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

Journal of Mathematical Analysis and Applications 256, 257᎐280 Ž2001.
doi:10.1006rjmaa.2000.7321, available online at http:rrwww.idealibrary.com on

Existence of a Periodic Solution for Some Partial
Functional Differential Equations with Infinite Delay 1
Rachid Benkhalti
Department of Mathematics, Pacific Lutheran Uni¨ ersity, Tacoma, Washington 98447

and
Hassane 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 nonlinear 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. ᮊ 2001 Academic Press

1. INTRODUCTION
The purpose of this work is to discuss the existence of a periodic
solution of the following partial functional differential equation with
infinite delay

¡d

¢x s ␾ g B ,

~ dt x Ž t . s Ax Ž t . q F Ž t , x . ,
t

t G 0,

Ž 1.

0

1

This work is supported by Moroccan Grant PARS MI 36.
257
0022-247Xr01 $35.00
Copyright ᮊ 2001 by Academic Press
All rights of reproduction in any form reserved.


258

BENKHALTI, BOUZAHIR, AND EZZINBI

where A is a nondensely defined linear operator on a Banach space
Ž E, < . <., and the phase space B is the linear space of functions mapping
from Žyϱ, 0x into E satisfying some fundamental axioms Žsee Section 2..
For every t G 0, the function x t g B is defined by
xt Ž ␪ . s x Ž t q ␪ . ,

for ␪ g Ž yϱ, 0 x .

The function F : w0, qϱ. = B ª E is continuous and ␻-periodic in t.
Several results were obtained for Eq. Ž1. when the delay is bounded and
in this case the phase space is the space of continuous functions from
wyr, 0x into E, especially we refer to w22, 25x. Equation Ž1. with A being
densely defined has been studied by several authors. In w9᎐11x, Henriquez
discussed the problem of existence of solutions defined in the sense of
classical semigroups theory. In w11x, regularity of solutions was obtained by
making sufficient conditions on the phase space, the perturbation F, and
on the C0-semigroup generated by A. In w9x, existence of periodic solutions
was investigated using the Sadovskii fixed point theorem on the well-known
Poincare
´ map. In w10x, a method of approximation of Eq. Ž1. by a family of
equations with finite delays was introduced and results on the existence of
periodic solutions and stability were deduced. In w15x, Murakami treated
the stability properties when A is the infinitesimal generator of a compact
semigroup. In w18x, Shin studied the existence and uniqueness of Eq. Ž1.
when A generates a C0 semigroup. In w13x, Hino et al. established results
on the existence of an almost periodic solution of Eq. Ž1. when A
generates a compact C0 semigroup by assuming the existence of a bounded
solution which is BC-uniformly stable. Recently in w20x, Shin and Naito
established the existence of a periodic solution for some nonhomogeneous
linear functional differential equations in a Banach space. The linear part
is assumed to generate a compact C0-semigroup and the authors obtained
several criteria on the existence of a periodic solution. The following
method is based on the perturbation theory of semi-Fredholm operators
and the fixed point theorem for affine map which has been obtained by
Chow and Hale in w6x.
In w1, 2, 4x, we considered Eq. Ž1. with A being nondensely defined and

satisfying the Hille᎐Yosida condition. More precisely, we addressed the
problem of existence, uniqueness, regularity, existence of global attractor,
and local stability by means of the integrated semigroups theory. For
similar approaches, we refer to w3x for equations with finite delay, to w5x for
equations without delay, and also to some references therein.
The aim of this paper is to investigate the existence of periodic solutions
for Eq. Ž1. with A satisfying the Hille᎐Yosida condition without being


FUNCTIONAL DIFFERENTIAL EQUATIONS

259

densely defined. Mainly, we recuperate the announced results in w9x. Then,
assuming that the phase space B satisfies an extra axiom, we prove the
existence of a periodic solution under slightly different conditions. Finally,
in the case where B is a Žuniform. fading memory space, we exhibit a
Massera type criterion for the nonhomogeneous linear case. Our main
results would be extensions of the results obtained in w6, 9, 20x.
In Section 2, we recall some basic results on existence and uniqueness of
integral solutions of Eq. Ž1.. In Section 3, we establish some results on the
existence of periodic solutions in the nonlinear case. In Section 4 we deal
with the nonhomogeneous linear case and we prove Massera’s criterion:
the existence of a bounded solution implies the existence of a periodic
solution. The remaining section is devoted to an example.

