INTEGRO-ORDINARY DIFFERENTIAL

PERIODIC BOUNDARY VALUE PROBLEM FOR SECOND ORDER

INTEGRO-ORDINARY DIFFERENTIAL

EQUATIONS

WITH GENERAL KERNEL AND

CARATHIODORY NONLINEARITIES*

JUAN J. NIETO

Departamento

deAnhlisis Matemhtico Facultad de Matemfiticas UniversidaddeSantiago^deCompostela

SPAIN

(Received March 8, 1994 and in revised form November 29, 1994)

ABSTRACT. Westudy the existenceofsolutionsfor the periodicboundary^valueproblemfor some second order integro-differential equations with a general kernel. Also we develop the monotonemethodtoapproximatetheextremalsolutionsof theproblem.

KEY

WORDSAND PHRASES: Integro-Differential Equation,

Upper

^and LowerSolutions, Carath6odoryfunction

1991AMS

SUBJECT

CLASSIFICATION CODES: /45J05

1. INTRODUCTION

Thepurpose ofthis paperis tostudy the followingperiodicboundary valueproblem fora second order nonlinearintegro-ordinarydifferentialequation

-u"(t)--

f(t,u(t),Ku(t)),u(O)--u(En),u’(O)--u’(2n)

(1.1) where

f:I

^x

R:’

Ris aCarath6odory function,Kis an integral operator in

La(I)

with kernel k

_La(J),J

-Ixl. Relatedto

(1.1)

weconsiderthelinearproblem

-u"(t)

+

Mu(t)

+

N[Ku](t)

h(t),

u(O)

u(Etr),

u’(0) u’(2n) (1.2)

where

M,N

ER,and h EL

2(1).

By

a solution u of

(1.1)

we mean a function u

Ha(I)

such that the function

t_I-*f(t,u(t),[Ku](t))

^{is a} function in

L2(I)

satisfying the equation ^{for a.e.} ff.l, and

u(0)

u(2),

u’(0) u’(2).

Problem

(1.1)

is consideredin

[7]

with

f

^{continuousand}

^K

^aVolterra integral operatorwith positive kernel. The authors developed the monotone iterative method for

(1.1)

^{based on a}

comparison result. As it ispointed^outin

[6],

the methodof

[7]

is notapplicable^to thegeneral situation. Erbe andGuostudiedproblem

(1.1)

with

f

continuous, and kcontinuousandpositive.

Theyfirst consideredthe linearproblem

(1.2)

^andgavean estimate on

kll

^{Wenotethat in}

^[6],

Ku

NTu

+

NxSu

^with

^N, N1

^{realnumbers, Tanintegral}operator of Volterra type, andSanintegral operator of Fredholm type. Problem

(1.1)

is studied in

[8]

^{under the}assumptionthat

f(t,u,v)

is continuousandincreasingin v, and in

[7]

^for

f

^{continuousandK}of Volterra type. Followingthe ideasof

[6]

^westudy

(1.1)

^{in the}generalcase,i.e.,

fis

^aCarath6odoryfunctionandk isanL kernel.

Also,wedonotrequire ktohaveconstantsignon

J.

Forthelinearproblem

(1.2)

^tohaveaunique solution,wegiveanestimate on

kll

^thatimprovesthe estimationgivenⁱⁿ

[6],

^andourestimateis the bestpossiblein thesensethat ifequality^isattained, then theexistence-uniqueness^resultfor

(1.2)

isnotvalidanymore.

*Research partially supported by

DGICYT,

ProjectPB91-0793.

