Indefinite eigenvalue problem with eigenparameter in the two boundary conditions

Internat. J. Math. & Math. Sci. VOL. 21 NO. 4 (1998) 775-784

775

INDEFINITE EIGENVALUE PROBLEM WITH EIGENPARAMETER

IN

THE

TWO

BOUNDARY CONDITIONS

S.F. M.IBRAHIM Mathematics

Department

Faculty ofEducation AinShams University

Roxy,Cairo,EGYPT

(ReceivedApril 4,1996and in revisedform July 23, 1996)

ABSTRACT. The object ofthis paperis to establish anexpansion theorem for aregularindefinite eigenvalue problem ofsecond order differential equation with an eigenvalue parameter ,k in the two boundaryconditions Weassociated with thisproblemaJ-selfadjoint operatorwith compact resolvent defined in asuitable Kreinspace andthen wedevelopanassociatedeigenfunction expansiontheorem

KEY WORDS AND PHRASES: An expansion theorem, a regular indefinite eigenvalueproblem,

eigenvalue parameterintheboundary conditions,aJ-selfadjoint operator,Kreinspaceformulation 1991AMS SUBJECTCLASSIFICATION CODES: 65NXX,65N25.

1. INTRODUCTION

Theregularfight-definite eigenvalueproblem witheigenparameterinthe twoboundaryconditions where

r(x)

andq(z)arepositivefunctionson

[a, b]

hasbeenstudiedbyZayedand Ibrahim[1]

Daho andLanger [2] havemade an extensionofEveritt’spaper [3]and havereplacedthe Hilbert spaceinsome casesbyaPontraginspacewith indexone. Everitt

[3]

hasshown thatfora 6

[0,

]

the singular Sturm-Liouvillewithindefiniteweightfunction

r(x)

canberepresentedbyaselfadjoint operator in asuitable Hilbertspace

Recently, Fleckinger and Mingarelli [4] have studied an indefinite problem with the usual homogeneousDirichletorNeumannboundaryconditions.

The object ofthispaperistostudythefollowing regularindefmiteeig6nvalueproblem ofordertwo

consisting of the ordinarydifferentialequation 1

=

--

[-(r’)’

+q()]

=,

e

[,]

( )

togetherwith theboundaryconditions

Ms(u)

:=

[alu(a)- a2(pu’)(a)]

AR.(u):= ,[a3’/2(a

-4(pt)(a)],

(1 2)

Mz(u

:=

[91u(b)- (pu’)(b)]

ARz(u

:=

A[u(b)-/94(pu’)(b)],

(l3) whereweassumethroughoutthat:

(i)The functionsp(x),

p’(x), q(x)

and

r(x)

arerealcontinuousfunctionson

[a,b]

withp(z)

>

0 and continuouslydifferentiable.

(ii)Both the weightfunction

r(x)

andthe potentialfunctionq(x)change signon

[a,

b]

in the sense that theproblem

(1

1)-(1.3)is an indefinite.

776 SF M IBRAHIM

Pl and P2

>o.

(4)

The parameter

A

isacomplexnumber.

In this paper, our approach is to give a Krein space formulation to problem

(1

1)-(1.3) and a J-selfadjoint operatordefinedin itsuch thatthisproblemcanbe considered asthe eigenvalue problem of thisoperator

2. KREIN SPACE FORMULATION

DEFINITION2.1. Wedefine a Kreinspace

H

of three componentsvectorsby

H

L[,q(a,

b)

$CgC

H+[

%

]H-

( )

with indefinite innerproduct

1 1

P P2

andthe norm

wheref,g6

H

suchthat

Ilfll

:-

Ifl

2dx

/

If2

/

--03

Ifz

f

6.

H

(23)

f

(fl,

f2,

f3)

(fl,/ (fl),

R

(fl))

(2 4)

in which

Ra(fl)

o3fl(a)

Ra(.fi)

f/’(b)-

Z4(r,.f)(b),

and

(25)

in which

while

R(ffl)

3.ql

(12)

c4(]9.q[)(a),

Ra(gl)

193g]

(b)

4

(pg’

)(b)

H

:=

P+(Ll(a,b

9C

9C)

L,.I+

(a, b)

(9

C+

C+,

C+

:=

{u

e

C

Im# > O}

and C_ :=

{#

6C

Im# <

O}.

(2.6)

P+

denote theorthogonal projectorsinHsuch that

J:=P+-P-

and

P++P-=I.

(2 7)

Both

H

+ _and

H-

_{are Hilbertspaceswithrespecttothescalarproduct}

[., .]