2. EXISTENCE AND CONTINUATION OF SOLUTIONS
Throughout this paper, we suppose that A satisfies the Hille᎐Yosida
condition:
ŽH1. There exist two constants M G 1 and ␮ g ‫ ޒ‬with Ž ␮ q ϱ. ;

␳ Ž A. and supÄ5Ž ␭ y ␮ . n RŽ ␭, A. n 5 : ␭ ) ␮ , n g ‫ގ‬4 F M, where ␳ Ž A. is
the resolvent set of A and RŽ ␭, A. s Ž ␭ I y A.y1 .
We assume that Ž B, 5. 5 B . is a seminormed abstract linear space of
functions mapping Žyϱ, 0x into E, which satisfies the fundamental axioms
first introduced in w8x:
ŽA. There is a positive constant H and functions K Ž . ., M Ž . . : ‫ޒ‬qª
‫ ޒ‬, with K continuous and M locally bounded, such that for any ␴ g ‫ޒ‬,
a ) 0, if x : Žyϱ, ␴ q ax ª E, x␴ g B, and x Ž . . is continuous on w ␴ , ␴ q
ax then for every t in w ␴ , ␴ q ax the following conditions hold:
q

Ži.
Žii.
Žii.X
Žiii.

x t g B;
< x Ž t .< F H 5 x t 5 B , which is equivalent to
for each ␸ g B, < ␸ Ž0.< F H 5 ␸ 5 B ;
5 x t 5 B F K Ž t y ␴ . sup␴ F s F t < x Ž s .< q M Ž t y ␴ .5 x␴ 5 B .

ŽA1. For the function x Ž . . in ŽA., t ¬ x t is a B-valued continuous
function for t in w ␴ , ␴ q ax.
ŽB. The space B is complete.
Notice that axiom ŽB. is equivalent to saying that the space of equivaˆ [ Br5 . 5 B s Ä ␸ˆ : ␸ g B4 is a Banach space.
lence classes B
For a deeper discussion on the phase space B, we refer to w12x.


260

BENKHALTI, BOUZAHIR, AND EZZINBI

DEFINITION 1. We say that a function x: Žyϱ, ax ª E, a ) 0, is an
integral solution of Eq. Ž1. in Žyϱ, ax if the following conditions hold:

Ž i.

x is continuous on w 0, a x ,

Ž ii .

H0 x Ž s. ds g DŽ A. , for t g w0, ax , and

Ž iii .

t

xŽ t. s

¡␾ Ž 0. q AH x Ž s . ds q H F Ž s, x . ds,

~

¢␾ Ž t . ,

t

t

0

0

s

0 F t F a,
yϱ - t F 0.

It follows from Žii. that for an integral solution x, we have x Ž t . g D Ž A . ,
for all t G 0. In particular, ␾ Ž0. g D Ž A . .
Define the part A 0 of A in D Ž A . by

½
LEMMA 1 w21x.

D Ž A 0 . s Ä x g D Ž A . : Ax g D Ž A . 4 ,
A 0 x s Ax

for x g D Ž A 0 . .

A 0 generates a C0-semigroup ŽT0 Ž t .. t G 0 on D Ž A . .

It is well known from w1, 21x that for ␾ g B such that ␾ Ž0. g D Ž A . , if
the associated integral solution x Ž . , ␾ . exists, then it is given by the
formula

¡T Ž t . ␾ Ž 0. q

xŽ t, ␾ . s

~

0

¢␾ Ž t . ,

t

H T Ž t y s . B F Ž s, x Ž . , ␾ . . ds,
␭ªqϱ 0
lim

0

␭

s

for t G 0,
for t g Ž yϱ, 0 x ,

where B␭ s ␭ RŽ ␭, A..
Assume the following compactness condition:
ŽH2. The semigroup ŽT0 Ž t .. t G 0 is compact on D Ž A . , that is, T0 Ž t . is
compact in D Ž A . whenever t ) 0.
PROPOSITION 1 w1x. Assume that ŽH1. and ŽH2. hold. Let ⍀ be a
nonempty subset of B and F be continuous on w0, ax = ⍀, a ) 0. If ␾ g ⍀
with ␾ Ž0. g D Ž A . , then Eq. Ž1. has an integral solution x Ž . . in Žyϱ, b x, for
some b g Ž0, a..

ŽH3.

F takes bounded sets of w0, qϱ. = B into bounded sets of E.


FUNCTIONAL DIFFERENTIAL EQUATIONS

261

PROPOSITION 2 w1x. Assume that the conditions ŽH1., ŽH2., and ŽH3.
hold and let ␾ g B such that ␾ Ž0. g D Ž A . . Then, Eq. Ž1. has an integral
solution x Ž . , ␾ . in a maximal inter¨ al Žyϱ, b␾ . and either b␾ s qϱ or
lim sup t ª b␾ < x Ž t, ␾ .< s qϱ.
Global existence and uniqueness is obtained under the following assumption:
ŽH4. F : w0, ϱ. = B ª E, a ) 0, is continuous in t and uniformly
Lipschitz continuous with respect to the second argument: there exists a
positive constant L such that < F Ž t, ␺ 1 . y F Ž t, ␺ 2 .< F L 5 ␺ 1 y ␺ 2 5 B for
every t G 0 and ␺ 1 , ␺ 2 g B.
THEOREM 1 w2x. Under the conditions ŽH1. and ŽH4., let ␾ g B with
␾ Ž0. g D Ž A . . Then Eq. Ž1. has a unique integral solution x Ž . , ␾ . which is
defined for t G 0. Moreo¨ er, there exist two locally bounded functions
mŽ . ., nŽ . . : ‫ޒ‬qª ‫ޒ‬q such that for ␾ 1 , ␾ 2 g B with ␾ 1Ž0., ␾ 2 Ž0. g D Ž A .
and t G 0, one has
xt Ž . , ␾1 . y xt Ž . , ␾2 .

