Periodic boundary value problem for second order integro ordinary differential equations with general kernel and Carathéodory nonlinearities

(1)

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 UniversidaddeSantiagodeCompostela

SPAIN

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

ABSTRACT. Westudy the existenceofsolutionsfor the periodicboundaryvalueproblemfor

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 ispointedoutin

[6],

the methodof

[7]

is notapplicableto 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, Tanintegraloperator of Volterra type, andSanintegral

operator of Fredholm type. Problem

(1.1)

is studied in

[8]

under theassumptionthat

f(t,u,v)

is

continuousandincreasingin v, and in

[7]

for

f

continuousandKof Volterra type. Followingthe ideasof

[6]

westudy

(1.1)

in thegeneralcase,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 ifequalityisattained, then theexistence-uniquenessresultfor

(1.2)

isnotvalidanymore.

(2)

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 bytheexpression

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 iseasytoseethatforany

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)

(3)

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 afixedpointof 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)

hasauniquesolution. 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

M

(4)

760 J.J. NIETO

Notethat in the case thatKisgiven by(3.1),Erbe andOuo gave theestimate

(see

formula

(4)

in

[6])

M

Soil

goll

/

N[I

ll

<

2-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 aneigenvalueofT.

Suppose

that+1isaneigenvalue

andchoose u

,

0 withTu u. Thus,bythe Fredholmalternativetheorem, theequationu w +Tu

haseither nosolutionoraninfinitenumber ofsolutions.

4.

MAXIMUM

PRINCIPLE

Weare now interestedinobtainingasimilarresultto

(2.4)

for thelinearintegro-differential problem

(1.2).

Usingthe representation

(3.6)

for thesolutionof

(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)

d

Ilnll

<

M(I_2d)

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)

(5)

PROOF.

Wefirstnotethat thefollowing inequalityholdstrue: me

e

’-

+2tme 2rt (4.4)

M

Hence,

11

-

Now,using

(4.2)

wehavefor a.e.(t,s)@Jthat G(t,s)+H(t,s)>a

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: andthis isusefultoprovethefollowingresult.

THEOREM

4.3. Supposethatk

@L(R)(J)

issuch thatk 0 a.e.onJand

m

kll

[vt(e

:’’‘’’

1):’

+16

2Me

2’’

-(e

2,,,,

1)]

r/.

8I2em

(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.

a

gll

zll-1-(2:d)

gz-4llk[l$

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+].

(6)

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 forany

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 satisfiedif, 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 arethe minimal and mimal solutions

of

(1.1)

respectivelyon

[

Moreover,

thesesequences

veri

a,

,

o.

(7)

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, let_N

[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,letNi G

[ct,[3],

u,

-AN,,

1,2, andw -ui-uz.

Hence,

-w"+Mw+NKw

f(t,

Nt

,Krl)

+

MN

+

NKrI,

f(t,

r12

,Krh) Mr

h

NKN2

0 andthen,w 0.

Wenow_{define 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 usedirectlythatG+

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 anduppersolutions

of

(1.1)

respectivelysuch 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)

respectivelybetweenaand

[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)

(8)

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 perturbationof theordinarydifferentialequation-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 thefollowingcondition: 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,thereexistsmonotonesequences a,

}

,

and

,

p

uniformlyonIwith a _cto

...

a,

...

3,

...

[3o

3.

Here and

ap

aretheminimal

andmaximalsolutions

of

(5.8)

respectivelyon

[a,

].

REFERENCES

[1]

APPELL, J. and ZABREIKO, P.

P.,

Nonlinear Superposition

Operators,

Cambridge University

Press,

Cambridge, 1990.

[2]

BECKER,

L.

C., BURTON,

T.

A.

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