758 J.J. NIETO

PBVP

FORSECOND ORDER ORDINARY

DIFFERENTIAL EQUATION

Werecallhere,forconvenienceof thereader,someresultsforthefollowing periodic boundary value

problem (PBVP)

foralinear secondorderordinarydifferentialequation. ^Theproblem

-u"(t)+Mu(t)---h(t); u(O)=u(2n); u’(0)-u’(2) (2.1) withM

-m2,m

>0and h

L2(I),

hasauniquesolutiongiven by^theexpression

2n

u(t)-- J0 G(t,s)h(s)ds,

theGreenfunctionGisgiven by

l.[e’l’-S)+e’(2-’/sr].O<8<<27r

G(t,s)- 2m(e

"-

¹⁾

/---- 1)[em(S-t)q" ^{e’a ’)]"}

^{0_< _<.}

^<_

^27r

Moreover,

Giscontinuous onJ-I

I,

2

min{G(t,s)’(t,s)J} e’

G(t,s)ds

--,

_m(e2’’-

1)"

^a,

/2m b.

and

max{G(t,s)"

(t,s)

.J} _{=2,,2,,_}

SinceG(t,s) ^a>0forevery(t,s)

J

weobtainthefollowingmaximumprinciples:

h>0 a.e. on I implies u>0on I,

(2.3)

h<0 a.e. on

I

implies ^u<0on

I (2.4)

Obviously, h-0 implies u-0, but if h >0 on a set of positive measure of I, then u(t) ^a

f

^{h(s)ds>0forevery}

3.

LINEAR

INTEGRO-DIFFERENTIAL

EQUATIONS

We now consider the integro-differential problem

(1.2)

with

M

> 0,

N R,[Ku](t)--fok(t,s)u(s)ds,k .L2(J)

^{and h}

Notethat

K

isanintegraloperator,anditcouldbeeitherof VolterraorFredholm type. Even

K

canbeofmixedtypeasin

[6]

^where

Ku

NTu

+

NSu ^(3.1)

with

N,N

^realnumbers,

T

an integraloperator of Volterra typewithkernelk0,andSanintegral operator of Fredholm typewithkernel

k.

^{Thus, k}

^N k0

⁺

N k.

In

what

follows, I!

^denotestheusual norm in

L

Accordingtotheresultsofsection 2,wehave thatuisasolutionof

(1.2)

^{if andonlyif}

u(t) fo2 G(t,s)[h(sl -N[Ku](s)]ds (3.2/

UsingFubini’stheoremit iseasy^toseethatforany

2n 2

-N

fo G(t,s)[Ku](s)ds fo

x(t,s)u(s)ds with

2

x(t,s)---N fo G(t,r)k(r,s)ds (3.3)

Therefore, equation

(3.2)

^isequivalenttothefollowingabstractequation

where

and

u(t)

w(t)+

[Tu](t)

^(3.4)

2

w(t)-

fo

G(t,s)h(s)ds

2

[Tu](t)-- J0

"r.(t,s)u(s)ds

In

consequence,u is asolutionof thelinearproblem

(1.2)

ifandonlyifuis afixedpoint^of theoperator

Tw

^:L

Z(l) L Z(l), Tw(u)

^w+Tu.

THEOREM3.1.

Suppose

that

lIz

^{< 1, (3.5)}

then

(1.2)

^{has a} unique ^solution u---lim,,._.(R)u,,

uoLZ(1), u,/t--T,(u,),

^{n >0.}

Moreover,

the solution isgivenbythefollowingrelations

u(t)- fo [G(t,s)+H(t,s)]h(s)ds,

(3.6)

2

H(t,s)--

J0 L(t,r)G(r,s)dr (3.7)

2

L----x’t,, ^x--x, "t,(t,s)--f x,_(t,r)x(r,s)dr,

n>2.

(3.8)

do

PROOF. Notethat T(u)

T(v)[[2 11"

^u

vll

^{foranyu,v L}

^{2(I). By}

^thecontraction principle of Banachwehavethat

T,

hasauniquefixedpointwhich isthesolutionof

(1.2).

^Now,

we choose u0 w. Using againFubini’stheorem, itiseasytoseethat

2

u,(t)

w(t)+

J0 H,(t,s)h(s)ds H.(t,s)-- fo L,(t,r)G(r,s)dr,

We have that

lix, ll2-: ^1111’2,

^{and taking} into account

(3.5)

we see that the series

X*-:

^is

convergent in

L2(J).