_and

[.,

.]

_respectively

and thesymbol

4

denotes the directsumwhich isorthogonalwithrespecttothe scalarproduct

[., .],

that is,we have

INDEFIZNITE EIGENVALUE PROBLEM WITH EIGENPARAMETER 7 7 7

Thedecomposition(2. I)givesrise to apositivedefinitescalarproduct

(.,.)

onH.

(f,g):=[f+,#+]-[f-,g-]

for f,#EH,

f+,#+/-EH

+/- (2 9)

where

f

f

+

+

f-

andg

REMARK 2.1. (i)AccordingtoDefinition 21,theclass ofKrein spacesincludesHilbertspace

(H-

0)

aswell asanti-spacesof Hilbertspaces

(H

+

0);

_(see[5]).

(ii)Thedecomposable, non-degenerateinnerproduct spaceHis a Kreinspace[5]if andonlyiffor every fundamental symmetric operator

J,

the J-innerproductturns

H

into a Hilbertspace,thatis, we have

If,

g]

(Jr,

g), f,g

_

H. (2 10)

DEFINITION2.2. Wedefine aclosedlinear operator

A D(A)

H

by

A/:=

(r/1,Mo(A),

Ma(A)),

Vy

e

D(A)

(2.11)

suchthat

M,(f)

cif (a)

a2(pf)(a),

MO(A)

xA

()

where the domain

D(A)

oftheclosedlinear operator

A

is defined asthesetofall

f

(fl,

f2,

f3)

H

whichsatisfythefollowingconditions:

(i) f,

f{

areabsolutelycominuous functions on

[a, b]

with

/.,

(,

b)

and

(a)

h

(A),

(iii)

f3

REMARK 2.2 (i) The domain

D(A)

isdense inH with respecttothe indefinite innerproduct

(2.2).

(ii) Ais aneigenvalueand

f

isacorresponding eigenfunctionofproblem(1.1)-(1.3)ifandonlyit"

f

(fl,h,

f3)

D(A)

and

Af

Af

Therefore, the eigenvalues and the eigenfunctions of problem

(1 1)-(1.3)areequivalenttothe eigenvalues and the eigenfunctions of theoperatorA.

Weconsiderthefollowing assumptions:

lim

[(pf)(z)g(z)

f](z)(p)(z)]

0, (2 12)

and

lim

[(pf)(z)

g

(z)

fl

(z) (p)(z)

0. (2.13)

Integrating thefirsttermof

(3.1)

by parts twice,weget 3. THE J-SELFADJOINTNESS OF OPERATOR

A

DEFINITION 3.1. IntheKrein space

H,

asymmetric operator and a selfadjoint operatorwith respecttoindefinitescalar productarecalled ,/-symmetric and J-selfadjoint respectively(see [5]).

LEMMA3.1. Theoperator

A

inHisJ-symmetric.

PROOF. Onusing(2.2),

(2.11)

and theboundaryconditions(1.2)-(1.3),weget

1 1

[Af g]H

r(rf)’l

dx

+

M(fl)’ff2

M(fl)3

(pf[)"

dx/ qfi-ffldx

+

-

(ARo(fl))Ro(gl)

+

778 S.F M.IBRAHIM

On applying the conditions

(2.12)-(2.13)

on the first two terms of (32) and using the boundary conditions(1.2)-(1.3),weobtain

1 1

[Af,

g]H

f,r(rgx)dx

+

Ra(It)Ma(g) Ra(Ii)Ma(gl)

Pl P2

[/,A].

(3 3)

Thisprovesthat the closedlinearoperator

A

inHisJ-symmetric.

LetJbeaconjugation operator on

H;

thismeans that

J

is aconjugate-linear involutionwith

[J

f,

J

f]H

[g,f]H

g f,g H. (34)

DEFINmON3.2. The closedlinear operator

A

in

H

iscalled aJ-selfadjointin

H

if

D(A)

is dense inHand

A JA*J. (3 5)

Asin Knowles[6]we candefine an innerproduct on thedomain

D(JA’J)

by

[f,g]*H

[af, Jg]H

+

[A*Jf, A*JglH,

V f,g6.H. (3 6)

SinceJis aconjugation operatoron

H,

wefindthat(3.6)isequivalentto

[f,g]*l-I

[f,g]H

+

[JA*Jg,

JA*J

f]H"

(3 7)

With this indefinite innerproduct,

D(JA’J)

becomes aKrein space (seeDunfordand Schwartz [7,

p.

1225]).

