A focal boundary value problem for difference equations

(1)

Internat. J. Math. & Math. Sci. VOL. 16 NO. (1993) 169-176

169

A FOCAL BOUNDARY VALUE

PROBLEM FOR

DIFFERENCE

EQUATIONS

CATHRYN DENNYandDARRELHANKERSON

Department

ofAlgebra,Combinatorics, and Analysis

AuburnUniversity Auburn, Alabama 36849-5307

(Received August 27, 1991 and in revised form April II, 1992)

ABSTRACT.The eigenvalueproblemindifferenceequations,

(--1)n-kAny(t)

A

l:,=o

-

p()’,(),

with

y(0)

0,0

< <

k,

Ak+iy(T

+

1)

0,0

_<

<

n k,isexamined. Undersuitableconditions

on the coefficients Pi,it isshown that the smallest positive eigenvalueis adecreasing functionof

T. Asaconsequence,results concerning thefirstfocal point for the boundary valueproblem with

A

areobtained.

KEY WORDS AND PHRASES.

Differenceequation, eigenvalue, boundary value problem, focal point,

Green’s

function.

1991

MATHEMATICS SUBJECT CLASSIFICATION CODES.

39A10.

1. INTRODUCTION

Letk andnbe integers with

<_

k

<

n. Forfunctionsydefinedon aninterval of integers, define thedifferenceoperator A by

Ay

y,

Ay(t)

y(t

+

1)

y(t),

and

zy

A(Z-ly)

for

>

1.

We

shall be concernedfirst withtheeigenvalue problemfor difference equations

k-I

(--1)"-kA"y(t)

A

’pi(t)Zy(t),

0

<

< T,

(1.1)

i=0

Aiy(0)=0,

0_<i_<k-1,

(1.2)

A+iy(T

+

1)

0, 0

<

n k- 1,

whereTisanonnegative integer. Throughoutthis paper, theinterval notation inexpressions such

as

(1.1)

denote intervals of integers; forexample,

[0,T]

{0,1,...,

T}.

Undersuitable conditions

onthe coefficientspi,weshow that the smallest positiveeigenvalueisadecreasingfunctionofT.

Next,

wewillconsidertheboundaryvalueproblem

(1.2),

k-1

(--1)"-A"y(t)

-pi(t)Aiy(t),

0

<_

T.

(1.3)

i=0

(2)

170 C. DENNY AND Do HANKERSON

Itcanbe shown that theGreen’sfunction

GT(t,s)

for thefocalboundaryvalueproblem

(-1)’-:A"/(t)

/x,y(o)

o,

o

_< _<

,

A:+’z/(T

+

1)=

0, O<i<n-k-1,

(1.4)

exists. ExtensivediscussionsconcerningGreen’sfunctionsfordifferenceequationscanbe foundin nartman

[1]

and Kelley and Peterson

[2];

seealso

[3].

In particular, if

/(t)

is a solution of

(1.3),

(1.2)

on

[0,

T

+ HI,

then

/(t)

solves the equation

T k-I

y(t)

GT(t,s)

pi(s)Aiy(s),

e [0,

T

+

n].

s=0 i=0

As a consequence, ifsign conditions on

Gr(t,s)

are known and certain positivity conditions are

placedonthepi’s, then questions concerning theeigenvalueof

(1.1), (1.2)

and theexistenceoffocal pointsfor

(1.3),

(1.2)

can be examined intermsofafamily oflinearoperators thatdependonT.

Manyauthors have applied the theory ofconesinaBanachspaceand positive operatorseither to demonstrate theexistenceof smallest positiveeigenvalues, to compare these eigenvalues, or to establishthe existence of firstconjugate pointsorfirst focal points ofboundaryvalueproblemsfor linearequations; see, forexample,Eloe and Henderson

[4, 5],

Gentry

andTravis

[6],

Hankerson and Henderson

[7],

nankerson and

Peterson

[8,

91,

KeenerandTravis

[10],

Tomastik

[11,

21,

andTravis

[13].

We also mention papers by Eloe

[14]

and Eloe and Henderson

[151

which examine criteria for disfocality ofdifferenceequations, two papers byHenderson

[16, 17]

onfocal boundary value problems for nonlineardifferenceequations, and apaper by Henderson andLee

[18]

oncontinuous dependenceanddifferentiationofsolutionsofdifferenceequations. Much ofourmotivationforthis studyarethe works ofKeenerandTravis

[19],

Schmittand Smith

[20],

and Tomastik

[21].