B F mŽ t . e

nŽ t . 5

␾1 y ␾2 5 B .

3. MAIN RESULTS
Set
X [ Ä ␾ g B : ␾ Ž 0. g D Ž A . 4 ,
and
E [ Ä ␾ g X : Eq. Ž 1 . has a unique integral solution
x Ž . , ␾ . defined on ‫ ޒ‬4 .
PROPOSITION 3. Assume that ŽH1. holds. If ␾ g E such that x␻ Ž . , ␾ . s
␾ then x Ž . , ␾ . is an ␻-periodic solution of Eq. Ž1..
Proof. Let x Ž . . [ x Ž . , ␾ . and y Ž . . [ x Ž . q␻ .. Then for t G 0
y Ž t . s T0 Ž t q ␻ . ␾ Ž 0 . q lim

tq ␻

H
␭ªqϱ 0

T0 Ž t q ␻ y s . B␭ F Ž s, x s . ds,


262

BENKHALTI, BOUZAHIR, AND EZZINBI

which implies that

y Ž t . s T0 Ž t . T0 Ž ␻ . ␾ Ž 0 . q lim

␻

H
␭ªqϱ 0

q lim

tq ␻

H
␭ªqϱ ␻

T0 Ž t q ␻ y s . B␭ F Ž s, x s . ds

s T0 Ž t . T0 Ž ␻ . ␾ Ž 0 . q lim

␻

H
␭ªqϱ 0

q lim

T0 Ž ␻ y s . B␭ F Ž s, x s . ds

T0 Ž ␻ y s . B␭ F Ž s, x s . ds

t

H T Ž t y s. B F Ž s q ␻ , x

␭ªqϱ 0

␭

0

sq ␻

. ds

t

s T0 Ž t . x Ž ␻ . q lim

H T Ž t y s . B F Ž s, y . ds
␭ªqϱ 0

s T0 Ž t . y Ž 0 . q lim

␭

0

s

t

H T Ž t y s . B F Ž s, y . ds.
␭ªqϱ 0
0

␭

s

Since y 0 s x␻ s ␾ , by the uniqueness property of x Ž . , ␾ . we have y Ž t . s
x Ž t . for all t g ‫ޒ‬q, which proves our claim.
Define the Poincare
´ map P␻ : E ª X by P␻ Ž ␾ . [ x␻ Ž . , ␾ ., where
x Ž . , ␾ . is the integral solution of Eq. Ž1.. To show that Eq. Ž1. 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.
of P␻ .

Each of the following conditions implies the continuity

Ži. Conditions ŽH1. and ŽH4. are satisfied.
Žii. Conditions ŽH1., ŽH2., and ŽH3. are satisfied and for each ␺ g E ,
there exists r␺ ) 0 such that the set Ä x s Ž . , ␾ . : s g w0, ␻ x, ␾ g B Ž ␺ , r␺ .4 is
bounded, where B Ž ␺ , r␺ . [ Ä ␾ g E : 5 ␾ y ␺ 5 B F r␺ 4 .
Proof. Ži. The continuity of P␻ follows from Theorem 1.
Suppose that the conditions in Žii. are satisfied. Let ␺ g E and Ž ␺ k . k ; E
be a convergent sequence with lim k ªqϱ ␺ k s ␺ . Set x Ž . . [ x Ž . , ␺ . and
x k Ž . . [ x Ž . , ␺ k .. We will prove that any subsequence Ž␩ n . n ; Ž ␺ n . n has a
subsequence Ž␩ n k . k such that P␻ ␩ n k ª P␻ ␺ , as k ª qϱ, from which we
will conclude that P␻ ␺ k ª P␻ ␺ . In fact, we will first show that Ä x k Ž . . : k
g ‫ގ‬4 is relatively compact in C Žw0, ␻ x; E .. Using axiom ŽA.Žii.X , we can see
that Ä ␺ k Ž0. : k g ‫ގ‬4 is relatively compact in E. Suppose that 5 ␺ k y ␺ 5 B
F r␺ . It’s clear that ÄT0 Ž t . ␺ k Ž0. : k g ‫ގ‬4 is relatively compact in E. Let

263

FUNCTIONAL DIFFERENTIAL EQUATIONS

0 - t F ␻ and 0 - ␧ - t. Then
t
T Ž t y s . B F Ž s, x . ds
H
␭ªqϱ 0

lim

k
s

␭

0

ty ␧

s T0 Ž ␧ . lim

T0 Ž t y ␧ y s . B␭ F Ž s, x sk . ds

H
␭ªqϱ 0

t

T Ž t y s . B F Ž s, x . ds.
H
␭ªqϱ ty ␧

q lim

k
s

␭

0

Since T0 Ž ␧ . is compact, there exists a compact set W␧ such that

½

T0 Ž ␧ .

ž

lim

ty ␧

H
␭ªqϱ 0

T0 Ž t y ␧ y s . B␭ F Ž s, x sk . ds : k g ‫ ގ‬: W␧ .