^Wenowdeduce that

{H,,} H

in

L:’(J)

andthe validity of formulas

(3.6), (3.7), (3.8).

I

Takingintoaccount

(3.3)we

^{have that}

xll kll: ^{Gl[2. In}

consequence,

k[l: all

^{< implies}

that thelinearproblem

(1.2)

^hasaunique^solution. Itispossibletogive differentestimatesfork that imply that

(3.5)

holds. Forinstance,if thereexists c >0such that

Ifo2k(/,s)dsl

^{<c forevery sl}

^(3.9)

then

I(,)1

^b

or

vy(,)j and

1i11

^{2b. Therefore,}

c-

^{impliesthat thelinear}

problem

(1.2)

isuniquelysolvable.

Il. 211.

Inthecasethatk

eL(R)(J)then I(t,s)l ---

^fora.e.(t,s)eJ. Thus,

IIll= <-w--

^{and, inthis}

situation,

kll <

M^impliesthat

^:[12

^<1.

760 J.J. NIETO

Notethat in the case thatKisgiven by(3.1),^{Erbe and}Ouo gave theestimate

(see

formula

(4)

in

[6])

Soil ^goll

^/

^N[I ll

^<

_2-

M whichobviously implies

(3.5).

Anatural questionisifTheorem 3.1 remains valid inthecritical case

11 -

^Thefollowing

exampleshowsthat in suchacase the linearproblem

(1.2)

mayhaveeither no solution or an infinite numberof solutions, thus showing that theestimate

(3.5)

isassharpaspossible.

EXAMPLE. Takek c G,c E

R

^{such that}

kll

^{1. Theintegraloperator}

^T

associated to

"tiscompact andselfadjoint. Thus,+1 or -1 is aneigenvalue^ofT.

Suppose

that+1isaneigenvalue andchoose u

,

_{0 withTu u. Thus,by}the Fredholmalternativetheorem, theequation^{u w +}Tu haseither nosolutionoraninfinitenumber ofsolutions.

4.

MAXIMUM

PRINCIPLE

Weare now interestedinobtainingasimilarresultto

(2.4)

for thelinearintegro-differential problem

(1.2).

Usingthe representation

(3.6)

^{for the}solutionof

(1.2),

we seethatit isequivalent toshow thatG

+H

a:0 a.e.onJ. SinceG(t,s)>a ^forany(t,s)EJ,we canaffirmthatG/H>0 a.e.onJif, for instance,

nil ^-:

^{a. Wefirstgiveanestimatefor}

nil

THEOREM4.1. Supposethat k

EL(R)(J)

^and

M (4.1)

I111 <2:n

Then,

(3.5)

holds and

n M(M

2

**)"

PROOF. Fromtherelation

(3.3)

^wededuce thatx

EL(R)(J),

^and

I1:11-

¹

11- --ff--

^d<

2"

Onthe otherhand,

x, L’(J)

^foranyn

N,

^and

11- ^a(2nay-x.

^Thisimpliesthat the series

x,

isconvergent in

L’(J).

Therefore,

d and

(4.2)

Ilnll

^<

M(I_2d)

d which ispreciselyestimate

(4.2).

Wenotethat therighthand side of

(4.2)

^tendsto 0when

11.

^{tendsto0. Thus,weobtainthe}

followingmaximumprinciplefor the linearequation

(1.2).

THEOREM4.2. Assumethatk

EL (R)(J)

^and

Mm

e

:11

^<_e2,,,,,

+ 2nme

"r.

(4.3)

Then,wehavethat

(3.5)

^{holds andG}+

H

a0a.e.on

J.

PROOF.

Wefirstnotethat thefollowing inequalityholdstrue:

me

e

’-

^{+2tme 2rt (4.4)}

Hence,

11 -

M ^Now,usingG(t,s)+H(t,s)>a

^(4.2)

^{wehavefor a.e.(t,s)@Jthat} M(M

211 kll ^)"

Combining

(4.2)

^and

(4.3)

^weobtainthat

nil

^a.