In

section2,weincludepreliminary notation, and fundamental results from the theory ofcones in aBanachspace.

In

section3, weshow,under suitableassumptions onthecoefficientsPi, that

the smallest positiveeigenvalueof

(1.1),

(1.2)

decreaseswithT. This willleadtoresults concerning thefirstfocal point of

(1.3), (1.2).

2.

PRELIMINARIES

Inthissection,wegivedefinitionsandauxiliary results fromconetheory. Much of thediscussion inthis sectioninvolving the theory ofconesinaBanachspace arisesfrom resultsinKrasnosel’skii"s book

[22].

OthergoodreferencesincludeKrenandRutman

[23],

andDeimling

[24].

Let

B

beaBanach space.

A

closed subset

K

of

B

is saidtobea coneprovided:

(i)

ifu,v_E

K

thencu+v

E Kforallcr,/

>

0, and

(ii)

ifu,-u

K

thenu 0.

A

cone

K

is said tobe reproducing provided everyz

B

can bewrittenas z u vfor someu,v fi K. If

K

isacone

and u,v

B,

thenwewriteu

<

v

(with

respect to

K)

providedv- uEK. IfLand

M

arelinear operatorson

B,

then

L

<

M (with

respectto

K)

provided

LZ/< My

forall/5K. Finally, given

a bounded linear operator

L

on

B,

wesaythat

L

is positive if

L(K)

C_

K,

and wesay that

L

is uo-positive ifgiven any nonzero u

K,

there exist kl,

k2

>

0, such that

kluo

<

Lu

< k2uo.

If L

B

isaboundedlinearoperator, weshalluse

r(L)

todenote the spectralradius ofL.

(3)

FOCAL BOUNDARY VALUE PROBLEM FOR DIFFERENCE EQUATIONS 171

THEOREM2.2

IlL

and

M

arecompactlinearpositiveoperatorssuch that

L

<

M,

then

r(L) <

r(M).

THEOREM2.3 Let

L

be a compact linearpositive operator, andsuppose

Lx

>_

px

for

some

p

>

0 andx E

B

with-x

K

andx

u-vforsomeu,

v K. Then

L

has an eigenvector xo

K

correspondingto an eigenvalue

,o

>_

P.

Theorems 2.5, 2.10, 2.11, and 2.13 ofKrasnosel’skiigive the following theorem.

THEOREM

2.4

Let

K

be areproducingcone.

If

L

is a compactuo-positivelinearoperatorthen

L

hasan essentiallyunique eigenvector in

K

and the corresponding eigenvalue issimple, positive, andlarger than the modulus

of

anyothereigenvalue.

Thefollowingtheorem appearsin Keenerand Travis

[19,

Theorem

2.3]

andis ageneralization ofTravis

[13,

Theorem

2.3].

THEOREM2.5 Let

L

and

M

be boundedlinearoperators andassumethat at least one

of

the operators is Uo-positive.

Assume L

<

M

and there ezist nonzero vectorsul, u2,

_

K

and scalars

,1,2

>

0 such that

Lul > Alui

and

Mu2

< A2u2.

Then

<

,2.

If

,

thenut isa scalar multiple

of

u2.

3.

EIGENVALUES AND FOCAL POINTS

Ourmainobjectivein this section is todescribe how the smallestpositiveeigenvalueof

(1.1),

(1.2)

changeswithT. Wewill transform these questions about theeigenvalueintoquestions about thespectral radiusofcertainoperatorson aBanachspace,and thenapplytheconetheory.

First, let

(c)(i)

denote the factorial polynomial defined by

(c){0

c(c- 1).-. (c-

+

1).

If

Gr(t,s)

istheGreen’sfunctionfor

(1.4),

then

mi.{,-l,,)

(t

r

1)

(k-j-l)

(-r

+

_s

+

_n _k

1)

zGr(t,,)

(k-

j-

1)

(n-

k-

1)!

1"-’0

for j

<

k,and

Ak+JGT(t,s

(n-k-l-j)!

-;<

s

O,

t>s

for 0

<

j

<

n k 1. The

Green’s

functioncanbefoundwiththeaidof

[25,

Lemma

1].

Next,

let

B

{y"

[0, c)

R ly

isbodd nd

ZU(0)

0,0

_<

<

k}.

Then

B

isaBanada spaceunder thesupnorm. Define

Lr

B

---}

B

by

E

GT(t,

s)

pi(s)Aiy(s),

0

<

T

-t-n,

LTy(t)

,=o

=

(4)

172 C. DENNY AND D. HANKERSON

We

assume

that

p,(t)

>

0for

>

0, 0

<

k 1.

