℎ = (ℎ1

0

0 ℎ2

)

(37)

The operator 119870 defined by

119870119891 = int

119887

119886

119866 (119909 120577) 119891 (120577) 119889120577 (38)

where 119891 isin 1198712(Λ) is the inverse operator of 119873 Hence the

completeness of the system of all eigen- and associated func-tions of 119873 is equivalent to the completeness of those of 119870 in1198712(Λ)Since 119866(119909 120577) is a continuous Hilbert-Schmidt kernel and

119866(119909 119909) is integrable on [119886 119887) the operator119870 is of trace classLet us consider the operator minus119870 Since119873 is dissipative in

1198712(Λ) minus119870 is also dissipative in 119871

2(Λ) Thus all conditions are

satisfied for Lidskiırsquos theorem So we have the following

Theorem 4 The system of all root functions of minus119870 (also 119870) iscomplete in 119871

2(Λ)

Since the completeness of the system of root functions(eigen- and associated functions) of 119870 is equivalent to thecompleteness of those of 119873 is obtained that the following

Theorem 5 All eigenvalues of the problem (14)ndash(19) lie in theopen upper half-plane and they are purely discrete The systemof all eigen- and associated functions of the problem (14)ndash(19)is complete in 119871

2(Λ)

ISRNMathematical Analysis 5

References

[1] H Weyl ldquoUber gewohnliche Differentialgleichungen mit Sin-gularitaten und die zugehorigen Entwicklungen willkurlicherFunktionenrdquo Mathematische Annalen vol 68 no 2 pp 220ndash269 1910

[2] E C TitchmarshEigenfunction Expansions Associatedwith Sec-ond Order Differential Equations Part 1 Oxford UniversityPress 2nd edition 1962

[3] E A Coddington and N Levinson Theory of Ordinary Differ-ential Equations McGraw-Hill New York NY USA 1955

[4] M A Naimark Linear Differential Operators Nauka MoscowRussia 2nd edition 1969 English translation Ungar New YorkNY USA 1st edition parts 1 1967 1st edition parts 2 1968

[5] I M Glazman Direct Methods of Qualitative Spectral Analysisof Singular Differential Operators Israel Program for ScientificTranslations Jerusalem Israel 1965

[6] F V Atkinson Discrete and Continuous Boundary ProblemsAcademic Press New York NY USA 1964

[7] W N Everitt ldquoThe Sturm-Liouville problem for fourth-orderdifferential equationsrdquo The Quarterly Journal of Mathematicsvol 8 pp 146ndash160 1957

[8] W N Everitt ldquoFourth order singular differential equationsrdquoMathematische Annalen vol 149 pp 320ndash340 1963

[9] W N Everitt ldquoSingular differential equations I The even ordercaserdquoMathematische Annalen vol 156 pp 9ndash24 1964

[10] W N Everitt ldquoSingular differential equations II Some self-adjoint even order casesrdquoTheQuarterly Journal of Mathematicsvol 18 pp 13ndash32 1967

[11] W N Everitt D B Hinton and J S W Wong ldquoOn the stronglimit-119899 classification of linear ordinary differential expressionsof order 2119899rdquo Proceedings of the London Mathematical Societyvol 29 pp 351ndash367 1974

[12] V I Kogan and F S Rofe-Beketov ldquoOn square-integrable solu-tions of symmetric systems of differential equations of arbitraryorderrdquo Proceedings of the Royal Society of Edinburgh A vol 74pp 5ndash40 1974

[13] C T Fulton ldquoThe Bessel-squared equation in the lim-2 lim-3and lim-4 casesrdquoThe Quarterly Journal of Mathematics vol 40no 160 pp 423ndash456 1989

[14] A M Krall ldquo119872(120582) theory for singular Hamiltonian systemswith one singular pointrdquo SIAM Journal on Mathematical Anal-ysis vol 20 no 3 pp 664ndash700 1989

[15] A M Krall ldquo119872(120582) theory for singular Hamiltonian systemswith two singular pointsrdquo SIAM Journal on MathematicalAnalysis vol 20 no 3 pp 700ndash715 1989

[16] D B Hinton and J K Shaw ldquoOn Titchmarsh-Weyl119872(120582)-func-tions for linear Hamiltonian systemsrdquo Journal of DifferentialEquations vol 40 no 3 pp 316ndash342 1981

[17] D BHinton and J K Shaw ldquoHamiltonian systems of limit pointor limit circle type with both endpoints singularrdquo Journal ofDifferential Equations vol 50 no 3 pp 444ndash464 1983