5

/

From the boundedness of F, there exists a positive constant a such that
t
T Ž t y s . B F Ž s, x . ds
H
␭ªqϱ ty ␧

lim

k
s

␭

0

F a␧ ,

uniformly in k g ‫ގ‬.

We deduce that the set Ä x k Ž t . : k g ‫ގ‬4 is totally bounded and therefore is
relatively compact in E. To establish the equicontinuity, let 0 - t 0 - t F ␻ .
Then
x k Ž t . y x k Ž t0 .
F Ž T0 Ž t . y T0 Ž t 0 . . ␺ k Ž 0 . q

t

T Ž t y s . B F Ž s, x . ds
H
␭ªqϱ t

lim

0

␭

k
s

0

t0

q Ž T0 Ž t y t 0 . y I . lim

H

␭ªqϱ 0

T0 Ž t 0 y s . B␭ F Ž s, x sk . ds .

The semigroup ŽT0 Ž t .. t G 0 is compact for t ) 0. It follows from w23x that
lim T0 Ž t . y T0 Ž t 0 . s 0,

tªt 0

which implies that
lim
tªt 0

Ž T0 Ž t . y T0 Ž t 0 . . ␺ k Ž 0 .

s 0,

uniformly in k g ‫ގ‬.

On the other hand there exists a positive constant b such that
t
T Ž t y s . B F Ž s, x . ds
H
␭ªqϱ t

lim

0

0

␭

k
s

F b Ž t y t0 . ,

uniformly in k g ‫ގ‬.

264

BENKHALTI, BOUZAHIR, AND EZZINBI

Moreover W0 s Ä lim ␭ ªqϱ H0t 0 T0 Ž t 0 y s . B␭ F Ž s, x sk . ds : k g ‫ގ‬4 is relatively
compact and it is well known that
lim Ž T0 Ž h . y I . u s 0,

uniformly in u g W0 .

hª0

Which implies that lim tªt < x k Ž t . y x k Ž t 0 .< s 0, uniformly in k g ‫ގ‬. For
0

t)t 0

0 F t - t 0 F ␻ , one can use a similar argument to show that lim tªt < x k Ž t .
0

t-t 0

yx t 0 .< s 0, uniformly in k g ‫ގ‬. It follows that Ž x k Ž . .. k g ‫ ގ‬is equicontinuous. By the Arzela᎐Ascoli
theorem, we deduce that Ä x k Ž . . : k g ‫ގ‬4 is
´
relatively compact in C Žw0, ␻ x; E .. Therefore, if ␩ n is a subsequence of ␺ k ,
there exists a subsequence x Ž . , ␩ n k . of x Ž . , ␩ n . which converges to a
certain function uŽ . . g C Žw0, ␻ x; E . with uŽ0. s ␺ Ž0.. Set uŽ ␪ . s ␺ Ž ␪ .

for ␪ - 0. Using axiom ŽA.Žiii., we infer that for all s g w0, ␻ x, x s Ž . , ␩ n k .
ª u s as k ª ϱ. Since Ä x s Ž . , ␩ n k . : 0 F s F ␻ , k g ‫ގ‬4 is a bounded set of
B and F takes bounded sets into bounded sets, then we have
kŽ

u Ž t . s T0 Ž t . ␺ Ž 0 . q lim

t

H T Ž t y s . B F Ž s, u Ž . , ␺ . . ds,

␭ªqϱ 0

␭

0

s

t G 0.

As Ž␩ n . nG 0 is an arbitrary subsequence of Ž ␺ n . nG 0 , this shows that
x␻ Ž . , ␺ n . ª x␻ Ž . , ␺ . as n ª qϱ, and P␻ is continuous on E .
Assume that
ŽH5. there exists a nonempty closed bounded and convex subset D
in E such that P␻ Ž D . : D and the set Ä x s Ž . , ␾ . : s g w0, ␻ x, ␾ g D4 is
bounded.
THEOREM 2.
inf

0- ␴ - ␻

Assume ŽH1., ŽH2., ŽH3., and ŽH5. hold. If
M Ž ␻ y ␴ . HK Ž ␴ . sup

0FtF ␴

T0 Ž t . q M Ž ␴ . - 1,

Ž 2.

then Eq. Ž1. has at least one ␻-periodic solution.
In order to prove this theorem, we need some preliminaries. The
Kuratowski’s measure of noncompactness ␣ Ž ⍀ . of ⍀ is defined by

␣ Ž ⍀ . s inf Ä d ) 0 : ⍀ has a finite cover of diameter - d 4 .
THEOREM 3 w17x. Let Z be a Banach space, D 0 a con¨ ex closed and
bounded set of Z, and P : D 0 ª D 0 an ␣-condensing map, that is, P is
continuous and for e¨ ery bounded set D : D 0 with ␣ Ž D . ) 0,

␣ Ž P Ž D. . - ␣ Ž D. .
Then P has at least one fixed point in D 0 .