Remark. Note that LTy depends only on the values of y on

[0,

T

+

k

1].

Hence, if

T

>

T

and

BT

{y’[0,

T1

+

k-

1]

R IA’y(0)

0, 0

<

k

1},

thenwecan regard

LT

as amap

BT

B. Wecanthendefinethe map

MT’BT

B.

by

MTZ

Lrzlto,r,+k_

1.

Now,

if

(A,z)

is an eigenpair for

MT,

then we can extend z to a function in

B

by setting y(t)

z(t)

for 0

<

TI +

k- and Ay(t)

Lrz(t)

for

>

T +

k; then

(A,y)

isan eigenpair for

LT.

Conversely,if

(A,y)

isan eigenpair for

Lr

with

A

0,thenz

Ylt0m,+k-l

0

(otherwise

LTy 0and

A

0)

and

(A,

z)

isaneigenpair for

MT.

Inaddition, forany function

x(t)

definedon

[0,

T

+

k-

1],

the expression

Lrx

makessenseand

wewillallow thisslight abuseofnotation. Then ify(t)

LTx(t),

itfollows that

y(t)

isasolution of theboundaryvalueproblem

(1.2),

k-1

(--1)’-kA’y(t)

-pi(t)Zx(t).

If

(A,x)

is an eigenpair for

Lr

with

A

0, then

(1/A,x)is

an eigenpair for

(1.1), (1.2),

and cqnversely. Note alsothatA 0isnotaneigenvalue of

(1.1),

(1.2).

To begin with, we wishto examine, undersuitable conditions, what happens to thespectral radiusof

Lr

as

T

increases. Ourfirstresultwillplayakeyroleinsubsequentwork.

THEOREM:3.1 Thespectralradius

r(Lr)

isanondecreasing

function

ofT.

Morever,

if

pio(T)

>

0

for

some

T

>

k io, then

r(LT_)

<

r(Lr).

PROOF. Define thecone

KT

in

BT

by

KT

{y

6-

Brlmiy(t)

>_

O, 6-

[0,

T

+

k-

i],

0

<

k-

1}.

Then

Kr

{y

6-

Br

zy(t)

> O,

6-

[k-

i,T

+

k-

i],

0

<

k-

11,

and

Kr

is reproducing. Wewillbeginby showing

r(Lr)

isnondecreasing; that is, wewill show

r(Lr_) < r(Lr).

Let z6_

KT

andregard

LT-I

as anoperatoron

BT.

Then

T-1

(8

-"

1 k

t)

(n-k-l)

-AkLT-’x(t)

(n

k

1)!

Pi(s)Aiz(s)

s=t i=0

r

(s

+

_n _k

t)

(’--)

k-<

(n-k-l)!

s=t i=0

AtLrx(t),

O<t<T-1.

Wecanrepeatedly sumbothsides of

ALr_Ix(t)

<_

ALTx(t)

and usetheboundary conditions to show that

A’LT_z(t)

<

ALTZ(t)

for 0

<

T

+

k- i, 0

<

k- 1. Then

LT-

<

LT

with respect to

KT

andby Theorem 2.2,

r(Lr_)

< r(Lr).

Now suppose that pio(T)

>

0 for some

T

>

k-

i0.

We first show that

r(Lr)

>

0. Let

u(t)

() _6-

K..

_Note_that

T k-I

ZLTu(t)

,

zGr(t,s)

-pi(s)zu(s)

>

ZGT(t,T)Pio(T)AiOu(T)

>

0

(5)

FOCAL BOUNDARY VALUE PROBLEH FOR DIFFERENCE EQUATIONS 173

for k j

_<

T

+

k j and 0

_<

j

_<

k 1. Thus,

Lru

G

Ky

r. Itfollows that thereexistsan

e

>

0suchthat

LTU

_

eu,andbyTheorem 2.3wehave

r(LT)

_

e

>

O.

Finally, we will show

r(LT_)

<

r(Lr).

Assume

r(LT_)

>

0.

By

Theorem 2.1,

r(Lz_)

is

an eigenvalueof

LT-

withcorrespondingeigenvector y G

K:r_.

Wecan use

r(LT_)y

L:r-y

to extend y to a function in

Br.

Weclaim that y G

If..