Therefore,we can write thatG(t,s)+H(t,s) a

-Ilnll

^{/0fora.e.(t,s)EJ,}completing the proofofthe theorem.

Asaconsequence,weobtainthat ifinequality

(4.3)

^{holds then}

h 0(< 0) a.e. on I implies ^u 0(<0) a.e. onJ (4.5) Wenowconsider the casewhenthe kernel hasconstantsignonJ. Ifk^:a0 a.e. onJ,then"t 0 a.e. onJand we have thatH 0 a.e. onJ. Then, triviallythe maximumprinciple

(4.5)

holds.

Ifk 0a.e. on

J,

then’t 0a.e.onJand theprevious reasoningisnotvalid. However,wehave that(-1)"’t,, 0 a.e. on

J,

^{na: and}this isusefultoprove^thefollowing^result.

THEOREM

4.3. Suppose^thatk

@L(R)(J)

issuch thatk 0 a.e.onJand

kll _8I2em

m

[vt(e

^{:’’‘’’}

1):’

+16

2Me

^2’’

-(e

^2,,,,

1)]

^{r/. (4.6)}

Then,(3.5)holds andG+H 0a.e. onJ.

M M

PROOF. Wefirstnotethatr/ < and then

k!l

^<

.

^{Onthe otherhand,}

and

Hence,

fora.e.(t,s)J,

for(t,s) J. In consequence,

,,, ^{:,,11 .., :}

^{d(2nd) -2.}1-(2:d)

^a gz-4llk[l$ ^{gll zll-}

nodd nodd

-MIIZII

L(t,s)

>M2

⁴

[I z:ll

H(t,s)

G+Ha-

M’-- 4llkll

^a

onJ.

Now, ax

^{>0if andonlyif}

4

e" _]l kll

⁺

^m(e

^2,,,

^1)II kll. ^-M2e’

^O,

andthis is truefor

ll ^[0,r+].

Notethatestimate

(4.6)

improves

(4.3)

^sincer

< _r+.

762 J.J. NIETO

5. MONOTONE ITERATIVE METHOD

Wenowconsider thenonlinearequation

(1.1).

Werecall that

fis

^aCarath6odoryfunction if

f(t,.,

.)iscontinuous fora.e.

l,f(.,u,v)

^ismeasurable foranyu,v

R,

^{and for}any

R

>0there exists o

o,

L

:’(I)

^suchthat

If(t,u,

^{v) o(t)fora.e. Iforanyu,v Rwithmax(u}

,I vl)

^R.

DefineH(/) {u H(1):u(O)

^u(2 n),u’(0)

u’(2n)}.

^Bya solution u

of(1.1)

we mean a function u

H(1)

such that

I [F(u)](t) f(t,u(t),[Ku](t))

is afunctionofL

(I)

satisfyingtheequation fora.e. I.

We saythata

H(I)

is alowersolutionfor

(1.1)

ifFa

CL(1),

and

-a"(t) f(t,a(t),[Ka](t))

for a.e. I.

(5.1)

Similarly,wedefineanuppersolution as afunction

H(I)

^{such that}

F ^{L (I),}

^and

-"(t)f(t,t),[K](t))

^fora.e,

I. (5.2)

In

u

L (I),

ⁱⁿgeneral, Fuis not a functionofL

z(1).

The condition that

F

maps

L z(1)

^into

L z(1)

isequivalent 1 totheexistenceofb

L (I),

a

R

such that

f(t,

^u,

v)[

^b(t)+

a([

^{u +}

]vl)

^fora.e.

Iandeveryu,v R.

Now supposethat condition

(3.9)

isverified. Obviously,

(3.9)

^{is satisfied}if, for instance, k

L "(J). However,

k(t,s)-s -aisakernelthat satisfies

(3.9)

^{but doesnotbelongto}

L’(J).