FUNCTIONAL DIFFERENTIAL EQUATIONS

265

Proof of Theorem 2. It is clear that the condition Žii. in Proposition 4 is
verified for E s D. Then the map P␻ Ž ␾ . s x␻ Ž . , ␾ . is continuous from D

ˆ ª Dˆ that
to D. Moreover, there exists $
an induced application Pˆ␻ : D
ˆ
ˆ
Ž
.
satisfies the condition P␻ ␸
ˆ s P␻ Ž ␸ . for all ␸ˆ g D and every ␸ g ␸ˆ. We
ˆ, there exists V : D
shall prove that Pˆ␻ is condensing. Since for all C : D
ˆ
Ž
.
Ž
.
such that C s V and ␣ C s ␣ V , it suffices to estimate ␣ Ž Pˆ␻ Ž Gˆ..
only for each subset G : D with ␣ Ž G . ) 0. If we let G w ␴ , ␻ x, 0 F ␴ F ␻ ,
be the set defined by
G w ␴ , ␻ x [ Ä x Ž . , ␸ . rw ␴ , ␻ x : ␸ g G 4 ,
then, using ŽH5. and axiom ŽA.Žii. we can see that the conditions of
Theorem 2.1 in w19x are satisfied. Hence

␣ Ž Pˆ␻ Ž Gˆ . . F K Ž ␻ y ␴ . ␣ Ž G w ␴ , ␻ x . q M Ž ␻ y ␴ . ␣ Ž Gˆ␴ . ,

Ž 3.

where G␴ s Ä x␴ Ž . , ␸ . : ␸ g G 4 . On the other hand, for all ␴ ) 0, G w ␴ , ␻ x
is relatively compact in C Žw ␴ , ␻ x; E .. To prove this assertion, we use the
fact that T0 Ž t . is compact for any t ) 0 and we can proceed as in the proof

of Proposition 4 in w2x. That is, we prove that the family G w ␴ , ␻ x is
equicontinuous and G w ␴ , ␻ xŽ t ., for each t g w0, ␻ x, is relatively compact
in E. Thus, Ž3. becomes

␣ Ž Pˆ␻ Ž Gˆ . . F M Ž ␻ y ␴ . ␣ Ž Gˆ␴ . .

Ž 4.

To estimate ␣ Ž Gˆ␴ ., we take w0, ␴ x instead of w ␴ , ␻ x in Ž3., and we infer
that

␣ Ž Gˆ␴ . F K Ž ␴ . ␣ Ž G w 0, ␴ x . q M Ž ␴ . ␣ Ž Gˆ . .

Ž 5.

Next we prove that

␣ Ž G w 0, ␴ x . F H sup

0FtF ␴

T0 Ž t . ␣ Ž G . .

Ž 6.

In fact, if G Ž0. s Ä ␸ Ž0. : ␸ g G 4 then axiom ŽA.Žii. gives

␣ Ž G Ž 0 . . F H␣ Ž G . .

Ž 7.

Let d ) 0 such that ␣ Ž G Ž0.. - d. By definition of ␣ Ž . ., G Ž0. has a finite
cover Ž Ci .1 F iF n : E, such that diamŽ Ci . - d. Next, for a subset C of E,
we denote by CU the set
CU s Ä T0 Ž . . u rw0, ␴ x : u g C 4 .
Then it is clear that
diam Ž CiU . - sup

0FtF ␴

T0 Ž t . d


266

BENKHALTI, BOUZAHIR, AND EZZINBI

and
n

U

G Ž 0. :

D CiU .
is1

Hence
U

␣ Ž G Ž 0 . . - sup

0FtF ␴

T0 Ž t . d.

The last inequality together with Ž7. and the choice d [ H␣ Ž G . q ␧ ,
␧ ) 0, imply that
U

␣ Ž G Ž 0 . . F H sup

0FtF ␴

T0 Ž t . ␣ Ž G . ,

since
U

G w 0, ␴ x : G Ž 0 . q Ä w Ž . , ␸ . w0, ␴ x : ␸ g G 4 ,
where w Ž t, ␸ . s lim ␭ªqϱ H0t T0 Ž t y s . B␭ F Ž s, x s Ž . , ␸ .. ds, for t g w0, ␴ x.
One can use similar arguments as in the proof of Proposition 4 to see that
Ä w Ž . , ␸ .rw0, ␴ x : ␸ g G 4 is relatively compact in C Žw0, ␴ x; E .. In fact, let
0 - t F ␴ and 0 - ␧ - t. Then
t

H T Ž t y s . B F Ž s, x Ž . , ␸ . . ds
␭ªqϱ 0
lim

␭

0

s

ty ␧

s T0 Ž ␧ . lim

H
␭ªqϱ 0
t

q lim

H

␭ªqϱ ty ␧

T0 Ž t y ␧ y s . B␭ F Ž s, x s Ž . , ␸ . . ds

T0 Ž t y s . B␭ F Ž s, x s Ž . , ␸ . . ds.

Since T0 Ž ␧ . is compact, there exists a compact set W␧ such that

½

T0 Ž ␧ .

ž

lim

ty ␧

H
␭ªqϱ 0

T0 Ž t y ␧ y s . B␭ F Ž s, x s Ž . , ␸ . . ds : ␸ g G : W␧ .