To seethis, sincey is an eigenvector correspondingto

r(LT_),

thereexistsSo

e [0,

T-

1]

such that,i=o-l

p,(So)y(So)

>

O. Then

T-I k-I

,(L_)()

G_,(,)

,

s=O i=0

k-I

GT-I(t’80)

Z

P’("qo)Y("qO) >

O,

-0

for

[k

j,

T

+

k j],0

_<

j

_<

k 1. Itfollows that y

e

K.

Hence

p,o(T)Aiy(T)

>

0 and

T-I k-I

=0 i=0

T k-1

<

ZAkGT(t’s)

Z

s=O i=0

ALTy(t),

O<_t<_T-1.

Using the boundary conditions at 0, weobtain

LTy-

LT-y

G

K.

Hence,

there exists an e

>

0 such that

LTy

LT-y

>_

ey, or

LTy

_

ey

+

LT-y

(e

+

r(LT-1))y. By

Theorem2.3,

r(LT)

>

r(Lr_).

We obtain a corresponding result for the smallest positive eigenvalue, A0, of the

eigenvalue

problem

(1.1),

(1.2).

COROLLARY3.2

Assume

pio(To)

>

0

for

someio,

To

suchthat

To

>_

k io. Then the smallest positive eigenvalue,

Ao(T),

for

the eigenvalue problem

(1.1),

(1.2)

decreases

for

T >

To.

If,

in addition, thereexist l,

T

such that

T

>

min

{k

i,

To

+

}

andp

(T)

>

O,

then

Ao(T)

<

Ao(T0).

PROOF.

By

Theorem3.1,

r(Lro) >

0and

r(Lr,)

isaneigenvalueof

LT,.

Fromthe correspon-dence between eigenpairs of theeigenvalue problemand

Lro,

weseethat

1/r(Lr,)

isthe smallest positiveeigenvalue of

(1.1),

(1.2).

Since

r(Lr)

is increasing, it follows that

Ao(T)

1/r(LT)

is decreasingfor

T

>

To.

Finally,

r(Lr_)

<

r(Lr)

by the previoustheorem, and hence

A0(T0) >

A0(T,

1)

>

Ao(T).

Wenowshall concentrateonthecharacterizationof the first focal point of theboundaryvalue problemandcorrespondingresults.

THEOREM

3.3 Assume

p,o(T)

>

0

for

some

T

)_ k-

io. Suppose

that the boundary value problem

(1.3),

(1.2)

has.a

nontrivial solution y

KT

(again, we mean

YI[0,T+-I]

G

KT).

Then

T

isthe

first

focal

point.

PROOF. Our first step will be toshow

r(LT)

1. Now

LTy

y implies

r(LT)

>

1. Let

(6)

174 C. DENNY AND D. HANKERSON

Finally, if the firstfocal point r/

< T,

then

r(L,) _>

1. But Theorem 3.1 shows that

r(L,)

<

r(LT),

contradicting

r(LT)

1. El

Wearealsointerested in theuniqueness of thefunctioncorrespondingtothefirstfocal point. If conditionscanbeplacedonthe

p,’s

sothat

LT

isu0-positivewithrespect to

KT,

then Krasnosel’skii’s Theorem 2.4 can be applied.

In

this direction, then, suppose that

p_(t) >

0 for 6

[0, T].

Since

K

is nonempty,

Kr

is reproducing.

To

show

Lr

is u0-positive, it is sufficient to show

LT(KT\{O})

C_

K.

Let z E

K\{0}.

First

A-z(t)

>

0 for some E

[0, T];

otherwise, z would be the trivial solution. Then forsomes0fi

[0, T],

T k-1 k-1

r(t)

(t,,)

p,(,)A’(,)

>_ (t,,0)

p,(,o)/’(,o)

>

o,

s=O i=0 i:0

for (/

[k-

j,T

+

k- -j], 0

_<

j

_<

k-1. Therefore,

LTZ

K

forallz

KT\{0},

and

LT

is u0-positive.

Now suppose, in addition, that

(1.3), (1.2)

has a nontrivial solution y E

KT.

Notethat by Theorem3.3,

T

is thefirstfocal point.

An

application of Theorem 2.4 shows thaty isunique up toscalar multiple.

Finally, the requirement thatp_be strictly positivecanbereplaced byothersimilarconditions. For example, ifpk_ is identically zero, we could requirepk-2

>

0 and change our interval to

[0,

T

+

k-

2]

inthe definition of

However,

underweakerconditionswecanbecertainthatnontrivial solutions in

KT

areactually in

K.

This condition isthe keytoshow uniqueness.