LEMMA

3.2. If

A

is aJ-symmetric operatorin

H,

then

D(JA’J)

D(A).

PROOF. Let

g

D(JA" J)

(9

D(A),

(3 8)

then

If, g]

0, for

f

.

D(A).

(3 9)

Makinguse

of(3.7),

(3.9)

and thefact that

JA’Jf

Af, for

f

e

D(A)

weget

[JA"

Jg,

Af]

_H g,

f]H"

(3.10) Onusingthe definitionofanadjoint operator,(3.1 O)implies

JA’JgED(A’).

Thisgives

g

e

D(A"

JA’J).

(3 I)

From

(3.8)

and (3.11), we can concludethatthevectorfunction 9 is thezerovectorfunction This impliesthat

D(JA’J)

D(A).

(3 12)

INDEFINH’EEIGENVALUEPROBLEM WITH EIGENPARAMETER 7 7 9 4. THE BOUNDEDNESS OF

THE

OPERATOR

A

Inthis sectionweshall show that the J-selfadjoint operatorAin

H

is unbounded fromaboveand boundedfrombelow. Tothisendweneed the following lemma

LEMMA 4.1. Let f,

f’

be continuousfunctions on

[a, b]

with

Tfl

E

Lrl(a,b

Sincep(x) is continuous on

[a,

b]

aswell asp(x)

>

0 on

[a,

b],

then thereexists apositiveconstant

co

withp(x)

>_ co

suchthat

bp(:)IfI(:r.)12dx

>_

(b

c

a) I.fl

(b)

fl

(a)]

2.

(4 1)

PROOF. Schwartz’s inequality gives

p(z)lf(x)12dx

>

co

lYI(x)

dx

>-

(b

a)

f[

(x)dx

(b

)

I/ (b)

-/

()

REMARK

4.1. Weassumethat there isareal number"7such that

(.) >

(),

e

[,b]

(4 2)

forallsigns of

r(x)

andq(x).

LEMMA4.2. TheJ-selfadjoimoperatorAin

H

isboundedfrombelow

PROOF. Onusing theboundaryconditions

(1.2)-(1.3),

weget

1 1

[Af

f]H

r(’rfl)

fl

dx -t-

Ma(f)

f2

M(fl) f3

Pl P2

)’

(f;

Td+

ql/l

d

P P (4 3)

Integratingthefirsttermof

(4.3)

by

pans,

wefindthat

(44)

Substituting(4.1), (4.2)into(4.4)andusing theboundaryconditions(1.2)-(1.3),weobtain

(4

5)

780 S F. MIBRAHIM

JAr, f],

>

_’7

rlAldz

"["--/91

fl(t/)fl(t)

13

+

(b

)

1

4

[/(a

f)(a) +

f(a)(pf)(a)] +

aa4(pf)(a)(pf)(a)

+4(p/:)(b)(pf:)(b)

}.

(46)

Choosethe

re

nbersa,

,;

1, 2, 3,4 proddedp, p

>

0 d the onstt

>

0dchoose the

vues

p(a)

>

O,p(b)

>

O,

f

(a),

f

(b),

f

(a),

f

(b)

such thatthefollonginequityis vid

[AI,

f]H

rlfl

1

P

+ a(pf)(a)(pf{)(b)

}

1

(pf)(b)]

P

+ (pS)(b)(pf)(b)

}.

(47)

The inequity(47)cmbe

retten

inthefo

1 1

P

c

Ilfll,

(48)

where

e

constmtc

n(7,1).

LE4.3. TheJ-selfadjointoperator

A

H

isunboundedfrom above.

PROOF. Let

X(Z)

be a test nction in the ein spaceH thcompact

suppo

on

[a, b]

d define a

suence

oftstestnctionby

X(Z)

:=X(mZ)

for z

[a,b],

m 1,2,3,..., (49) Onusingthe same

ments

ofLena4.2,wecshow that

where

c

is a constt. Lettingm

,

weget

lim

[Ax,

X]n

.

(4 11)

Thisprovesthat

A

in

H

isunbounded from above. 5. THE EIGENVALUES OF OPERATOR A

Theproblem(1

1)-(1.3)

inthe indefinitecasegivesuspositiveand negative eigenvalues Thuswe consider theinfinitesequenceof the eigenvalues ofA:

--OO

< A

_

A

_

A₃

_ _

An

<

An+l

__

An+2

__

(5 1)

whereA,,

<

0

<

A=+I.