[18] D B Hinton and J K Shaw ldquoParameterization of the 119872(120582)

function for a Hamiltonian system of limit circle typerdquo Proceed-ings of the Royal Society of Edinburgh A vol 93 no 3-4 pp 349ndash360 1983

[19] D B Hinton and J K Shaw ldquoOn boundary value problems forHamiltonian systems with two singular pointsrdquo SIAM Journalon Mathematical Analysis vol 15 no 2 pp 272ndash286 1984

[20] P W Walker ldquoA vector-matrix formulation for formally sym-metric ordinary differential equations with applications to solu-tions of integrable squarerdquo Journal of the London MathematicalSociety vol 9 no 2 pp 151ndash159 197475

[21] I C Gohberg and M G Kreın Introduction to the Theoryof Linear Nonselfadjoint Operators American MathematicalSociety Providence RI USA 1969

[22] B S Pavlov ldquoSpectral analysis of a dissipative singularSchrodinger operator in terms of a functional modelrdquo in PartialDifferential Equations vol 65 of Itogi Nauki i Tekhniki SeriyaSovremennye Problemy Matematiki Fundamentalrsquonye Naprav-leniya pp 95ndash163 1991 English translation in Partial Differen-tial Equations VIII vol 65 of Encyclopaedia of MathematicalSciences pp 87ndash163 1996

[23] G Guseinov ldquoCompleteness theorem for the dissipative Sturm-Liouville operatorrdquo Doga Turkish Journal of Mathematics vol17 no 1 pp 48ndash54 1993

[24] E Bairamov and A M Krall ldquoDissipative operators generatedby the Sturm-Liouville differential expression in the Weyl limitcircle caserdquo Journal of Mathematical Analysis and Applicationsvol 254 no 1 pp 178ndash190 2001

[25] E Ugurlu and E Bairamov ldquoDissipative operators with impul-sive conditionsrdquo Journal of Mathematical Chemistry vol 51 pp1670ndash1680 2013

[26] Z Wang and H Wu ldquoDissipative non-self-adjoint Sturm-Liouville operators and completeness of their eigenfunctionsrdquoJournal of Mathematical Analysis and Applications vol 394 no1 pp 1ndash12 2012

[27] F Smithies Integral Equations Cambridge University PressNew York NY USA 1958

[28] F Riesz and B Sz-Nagy Functional Analysis Frederick UngarNew York NY USA 6th edition 1972

[29] E Prugovecki Quantum Mechanics in Hilbert Space vol 92Academic Press New York NY USA 2nd edition 1981

[30] M S P Eastham ldquoThe limit-2119899 case of symmetric differentialoperators of order 2119899rdquo Proceedings of the London MathematicalSociety vol 38 no 2 pp 272ndash294 1979

[31] E L Ince Ordinary Differential Equations London UK 1927

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRNMathematical Analysis 5

References

[1] H Weyl ldquoUber gewohnliche Differentialgleichungen mit Sin-gularitaten und die zugehorigen Entwicklungen willkurlicherFunktionenrdquo Mathematische Annalen vol 68 no 2 pp 220ndash269 1910

[2] E C TitchmarshEigenfunction Expansions Associatedwith Sec-ond Order Differential Equations Part 1 Oxford UniversityPress 2nd edition 1962

[3] E A Coddington and N Levinson Theory of Ordinary Differ-ential Equations McGraw-Hill New York NY USA 1955

[4] M A Naimark Linear Differential Operators Nauka MoscowRussia 2nd edition 1969 English translation Ungar New YorkNY USA 1st edition parts 1 1967 1st edition parts 2 1968

[5] I M Glazman Direct Methods of Qualitative Spectral Analysisof Singular Differential Operators Israel Program for ScientificTranslations Jerusalem Israel 1965

[6] F V Atkinson Discrete and Continuous Boundary ProblemsAcademic Press New York NY USA 1964

[7] W N Everitt ldquoThe Sturm-Liouville problem for fourth-orderdifferential equationsrdquo The Quarterly Journal of Mathematicsvol 8 pp 146ndash160 1957

[8] W N Everitt ldquoFourth order singular differential equationsrdquoMathematische Annalen vol 149 pp 320ndash340 1963

[9] W N Everitt ldquoSingular differential equations I The even ordercaserdquoMathematische Annalen vol 156 pp 9ndash24 1964