/

5

From the boundedness of F, there exists a positive constant a such that
t

H T Ž t y s . B F Ž s, x Ž . , ␸ . . ds
␭ªqϱ ty ␧
lim

0

␭

s

F a␧ ,

uniformly in ␸ g G .

Hence the set Ä w Ž . , ␸ . relatively compact in E. To establish the equicontinuity, let 0 - t 0 - t F ␴ .


FUNCTIONAL DIFFERENTIAL EQUATIONS

267

Then
w Ž t , ␸ . y w Ž t0 , ␸ .
F

t

H T Ž t y s . B F Ž s, x Ž . , ␸ . . ds
␭ªqϱ t
lim

␭

0

s

0

q Ž T0 Ž t y t 0 . y I . lim

t0

H
␭ªqϱ 0

T0 Ž t 0 y s . B␭ F Ž s, x s Ž . , ␸ . . ds .

There exists a positive constant b such that
lim

t

H T Ž t y s . B F Ž s, x Ž . , ␸ . . ds

␭ªqϱ t 0

␭

0

s

F b Ž t y t0 . ,
uniformly in ␸ g G .

Moreover, W0 s Ä lim ␭ ªqϱ H0t 0 T0 Ž t 0 y s . B␭ F Ž s, x s Ž . , ␸ .. ds : ␸ g G 4 is relatively compact and it is well known that
lim Ž T0 Ž h . y I . u s 0,

hª0

uniformly in u g W0 ,

which implies that lim tªt < w Ž t, ␸ . y w Ž t 0 , ␸ .< s 0, uniformly in ␸ g G .
0

t)t 0

Using a similar argument for 0 F t - t 0 F ␴ , one can show that
lim tªt < w Ž t, ␸ . y w Ž t 0 , ␸ .< s 0, uniformly in ␸ g G . It follows that
0

t-t 0

Ä w Ž . , ␸ . theorem,
´
we deduce that Ä w Ž . , ␸ . We complete the proof of Ž6. by the properties of ␣ Ž . ..
The inequalities Ž4., Ž5., and Ž6. show that for all 0 - ␴ F ␻ ,

␣ Ž Pˆ␻ Ž Gˆ . . F M Ž ␻ y ␴ . HK Ž ␴ . sup

0FtF ␴

T0 Ž t . q M Ž ␴ . ␣ Ž Gˆ . .

Finally, condition Ž2. implies that Pˆ␻ is ␣-condensing.
By Theorem
3, we conclude that there exists ␸
ˆ g Dˆ such that Pˆ␻ ␸ˆ s ␸ˆ.

$
n .
nŽ .
ˆ
Ž
Then P␻ ␸ s P␻ ␸
ˆ s ␸ˆ for all n g ‫ގ‬. Since
$

$

5 P␻m␸ y P␻n␸ 5 B s 5P␻m␸ y P␻n␸5 B ,

for m, n g ‫ގ‬.

Ž P␻n␸ . n is a Cauchy sequence of B, which implies that there exists ␺ g D
such that P␻k␸ ª ␺ , as k ª qϱ. Consequently P␻ ␺ s ␺ . This completes
the proof by use of Proposition 3.


268

BENKHALTI, BOUZAHIR, AND EZZINBI

Remark 1. Assumption ŽH5. seems to be strong, but in the nonhomogeneous linear case, we will prove that the existence of a bounded solution
implies that ŽH5. is true.
In the sequel, we suppose that ŽH4. is satisfied.
Define UŽ t . on C for t G 0 by
UŽ t . ␾ s xt Ž . , ␾ . ,
where x Ž . , ␾ . is the integral solution of Eq. Ž1..

The following result on the solution operator UŽ t ., t G 0, is a key
property to the existence of periodic solutions.
Assume that ŽH1., ŽH2., and ŽH4. hold. Then UŽ t . is

PROPOSITION 5.
decomposed as

U Ž t . s U1 Ž t . q U2 Ž t . ,

t G 0,

where ŽU1Ž t .. t G 0 is the solution semigroup of
x Ž t . s T0 Ž t . ␾ Ž 0 . ,
x0 s ␾ g X ,

½

tG0

Ž 8.

and U2 Ž t . is compact for t ) 0.
Proof. Let Ž ␺n . nG 0 be a bounded sequence in X . Then for t ) 0 one
has

Ž U2 Ž t . ␺n . Ž ␪ . s

½

Ž U2 Ž t q ␪ . ␺n . Ž 0.

if ␪ g w yt , ␪ x

0

if ␪ g Ž yϱ, yt . .

We will first show that for all ␪ g Žyϱ, 0x, ÄŽU2 Ž t . ␺n .Ž ␪ . : n G 04 is relatively compact in E. Let 0 - t q ␪ , choose ␧ such that ␧ - t q ␪ . Then

Ž U2 Ž t . ␺n . Ž ␪ .
tq ␪

s lim

H
␭ªqϱ 0

T0 Ž t q ␪ y s . B␭ F Ž s, x s Ž . , ␺n . . ds
tq ␪ y ␧

s T0 Ž ␧ . lim

H
␭ªqϱ 0

T0 Ž t q ␪ y ␧ y s . B␭ F Ž s, x s Ž . , ␺n . . ds

tq ␪

H T Ž t q ␪ y s . B F Ž s, x Ž . , ␺ . . ds.