Forbrevity, the eigenvalues and eigenfunctionsaretogether called eigenpairs

DEFINITION 5.1. The J-selfadjoint operator A is called J-non-negative if

JAr,

f]

_n

>

O,

INDEFINITE EIGENVALUE PROBLEM WITH EIGENPARAMETER 7 81 THEOREM5.1. Considertheproblem

(I.1)-(1.3).

There existsatleasta finite number of’distinct positive eigenvalues

A,+I,

A,+_,

A+s

whoseassociatedeigenfunctionsCj,j n

+

1,n

+

2, n

+

s, satisfy

[ACj,

]

H

-->

0

PROOF. Suppose there exists h

<

n

+

s and the eigenpairs of operator A are

h,"

j n

+

1,n

+

2, such that

[A,

OJ]H

>

0. (5 2)

Wehave 0

<

{[A’.,.]H/[O:,O]H}

3,,

j n

+

1,n

+

2,n

+

3,...,h. Since h

<

n

+

s, then 0

<_

([Ah,

Ch]H/[h,

bh].}

Ah <

Therefore,thereexists atleastafinitenumberh; n

+

1

<

h

<

n

+

s. Thus thereexists atleast a finite number of distinct positive eigenvalues A,/I,A,+2,...,A,+s;s=1,2,3,.... This admits the existenceofinfinite numberpositive eigcnvalues

"n+,

Rn+2,

’,+s,

as oo.

TIIEOREM5.2. Considertheproblem(l. l)-(1.3). Thereexistsat mostfinitenumber of distinct negative eigenvalues

At,

A2,R3, A, whose associated cigenfunctions

;

1,2, 3, n, satiffy

From above we deduce that the set of real eigenvalues of operator

A

(negative and positive eigenvalues),isbounded from below and unbounded from above.

6.

TIIE

RESOLVENT OPERATOR AND TIIE EXPANSION THEOREM

Supposethat

@l(X;,X), @2(x;)),

arethefundamental solutionsof(I.I)on

[a,b]

whichsatisfythe initial conditions:

(6

l)

whereAECisnotaneigenvalue ofthe operator

A,

andput

which isindependent ofx6

[a, b].

Puttingx a, wetherefore have:

Mo((a; a))

+

a((; )).

(6 3)

Similarly, puttingz b,wetherefore have

Itisalso from(6. l)that

P((a;A))

p and

R(2(b;R))

p2 (6 5)

wherep,p2aregiven by(1.4).

From (6.1)it isclear thatforallA6C, M(q1

(a; ,))

,X/(@I

(a; R)),

whichgives thatq1

(x;

A)

satisfies the firstboundarycondition(1.2)atz a and

M(q;2(b;

A))

RR(@2(b;

R)),

whichgives that

@2(z;R)

satisfies the second boundary condition (1.3) at x b. Employing the same type of argumentasinthe regularSturm-Liouvilleproblem[9, Sec.

1.8]

itfollows thatthezeros of

o(R)

are real andthatif

A,,

n 1, 2, 3, denotes these zeros, the three-componentvectors

782 s.F.M.IBRAHIM

[,,,,]H=0

for n#rn, (67)

wheretheindefinite innerproductisgivenby(2.2). The initial conditions(6.1)also servetoguarantee that/’a

(z; A)

and

2

(z; A)

areentire inAforfixedz, andsoitfollowsthat

wx (A)

is an entirefunction ofA.

Welet

,I,

I111

(,I,(x), R(,I,(x)),

(,I,(x)))

(6.8)

denotethe normalizedeigenfunctions and let

k

#

0denote therealconstantforwhich

(z;)

a(z;)

v

[a,b]

(6.9)

for eachzero

ofwx(A). By

Green’sformulawith

r(z)

whichchangessign on

[a,

hi,

wehave

w( (b;

.),

(b;

))

w(

(a;

),

(;

))

(; A)dx

(6.10)

Wefindthat

W

(1 (b;)n),

1

(b;)))

k"

1[(2

+

ni4

)p(b)t (b; )

(1

+

An/3)l (b;

A)]

k;[,o() +

(

)Ra(x(; ))]

and

W(l(a;)n),

]31

(a; ))--

kl

[(01

.03)32(a;/n)- (0

;[o(.) +

(

.)R(*2(; X.))]

;(-

)Ro(2(;.))

(6 12)

where

wa (A,,)

0.

Substituting(6.11)and

(6.12)

into

(6.10)

weobtain

r(x)ba

(x;

A,)q

(x;

A)dx

=t=

k:

wb(A)

R(C(a; .))-

aa((b;

))}.

(6.13)

LettingA

A,,

weget

r(x){(x;A.)}2dx 5=

k-X{w(A,)

P(b(a;

A.))