[10] W N Everitt ldquoSingular differential equations II Some self-adjoint even order casesrdquoTheQuarterly Journal of Mathematicsvol 18 pp 13ndash32 1967

[11] W N Everitt D B Hinton and J S W Wong ldquoOn the stronglimit-119899 classification of linear ordinary differential expressionsof order 2119899rdquo Proceedings of the London Mathematical Societyvol 29 pp 351ndash367 1974

[12] V I Kogan and F S Rofe-Beketov ldquoOn square-integrable solu-tions of symmetric systems of differential equations of arbitraryorderrdquo Proceedings of the Royal Society of Edinburgh A vol 74pp 5ndash40 1974

[13] C T Fulton ldquoThe Bessel-squared equation in the lim-2 lim-3and lim-4 casesrdquoThe Quarterly Journal of Mathematics vol 40no 160 pp 423ndash456 1989

[14] A M Krall ldquo119872(120582) theory for singular Hamiltonian systemswith one singular pointrdquo SIAM Journal on Mathematical Anal-ysis vol 20 no 3 pp 664ndash700 1989

[15] A M Krall ldquo119872(120582) theory for singular Hamiltonian systemswith two singular pointsrdquo SIAM Journal on MathematicalAnalysis vol 20 no 3 pp 700ndash715 1989

[16] D B Hinton and J K Shaw ldquoOn Titchmarsh-Weyl119872(120582)-func-tions for linear Hamiltonian systemsrdquo Journal of DifferentialEquations vol 40 no 3 pp 316ndash342 1981

[17] D BHinton and J K Shaw ldquoHamiltonian systems of limit pointor limit circle type with both endpoints singularrdquo Journal ofDifferential Equations vol 50 no 3 pp 444ndash464 1983

[18] D B Hinton and J K Shaw ldquoParameterization of the 119872(120582)

function for a Hamiltonian system of limit circle typerdquo Proceed-ings of the Royal Society of Edinburgh A vol 93 no 3-4 pp 349ndash360 1983

[19] D B Hinton and J K Shaw ldquoOn boundary value problems forHamiltonian systems with two singular pointsrdquo SIAM Journalon Mathematical Analysis vol 15 no 2 pp 272ndash286 1984

[20] P W Walker ldquoA vector-matrix formulation for formally sym-metric ordinary differential equations with applications to solu-tions of integrable squarerdquo Journal of the London MathematicalSociety vol 9 no 2 pp 151ndash159 197475

[21] I C Gohberg and M G Kreın Introduction to the Theoryof Linear Nonselfadjoint Operators American MathematicalSociety Providence RI USA 1969

[22] B S Pavlov ldquoSpectral analysis of a dissipative singularSchrodinger operator in terms of a functional modelrdquo in PartialDifferential Equations vol 65 of Itogi Nauki i Tekhniki SeriyaSovremennye Problemy Matematiki Fundamentalrsquonye Naprav-leniya pp 95ndash163 1991 English translation in Partial Differen-tial Equations VIII vol 65 of Encyclopaedia of MathematicalSciences pp 87ndash163 1996

[23] G Guseinov ldquoCompleteness theorem for the dissipative Sturm-Liouville operatorrdquo Doga Turkish Journal of Mathematics vol17 no 1 pp 48ndash54 1993

[24] E Bairamov and A M Krall ldquoDissipative operators generatedby the Sturm-Liouville differential expression in the Weyl limitcircle caserdquo Journal of Mathematical Analysis and Applicationsvol 254 no 1 pp 178ndash190 2001

[25] E Ugurlu and E Bairamov ldquoDissipative operators with impul-sive conditionsrdquo Journal of Mathematical Chemistry vol 51 pp1670ndash1680 2013

[26] Z Wang and H Wu ldquoDissipative non-self-adjoint Sturm-Liouville operators and completeness of their eigenfunctionsrdquoJournal of Mathematical Analysis and Applications vol 394 no1 pp 1ndash12 2012

[27] F Smithies Integral Equations Cambridge University PressNew York NY USA 1958

[28] F Riesz and B Sz-Nagy Functional Analysis Frederick UngarNew York NY USA 6th edition 1972

[29] E Prugovecki Quantum Mechanics in Hilbert Space vol 92Academic Press New York NY USA 2nd edition 1981

[30] M S P Eastham ldquoThe limit-2119899 case of symmetric differentialoperators of order 2119899rdquo Proceedings of the London MathematicalSociety vol 38 no 2 pp 272ndash294 1979

[31] E L Ince Ordinary Differential Equations London UK 1927

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of