␭ªqϱ tq ␪ y ␧

q lim

0

␭

s

n

By Theorem 1, Ä F Ž s, x s Ž . , ␺n .. : s g w0, t x, n G 04 is bounded. Since T0 Ž ␧ .
is compact, there exists a compact set W␧ such that
T0 Ž ␧ .

½

lim

tq ␪ y ␧

H
␭ªqϱ 0

: W␧ .

T0 Ž t q ␪ y ␧ y s . B␭ F Ž s, x s Ž . , ␺n . . ds : n G 0

5

269

FUNCTIONAL DIFFERENTIAL EQUATIONS

Furthermore, there exists a positive constant c such that
lim

tq ␪

H

␭ªqϱ tq ␪ y ␧

T0 Ž t q ␪ y s . B␭ F Ž s, x s Ž . , ␺n . . ds F c ␧ ,
uniformly in n g ‫ގ‬.

Thus ÄŽU2 Ž t . ␺n .Ž ␪ . : n G 04 is totally bounded and therefore relatively
compact. To establish the second assertion, it is sufficient to show that
ŽU2 Ž t . ␺n . nG 0 is equicontinuous in Žyϱ, 0x. Let ␪ 0 g Žyϱ, 0x. For ␪ g
Žyϱ, 0x close enough to ␪ 0 such that ␪ 0 - ␪ , we see that

Ž U2 Ž t . ␺n . Ž ␪ . y Ž U2 Ž t . ␺n . Ž ␪ 0 .

¡ŽU Ž t q ␪ . ␺ . Ž 0. y ŽU Ž t q ␪ . ␺ . Ž 0.
s~ Ž U Ž t q ␪ . ␺ . Ž 0.
¢0
2

n

2

n

2

0

n

if ␪ 0 ) yt
if ␪ 0 s yt
if ␪ 0 - yt.

For yt - ␪ 0 - ␪ F 0, one has

Ž U2 Ž t . ␺n . Ž ␪ . y Ž U2 Ž t . ␺n . Ž ␪ 0 .
s lim

tq ␪ 0

H
␭ªqϱ 0

Ž T0 Ž t q ␪ y s . y T0 Ž t q ␪ 0 y s . .

= B␭ F Ž s, x s Ž . , ␺n . . ds
q lim

tq ␪

H

␭ªqϱ tq ␪ 0

T0 Ž t q ␪ y s . B␭ F Ž s, x s Ž . , ␺n . . ds.

By the boundedness of Ä F Ž s, x s Ž . , ␺n . : n G 0, s g w t q ␪ 0 , t x4 , we deduce
that there exists a positive constant d such that

Ž U2 Ž t . ␺n . Ž ␪ . y Ž U2 Ž t . ␺n . Ž ␪ 0 .
F Ž T0 Ž ␪ y ␪ 0 . y I .
= lim

tq ␪ 0

H
␭ªqϱ 0

qd Ž ␪ y ␪ 0 . ,

T0 Ž t q ␪ 0 y s . B␭ F Ž s, x s Ž . , ␺n . . ds
for all n G 0.

On the other hand W1 s Ä lim ␭ ªqϱ H0tq ␪ 0 T0 Ž t q ␪ 0 y s . B␭ F Ž s, x s Ž . ,
␺n .. ds : n G 04 is relatively compact in E and using the fact that
ŽT0 Ž . . x . x g W is equicontinuous at the right in 0, we get
1

lim

␪ª ␪ 0
␪) ␪ 0

Ž U2 Ž t . ␺n . Ž ␪ . y Ž U2 Ž t . ␺n . Ž ␪ 0 .

s 0,

uniformly in n g ‫ގ‬.


270

BENKHALTI, BOUZAHIR, AND EZZINBI

A similar argument shows that
lim

␪ª ␪ 0
␪- ␪ 0

Ž U2 Ž t . ␺n . Ž ␪ . y Ž U2 Ž t . ␺n . Ž ␪ 0 .

s 0,

uniformly in n g ‫ގ‬,

and we deduce the claimed equicontinuity. By the Arzela᎐Ascoli
theorem,

´
there is a continuous function ␾ : Žyϱ, 0x ª E and a subsequence ␾n of
ŽU2 Ž t . ␺n . nG 0 which converges compactly to ␾ in Žyϱ, 0x. Since ␾nŽ ␪ . s 0
for ␪ F yt and n G 0, then ␾ is continuous and ␾ Ž ␪ . s 0 if ␪ F yt.
Hence by axiom ŽA.Ži., ␾ belongs to B. Moreover, by axiom ŽA.Žiii.,
5 ␾n y ␾ 5 B F K Ž t . supyt F ␪ F 0 < ␾nŽ ␪ . y ␾ Ž ␪ .< and we have 5 ␾n y ␾ 5 B ª
0 as n ª ϱ. So the image of any bounded sequence contains a converging
subsequence in B with respect to the seminorm.
ŽH6. There exists a nonempty closed bounded and convex subset D
in X such that P␻ Ž D . : D.
THEOREM 4. Assume that ŽH1., ŽH2., ŽH4., and ŽH6. hold. If there
exists a positi¨ e constant ␩ such that
U1 Ž t . F ey␩ t ,

t G 0.