THEOREM3.4 Let

Pio(T) >

0

for

some

T

>_

k-

io

and lety

IfT

be anontrivialsolution

of

(1.3), (1.2).

Theny is unique up to scalarmultiple.

PROOF.

Since y6-

KT

isasolutiontotheboundary value problem

(1.3),

(1.2),

theny

LTy.

Letzalso beasolution of theboundary valueproblem

(1.3),

(.2)

and assume -z

Kr.

Choose

cmaximal so that y

>_

cz. Weknow that c

>

0sincey 6-

K,

by earlier arguments.

Suppose

y cz 0.

From

previous workweknow that

LTy

aLz

6_

Kr.

Then thereis an

>

0so that

LTy

cLTz

>_

ez.

Hence

y cz

>_

ez impliesy

>_ (a

+

e)z,

which contradicts the maximalityof

The next two theorems will examineconditions which guaranteethat thefirst focal point of equation

(1.3)

isgreaterthanorequalto

T.

THEOREM3.5 Assume thatpio(T)

>

0

for

some

T

>_

k-io.

If

thesmallestpositive eigenvalue

Ao(T)

of

(1.1),

(1.2) satisfies A0(T) _>

1, then the

first

focal

point ?

>_

T.

PROOF. Suppose,

on the contrary, that r/

<

T.

By

Corollary 3.2,

Ao(r/) >

o(T).

Then

o(rt)

>

1 since

o(T) >_

1. Thissays

r(Ln)

1/o(1)

<

1.

But

for z, asolutionof the boundary valueproblem,

Lnz

zimplies

r(Ln) _>

1,which isacontradiction. Therefore,

THEOREM

3.6 Letv be anontrivial

function

in

KT

such that

T k-1

GT(,

)

,,(),,()-<

"()

(7)

FOCAL BOUNDARY VALUE PROBLFI FOR DIFFERENCE EQUATIONS 175

withrespect to

KT,

where

p_(t) >

0

for

all E

[0,T].

Then the

first

focal

point

>

T.

PROOF. Underthe given hypothesis, weget that

LTV

<

Ivwithrespectto

KT.

Let

Ao(T)

be the smallest positiveeigenvalueof the eigenvalueproblem

(1.1),

(1.2),

and letzbeacorresponding eigenfunctionin

Ka".

Then

r(La")z

Lrx

(1/A0(T))z.

By

remarksprecedingTheorem 3.4,

LT

is u0-positive, and itfollows by Theorem2.5that

1/,0(T) <

1.

Therefore,

by Theorem 3.5, the first

focal point q

>_

T. 13

As

in the discussion preceding Theorem 3.4, the result still holds ifwe require that the last nontrivial coefficientfunction be strictly positive, and then modify the Banachspaceaccordingly. Theseconditionsguaranteed that

Lr

wasu0-positive and allowedtheapplicationofTheorem 2.5.

However,

wecan relaxthe requirement of u0-positivity, providedweadd thecondition

Lrv

O. To see this, consider thesituation in Theorem 2.5 where

L

<

M

and there are nonzerovectors

ul,u

K

and scalars

A,A

>

0such that

Lu

>

Alu

and

Mu

<

Auz.

In

theproofof Theorem

2.5, u0-positivity of

L

wasused to show e0

sup{,

L(u

eul) >

0}

>

0.

In

thiscase,

0

<_ L(u

,oU,)

Lug.

,oLu,

<_

Mu

,oA,U,

<_

Au

,oA,u,

A(u2

oU,).

Hnce

L(u

-,ou,)

KT

implies

Nowsuppose thatpi0(T)

>

0for some

T

>

k-i0, andthat thereis v

Ka"

such that

Lrv

is nontrivialand

Lrv

<

v. Using techniques similartothosein theproofof Theorem 3.1,it canbe shown that

La"v

K.

Ifz q

Ka-

isaneigenfunction for

(1.1), (1.2)

corresponding to

Ao(T),

then ,o

sup{,

Lr(v- ,z) > 0}

>

0.

Hence,

_< A0(T),

and thefirstfocal pointr/>

T

from thesame

arguments used at the end of theproofof Theorem 3.6.

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1.

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Green’s

functions,

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Amev.

Math.

Soc.

246

(1978),

1-30.

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KELLEY,

W. AND

PETERSON,

A.

Difference

Equations. Academic

Press,

New

York, 1991. 3.

AGARWAL,

R. P. AND

LALLI,

B. S. Discrete: Polynomial interpolation,

Green’s

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ELOE,

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