existence of a periodic solution for some partial functional differential equations with infinite delay

doi:10.1006 rjmaa.2000.7321, available online at http:rrwww.idealibrary.com onExistence of a Periodic Solution for Some Partial Functional Differential Equations with Infinite Delay1 Rac

doi:10.1006 rjmaa.2000.7321, available online at http:rrwww.idealibrary.com on

Existence of a Periodic Solution for Some Partial

Functional Differential Equations with Infinite Delay1

Rachid Benkhalti

Department of Mathematics, Pacific Lutheran Uni¨ersity, Tacoma, Washington 98447

andHassane Bouzahir and Khalil Ezzinbi

Departement de Mathematiques, Uni´ ´ ¨ersite Cadi Ayyad, Faculte des Sciences Semlalia,´ ´

B.P 2390, Marrakech, 40000 Morocco Submitted by J Henderson

Received April 17, 2000

This paper deals with the existence of periodic solutions for some partial functional differential equations with infinite delay We suppose that the linear part is nondensely defined and satisfies the Hille ᎐Yosida condition In the nonlin- ear case we give several criteria to ensure the existence of a periodic solution In the nonhomogeneous linear case, we prove the existence of a periodic solution under the existence of a bounded solution 䊚 2001 Academic Press

1 INTRODUCTIONThe purpose of this work is to discuss the existence of a periodicsolution of the following partial functional differential equation withinfinite delay

where A is a nondensely defined linear operator on a Banach space

ŽE, , and the phase space B< < B is the linear space of functions mapping

in this case the phase space is the space of continuous functions from

wyr, 0 into E, especially we refer to 22, 25 Equation 1 with A beingx w x Ž

densely defined has been studied by several authors In 9᎐11 , Henriquezdiscussed the problem of existence of solutions defined in the sense of

w xclassical semigroups theory In 11 , regularity of solutions was obtained by

making sufficient conditions on the phase space, the perturbation F, and

w x

on the C -semigroup generated by A In 9 , existence of periodic solutions0

was investigated using the Sadovskii fixed point theorem on the well-known

Poincare map In 10 , a method of approximation of Eq 1 by a family of´

equations with finite delays was introduced and results on the existence of

w xperiodic solutions and stability were deduced In 15 , Murakami treated

the stability properties when A is the infinitesimal generator of a compact

is assumed to generate a compact C -semigroup and the authors obtained0

several criteria on the existence of a periodic solution The followingmethod is based on the perturbation theory of semi-Fredholm operatorsand the fixed point theorem for affine map which has been obtained by

w xChow and Hale in 6

In 1, 2, 4 , we considered Eq 1 with A being nondensely defined and

satisfying the Hille᎐Yosida condition More precisely, we addressed theproblem of existence, uniqueness, regularity, existence of global attractor,and local stability by means of the integrated semigroups theory For

w xdensely defined Mainly, we recuperate the announced results in 9 Then,

assuming that the phase space B B satisfies an extra axiom, we prove theexistence of a periodic solution under slightly different conditions Finally,

in the case where B B is a uniform fading memory space, we exhibit aMassera type criterion for the nonhomogeneous linear case Our main

results would be extensions of the results obtained in 6, 9, 20

In Section 2, we recall some basic results on existence and uniqueness of

Ž integral solutions of Eq 1 In Section 3, we establish some results on theexistence of periodic solutions in the nonlinear case In Section 4 we dealwith the nonhomogeneous linear case and we prove Massera’s criterion:the existence of a bounded solution implies the existence of a periodicsolution The remaining section is devoted to an example

2 EXISTENCE AND CONTINUATION OF SOLUTIONS

Throughout this paper, we suppose that A satisfies the Hille᎐Yosidacondition:

ŽH1 There exist two constants MG 1 and ␮ g ⺢ with ␮ q ⬁ ;Ž

ŽA There is a positive constant H and functions K , M :Ž Ž ⺢ ªq

⺢q, with K continuous and M locally bounded, such that for any␴ g ⺢,

for all tG 0 In particular, ␾ 0 g D A Ž

Define the part A of A in D A by0 Ž

Assume the following compactness condition:

ŽH2 The semigroup T tŽ 0Ž tG 0 is compact on D A , that is, T t isŽ 0Ž

as-ŽH4 F : 0,w ⬁ = B B ª E, a ) 0, is continuous in t and uniformly

Lipschitz continuous with respect to the second argument: there exists a

␾ 0 g D A Then Eq 1 has a unique integral solution x , ␾ which isŽ

defined for t G 0 Moreo¨er, there exist two locally bounded functions

m , n : ⺢ ª ⺢ such that for ␾ , ␾ g B1 2 B with ␾ 0 , ␾ 0 g D A1 2 Ž

and t G 0, one has

x tŽ ,␾ y x , ␾1. tŽ 2. B F m t eŽ ␾ y ␾1 2 B

3 MAIN RESULTSSet

which implies that

x ,␾ is the integral solution of Eq 1 To show that Eq 1 has an

␻-periodic solution, it suffices to show that, by Proposition 3, P has a␻fixed point Before doing so, let us study the continuity of P ␻

PROPOSITION 4 Each of the following conditions implies the continuity

of P ␻

Ž i Conditions H1 and H4 are satisfied.Ž Ž

Ž ii Conditions H1 , H2 , and H3 are satisfied and for eachŽ Ž Ž ␺ g EE,

We deduce that the set x t : kg ⺞ is totally bounded and therefore is

relatively compact in E To establish the equicontinuity, let 0 - t - t F0 ␻.Then

 t0 Ž Ž k 4

Moreover W0s lim␭ªq⬁ 0H T0 t0y s B F s, x ds : k g ⺞ is relatively␭ s

compact and it is well known that

ª u as k ª ⬁ Since x , s s ␩ : 0F s F ␻, k g ⺞ is a bounded set of

B and F takes bounded sets into bounded sets, then we have

then Eq 1 has at least one ␻-periodic solution.