Then foranyu

L’(1)

wehave that

I[Ku](t)l

^{k(t,s)u(s) <}

cll ^ull-,

andKu

L(I).

In consequence,

If(t,

^u,

[Ku ](/))l a(t)

for a.e. I,where

R max(ll

^{u c u}

II-),

andFu L

(I).

Thus,ifcondition

(3.9)

^holdsanda

H(1),

^thenKa

L’(1)

andFa

EL2(1).

If

fi

^{arelower anduppersolutionsof}

(1.1)

respectively,weshallassumethat

afi

^{on I.}

^(5.3)

Thus, k 0( 0) impliesthatKa

(

^{a.e. on}

L

Inthecasethatk 0 a.e. onJ,weintroducethefollowingcondition: there exist

M

>0,N>0 such that

f(t,u,v)- f(t,w,x)

^-M(u

w)-N(v

-x)

(5.4)

fora.e.

l,a(t)w

^u

(t),and[Ka](t)x

^v

[K](/).

Ifk 0a.e. on

J,

weshallusethe condition: thereexist

M

>0,N>0such that

f(t,u, v)- f(t, w,x) -M(u w)-N(x v) (5.5)

fora.e.

I, (t)

^w u

t),

and

[K] (t)

^v x

[Ka] (/).

THEOM5.1.Assumethatk

L’(J),k

0 a.e.

onL adll ll- . ^{Zn aitM}

^upeose

that thereexist

H(I)

lower anduppersolutions

of(1.1)

respectivelysuchthat

0.3)

and0.4) hold. Then, thereexitsmonotonesequences

{a,} ,

^and

{}

uniformlyon

I

^with

-

^a

and

o .

^{Here and are}the minimal and mimal solutions

of ^(1.1)

respectivelyon

[

Moreover,

thesesequences

veri ^a, , _o.

PROOF. For

[ ],

^letusconsiderthefollowinglinearperiodic boundaryvalueproblem

u"

+Mu

+N[Ku]- hn(t), ^u(0)-

^{u(2n), u’(0)-}

^{u’(2n) (5.6)}

whereh(t)

h(t) f(t,rl(t),[KN](t))

+

Mrl

⁺

^N[KN](t).

This linearproblemhasauniquesolution u

--Arl

^inviewof Theorem 3.1 since N.

k]]

^r.

Moreover,in this caseG+H a0 a.e. onJ.

The operatorAiswelldefinedfrom[ct,

[3]

^to[ct,

[3]

^andAisincreasing.

Indeed, letN

[ct, 13]

anddefinev u ct. Thus, using

(5.4),

we obtain -v"^+My+NKv

f(t,N,Krl)

+

Mr

+

NKrl ^(-cf’

^+Mct+

^NKct)

f(t, N,KN)

+

Mri

⁺

NKrl f(t,

ct,

Kct)

^Mct NKctaO

Hence, byvirtueof Theorem4.3 wededuce thatva:0 onL Similarly,one can show thatu

[3

^on I.

Toshow themonotonicityofA,let_{Ni G}

[ct,[3],

u,

-AN,,

1,2, and_{w -ui-uz.}

Hence,

-w"+Mw+NKw

f(t,

_Nt

,Krl)

+

MN

⁺

^{NKrI, f(t,}

r12

,Krh) Mr

_h

NKN2

⁰

andthen,w 0.

Wenowdefine ct0

,

ct, Aet,, n 0.

By

thepropertiesof theoperator

A,

thesequence

{ct,

^isincreasingand uniformly boundedonI. Then,

{t, }

’

^pointwiseon

^L

Writing theintegral representation forAa,and using standard argumentsweobtainthat isactuallyasolution of

(1.1).

Analogously,defining

I-

13, 15, /-AI3,,ⁿ 0,

{13,,} ,I,

P, where

xp

is solutionof

(1.1).

Toshow that

q

and

q.,

aretheminimal and maximal solutionsof

(1.1)

in

[ct,13],

let u

be a solutionof

(1.1).