R((b"

(6.14)

From

(6.5)

and

(6.9)

with

A

),we get

and

R((;))

J

(6 15)

Onusing(6.15)toeliminate/ from(6.14),we canfindthat

1 1

[2

II,,ll

2

,-(z)lCa(z;,L,)12dx

/ Io(q,(a;

,,))12

/

zl=

plR#((b;An))w(An).

(616)

Now,for

f

(fl(x),f2,f3)

E

H,

wedefine (ql(x),2,q3) E

D(A)

astheunique solution of inhomogeneous operator equation

(hi-

A)

f.

(617)

INDEFINITEEIGENVALUEPROBLEMWITH EIGENPARAMETER 783

(6 18)

Onapplyingthemethod ofvariationof parameters,weget

dt

+

dll(Z;,)

+

(6 19)

where all,d2 are constants and

r(z)

changes sign on

[a,b].

From

(6.18)

and (6

19)

togetherwiththe initial conditions(6.1),we canget theconstantsall,

d2

intheform

1

Odz()

f3

"f-

l2(t; A)(rfl)(t)dt

(6.20a) and

(6.20b)

providedthat

r(z)

changesitssignon

[a,

b]

accordingtoourindefiniteproblem(1 1)-(1.3)

Consequently,wededucethat

l(z)

;0x(A)

[f3Pl(Z;

A)

+

fg2(a:;

A)]

+

a(z,t;A)(ryx)(t)dt, (6.21)

Ro(X),

(6.22)

(6 23)

where

G(z,

t;

A)

is theGreen’sfunctionof our indefiniteproblem(1.1)-(1.3)definedby

G(z, t;,)

()

for a

<

<

x

<

b for a

<

x

<

<

b

(6 24)

Theform oftheequations

(6.21)-(6.24)

showsthatthe resolvent operator

R(,; A)

(M

A)

-1

is actually compact; fordetailsof arguments ofTheorem 5 inHellwig[10,p.

1:0]

canbeused

REMARK6.1. (i),k 0 is not aneigenvalue of

A

in KreinspaceH.

(ii) Since A is a J-selfadjoint operator, then it has real eigenvalues andthe correspondingreal eigenfunctionsareorthonormal.

(iii) Onusing Theorem3 inHellwig

[10,

p. 30],wededuce that the density of

D(A)

in Kreinspace

H

gives thecompletenessof the orthonormal system of the real eigenfunctions ofA.

Theresultsofourinvestigationsaresummarized in thefollowing expansiontheorem

THEOREM6.1. Theclosed linear operator

A

in Krein space H hasanunbounded set ofreal eigenvalues offinite multiplicity without accumulationpoints in

(-

oo,

oo),

and theycan beordered accordingtosize

A,+,oo

as s--oa with

784 SF M.IBRAHIM

inthesense ofstrongconvergencein

H

f

If,

:I:’n]HO,

(6 2:5)

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ZAYED,

E.M.E.and IBRAHIM, S.F.M.,Regular eigenvalue problemwitheigenparameterin the boundary conditions, Bull.Calcu.Math.

Soc.

84(1992),379-393.

[2]

DAHO, K. and

LANGER,

H.,

Someremarkson apaperbyW N.Everitt,Proc.RoyalSoc.Edin. 78A(1977),71-79.

[3]

EVERITT, W.N.,

Some remarks on a differential expressionwith anindefinite weight function, Math. Stud. 13(Amsterdam North

Holland)

(1974), 13-28.

[4] FLECKINGER,

J. and

MINGAKELLI, A.B.,

Onthe eigenvalues of non-definite elliptic operators,

Dff.

Equ.,

Editors:Knowles and Lewis, ElsevierSciencePub.B.V.

(North-Holland)

(1984), 219-227.

[5]

BOGNAK,

J.,

Indefinite

InnerProduct

Spaces,

Springer-Verlag,Berlin-N.Y.

(1974)

[6] KNOWLES,

I.W.,

On the boundary conditions characterizing J-selfadjoint extensions of J-symmetric operators,J.

Diff. Equ.

40(1981), 193-216.

[7] DUNFORD,N. andSCHWARTZ, J.T.,Linear

Operators

Part11, Wiley,N.Y. (1963).

[8]

KACE, D., Thetheory of J-selfadjoint extensionsof J-symmetric operators, d.

D/ft.

Equ.

57, 2

(1985),258-274.

[9] TITCHMARSH,

E.C., Eigenfunc#on Expansions Associated With Second Order

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