In order to prove this theorem, we need some preliminaries The

Ž Kuratowski’s measure of noncompactness ␣ ⍀ of ⍀ is defined by

only for each subset G G : D D with ␣ G G ) 0 If we let G G ␴ , ␻ , 0 F ␴ F ␻,

be the set defined by

cover C i 1 F iF n : E, such that diam C - d Next, for a subset C of E, i

we denote by CU the set

where w t,␸ s lim␭ªq⬁ 0 0H T t y s B F s, x ,␭ s ␸ ds, for t g 0, ␴

One can use similar arguments as in the proof of Proposition 4 to see that

w ,Ž ␸.rw0,␴ x:␸ g G G4 is relatively compact in C 0,Žw ␴ ; E In fact, letx

Hence the set w ,␸ <w0,␴ x:␸ g G G is totally bounded and therefore is

relatively compact in E To establish the equicontinuity, let 0 - t - t F␴

ŽP␻n ␸ is a Cauchy sequence of B.n B, which implies that there exists ␺ g D D

such that P␻k ␸ ª ␺ , as k ª q⬁ Consequently P ␺ s ␺ This completes␻

the proof by use of Proposition 3

By Theorem 1, F s, x , s ␺ : s g 0, t , n G 0 is bounded Since T ␧ n 0

is compact, there exists a compact set W such that␧

T0Ž␧ ½␭ªq⬁ 0lim H T t0Ž q␪ y ␧ y s B F s, x , ␺. ␭ Ž sŽ n .ds : nG 05

: W ␧

Furthermore, there exists a positive constant c such that

Thus U t2 ␺ ␪ : n G 0 is totally bounded and therefore relatively n

compact To establish the second assertion, it is sufficient to show that

ŽU t2Ž ␺n n G 0 is equicontinuous in Žy⬁, 0 Letx ␪ g y⬁, 0 For ␪ g0 Ž x

Žy⬁, 0 close enough tox ␪ such that ␪ - ␪, we see that0 0

By the boundedness of F s, x , s ␺ : n G 0, s g t q ␪ , t , we deduce n 0

that there exists a positive constant d such that

␺n ds : n G 0 is relatively compact in E and using the fact that

ŽT x0Ž x g W1 is equicontinuous at the right in 0, we get

lim ŽU t2Ž ␺n Ž␪ y U t ␺ Ž 2Ž n Ž␪0. s 0, uniformly in ng ⺞

␪ª␪ 0

␪)␪ 0

A similar argument shows that

there is a continuous function ␾ : y⬁, 0 ª E and a subsequence ␾ of n

ŽU t2Ž ␺n n G 0 which converges compactly to ␾ in y⬁, 0 Since ␾ ␪ s 0Ž x nŽ

Ž for ␪ F yt and n G 0, then ␾ is continuous and ␾ ␪ s 0 if ␪ F yt.

Hence by axiom A i , ␾ belongs to B B Moreover, by axiom A iii ,

5␾ y ␾n 5B F K t supŽ yt F␪ F 0< Ž ␾ ␪ y ␾ ␪ and we have ␾ y ␾n Ž < 5 n 5Bª

0 as nª ⬁ So the image of any bounded sequence contains a converging

subsequence in B B with respect to the seminorm

ŽH6 There exists a nonempty closed bounded and convex subset D D

Then Eq 1 has at least one ␻-periodic solution.

Proof. We will use Sadovskii’s fixed point theorem as in the proof of

-␣ GŽG.

$

Thus P is␻ ␣-condensing on D Dand Eq 1 has an ␻-periodic solution

In relation with the above condition, we have obtained the followingresult:

together with the norm ␾ [ sup␥ ␪ F 0 e ␾ ␪ , ␾ g C , satisfies axioms␥

ŽA , A1 , and B Moreo Ž Ž ¨er if 5 Ž 5T t0 F ey␦ t, t G 0, where ␦ is a positi¨e

Ž

constant, then condition 9 is true.

Now we will give some sufficient conditions ensuring the assumption

ŽH5 Consider the phase space B B s C = L g to be the space of1Ž

Ž x w x

functions ␸ : y⬁, 0 ª E such that ␸ is continuous on yr, 0 , for some

Ž < Ž <

r ) 0, Lebesgue measurable, and g ␸ is Lebesgue integrable on

Žy⬁, yr , where g : y⬁, yr ª ⺢ is a positive Lebesgue measurable Ž

We suppose that g satisfies:

Ž i g is integrable onŽyd, yr for any d G r, and

Ž ii there exists a locally bounded function G :Žy⬁, 0 ª 0, q⬁x w such that

gŽ␰ q ␪ F G ␰ g ␪ , Ž Ž for all ␰ F 0 and ␪ g y⬁, yr _ N ,Ž ␰

4 MASSERA TYPE CRITERIONConsider the following nonhomogeneous linear equation

d

¡ x tŽ s Ax t q L t, x q f t ,Ž Ž . Ž tG 0,

t

where L is a continuous function form ⺢q= B B into E, linear with the

second argument and ␻-periodic in t, and f is a continuous ␻-periodic

uniformly bounded if supn g N supy⬁- ␪ F 0 ␸ ␪ - q⬁

We suppose that B B is a fading memory space in the sense that B B

satisfies the following two axioms

Ž C If a uniformly bounded sequence Ž␸n.n in C00 converges to a

Now assume that

ŽH7 Equation 11 has a bounded solution y onŽ ⺢ in the senseq

< Ž <

q

that suptg ⺢ y t - q⬁

Let BC be the space of bounded continuous functions mapping y⬁, 0

into E with the uniform norm topology Then one has

Axiom A can be chosen bounded on 0,q⬁ Using iii in Axiom A we

obtain the existence of a positive constant N such that1

Proof Set D D [ co y : n g ⺞ where co denotes the closure of the n␻

convex hull Then D D is a nonempty closed convex subset of X X D D is

ŽH6 is true and by Theorem 3, we conclude that Eq Ž11 has an

␻-periodic solution

COROLLARY1 Assume that B B s C ,␥ ␥ ) 0 Under the conditions H1 ,

ŽH2 , and H7 , Eq 11 has a m Ž Ž ␻-periodic solution, for a certain m g ⺞ U

w x

Proof. It is known from 12 that if␥ ) 0, the phase space C satisfies␥

Ž C and D1 Moreover KŽ Ž␴ s 1 and M ␴ s e Ž y␥ ␴ Hence

all ␭ in ␴ HH for which at least one of the following holds:

Ž i ImŽ␭I y HH s ␭I y HH z : z g Z is not closed. Ž 4

Ž ii the generalized eigenspace M H␭ŽH.s DkG1 kerŽ␭I y HH.k of ␭

THEOREM 6 Suppose that B B is a uniform fading memory space Under

conditions H1 , H2 , and H7 , Eq 11 has an ␻-periodic solution.

The proof is based on the following two results

Ž Then ⍀ ; D ⍀ Fori, j i, j ␸, ␾ g ⍀ by 18 we havei, j

U t1Ž ␸ y U t ␾1Ž B F K t yŽ ␧ ␦ q M t y ␧ C ␤ q ␦ ␸ y ␾ Ž ␧Ž B,and letting ␦ go to zero we obtain

␣ U t F C M t y ␧ ,Ž 1Ž ␧ Ž for t)␧ This completes the proof of the lemma

w x

THEOREM7 6 Let Z be a Banach space and P : Z ª Z a linear affine

map, that is, Px s Lx q y where L is a bounded linear map and y g Z is

fixed If Im I y L is closed and there exists an x g Z such that P x0 0 nG 0

is bounded, then P has at least a fixed point.

Then P␻␾ nG 0s y n␻ ,␾, f nG 0 is bounded Hence, by Theorem 7,

the map P has a fixed point in B␻ B, which gives an ␻-periodic solution of

Ž

Eq 11

where b :⺢ ª ⺢ is a continuous ␻-periodic function, g : ⺢ = 0, ␲ ª ⺢

is a continuous function, ␻-periodic with respect to time, G is a positive

It is known from 12 that the space C ,␥ ␥ ) 0, introduced in Proposition 6,

is a normed uniform fading memory space

We suppose that:

Ž i G eŽ y␥ is integrable onŽy⬁, 0 ,x

Ž ii w0g C y⬁, 0 = 0,ŽŽ x w ␲ : ⺢ , with limx ␪ ªy⬁Že␥␪sup0 F␰ F␲< Žw0 ␪,

␰ exists, and w 0, 0 s w 0,␲ s 0

Then L is continuous form ⺢q= C into E, linear with the second␥

argument and ␻-periodic in t and f : ⺢qª E is continuous ␻-periodic If

We can see that assumption ii implies that ␸ g C , and ␸ 0 g D A ␥ Ž

Let A be the part of the operator A in D A given by0 Ž

Assume further that

Žiii there exists d g 0, 1 such that 0 - lH GŽ y⬁0 Ž␪ d␪ - 1 y d,

Proof Let x , ␸ be the integral solution of Eq 20 Then x t, ␸

-␳ for t G 0 We proceed by contradiction Suppose that there exists t ) 00

< Ž <

such that x t ,0 ␸ ) ␳ Let

t1s inf t ) 0 : x t, Ž ␸ ) ␳ 4

By continuity, it follows that

Ž Hence, by assumption iii

Finally, all conditions of Theorem 6 are satisfied, and consequently, Eq

Ž20 has an ␻-periodic solution

ACKNOWLEDGMENT

The authors are deeply indebted to the referee for the careful reading of the previous version of the manuscript The valuable suggestions and remarks led to several improvements

in the exposition.

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