Then,Au u, andusingthepropertiesof the operatorAwehavethata.,_<u_</3, foreveryn

N.

Passingtothe limit whenn ^ooweobtainthat

9

^u<

^p.

Ifk<0a.e. on

J,

then we can usedirectly^thatG+

H

^>0a.e.

onJ

toobtain thefollowingresult.

u In addition, THEOREM5.2. Suppose thatk

L(R)(J),

k 0 a.e. on

J,

^and

assume thatthereexistct,

fJ H,(I)

^{lower and}upper^solutions

of ^(1.1)

respectively^{such that}(5.3) and(5.5)hold. Then,thereexist monotonesequences

{a,,}

’ ^,

^and

^{{1,,} xp}

^uniformlyon

^I

^with

%

, f, fo f

and

t, p

aretheminimaland maximal solutions

of ^(1.1)

respectivelyon

[o.,

PROOF.

ForN ^[a,13],

^{letusconsiderthe linearproblem}

^{(5.6). In}

^{thisease, Kt>u}

KI

^>

on/. Asin theproofof Theorem 5.1wehave that

(5.6)

^hasauniquesolution u

=AN.

The operator A is well definedfrom

[ct,[3]

^to

[ct,[3],

^{andit is} increasing. Asin theproofof Theorem 5.1 we

construct monotonesequences

{a, }

’ ⁹

^and

^{{13, } ap,}

^{where and}

^p

^arethe minimal and maximal solutionsof

(1.1)

respectively^betweenaand

[3.

It mayoccurthat

Ks K[3 a.e. on I (5.7)

evenifct

[3

^andkchangessignonJsince

[Kct](t)

dependsonthe valueofk(t,s)ct(s),0 s 2.

(For

^{example,take e>0,}

k(t,s)=

1 for 0<s<2n-e,k(t,s)=-I for 2n-e<s2n, andt.1,

[3 2.) In

such asituation,we canapplythepreviousresultstoobtain.

THEOREM 5.3.

Suppose

thatk

L(R)(J),N kl]

^{<r, and}

(5.3), (5.4), (5.7)

hold. Then, there exist monotone sequences

{,}

’ ^,

^and

^{{[3,} p}

^{uniformly on}

^I

with Cto-et,

o-[

^and

et0 a, [3, ^<

[30.

^{Here and}

ap

are the minimal and maximal solutions of

(1.1)

respectivelyon

[ct, I].

764 J.J. NIETO

If Kct

KI5

^{a.e. onJ}

^(even

^ifkhas no constantsignon

J),

then we have ananalogousresult using

(5.5).

In

^thegeneral case

(k

^hasno constant sign ^on

J)

we are ableto deal with a linearintegral perturbation^{of the}ordinarydifferentialequation-u"--f(t,u), beingsuchaperturbationof the type N

Ku,N

5R.

In

concrete, we consider theproblem

-u"-f(t,u)+N.Ku, u(O)-u(2n), u’(O)-u’(2n). (5.8)

Wenowrequire thefollowing^condition: there exists

M

>0such that

f(t,u)

f(t, v) -M(u

v) (5.9)

for a.e.

1,et(t)

^{v u}

[5(t). In

thiscase, for rl

[ct, l],

thelinearproblem

(5.6)

^reads

u"

+Mu f(t,

rl)

+Mr +NKr

THEOREM

5.4. Consider the nonlinearproblem (5.8) were

f satisfies

^{condition (5.9).}

Supposethatk

L(R)(1)

issuch that

IN[ k[[

^{<r, andthatthereexistct,}

f3 H"(1)

^{lower andupper}

solution

of

^(5.8)respectivelywith a

f3

^{onI. Then,thereexistsmonotone}sequences a,

} ,

^and

, ^p

^{uniformlyonIwith a} cto

...

^a,

...

^3,

...

^[3o

^3.

^{Here and}

^ap

^{aretheminimal}

andmaximalsolutions

of ^(5.8)

respectively^on

[a, ].

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