Ž 9.

Then Eq. Ž1. has at least one ␻-periodic solution.
Proof. We will use Sadovskii’s fixed point theorem as in the proof of
Theorem 2. Since U2 Ž ␻ . is compact, then for each subset G : D with
␣ Ž G . ) 0, one has
$

$

␣ Ž P␻ Ž Gˆ . . s ␣ Ž UˆŽ ␻ . Gˆ . s ␣ Ž U1 Ž ␻ . Gˆ .
F ey␩ ␻␣ Ž Gˆ .
- ␣ Ž Gˆ . .
$

ˆ and Eq. Ž1. has an ␻-periodic solution.
Thus P␻ is ␣-condensing on D
In relation with the above condition, we have obtained the following
result:
PROPOSITION 6 w2, 12x. Let ␥ ) 0 and C␥ be the space of all continuous
functions ␾ : Žyϱ, 0x ª E such that lim ␪ ªyϱ e␥␪␾ Ž ␪ . exists in E. Then C␥
together with the norm 5 ␾ 5 ␥ [ sup␪ F 0 e␥␪ < ␾ Ž ␪ .<, ␾ g C␥ , satisfies axioms
ŽA., ŽA1., and ŽB.. Moreo¨ er if 5 T0 Ž t .5 F ey␦ t , t G 0, where ␦ is a positi¨ e
constant, then condition Ž9. is true.
Now we will give some sufficient conditions ensuring the assumption
ŽH5.. Consider the phase space B s Cr = L1 Ž g . to be the space of


271

FUNCTIONAL DIFFERENTIAL EQUATIONS

functions ␸ : Žyϱ, 0x ª E such that ␸ is continuous on wyr, 0x, for some
r ) 0, Lebesgue measurable, and g Ž . .< ␸ Ž . .< is Lebesgue integrable on
Žyϱ, yr ., where g : Žyϱ, yr . ª ‫ ޒ‬is a positive Lebesgue measurable
function. A seminorm in Cr = L1 Ž g . is defined by
5␸5 B [

sup
yrF ␪ F0

␸Ž␪ . q

yr

Hyϱ g Ž ␪ .

␸ Ž ␪ . d␪ .

We suppose that g satisfies:
Ži. g is integrable on Žyd, yr . for any d G r, and
Žii. there exists a locally bounded function G : Žyϱ, 0x ª w0, qϱ.
such that
g Ž ␰ q ␪ . F GŽ ␰ . g Ž ␪ . ,

for all ␰ F 0 and ␪ g Ž yϱ, yr . _ N␰ ,

where N␰ : Žyϱ, yr . is a set with Lebesgue measure 0.
Thus, Theorem 1.3.8 in w12x asserts that Cr = L1 Ž g . is a phase space
which verifies axioms ŽA., ŽA1., and ŽB..
Define ␳ Ž t ., t G 0, by

␳ Ž t . [ sup

␪ Fyr

gŽ␪ y t.
gŽ␪ .

q ␳0 Ž t . q

yt
g Ž ␪ . d␪ ,
Hytyr

where t s maxÄ r, t 4 and ␳ 0 [ 1Žyϱ, r x.
In addition, we suppose that ␳ Ž t . ª 0, as t ª qϱ.
PROPOSITION 7. Assume that F : w0, qϱ. = Ž Cr = L1 Ž g .. ª E is a continuous and locally Lipschitz with respect to the second ¨ ariable function and
there exist N1 , N2 ) 0 such that
F Ž t , ␸ . F N1 5 ␸ 5 B q N2 ,

for ␸ g Cr = L1 Ž g . .

Ž 10 .

If there exist ␮ 0 ) 0 and k 0 ) 0 such that the following conditions hold:
Ži.
Žii.
Žiii.

5 T0 Ž t .5 F expŽy␮ 0 t ., t G 0,
H0tyr g Ž s y t . expŽ ␮ 0 Ž t y r y s .. ds F k 0 for t G r,
N1 expŽ ␮ 0 r .Ž1 q k 0 . - ␮ 0 .

Then there exist R ) 0 and ␴ ) 0 such that P␴ Ž B Ž0, R .. : B Ž0, R . and the
set Ä x t Ž . , ␾ . : ␾ g B Ž0, R ., 0 F t F ␴ 4 is bounded in Cr = L1 Ž g ..
The proof of the proposition can be done in a similar way as in w9x.


272

BENKHALTI, BOUZAHIR, AND EZZINBI

4. MASSERA TYPE CRITERION

Consider the following nonhomogeneous linear equation

¡d
¢x s ␾ g B ,

~ dt x Ž t . s Ax Ž t . q L Ž t , x . q f Ž t . ,
t

t G 0,

Ž 11 .

0

where L is a continuous function form ‫ޒ‬q= B into E, linear with the
second argument and ␻-periodic in t, and f is a continuous ␻-periodic
function.
For ␸ g B, t G 0, and ␪ F 0 we define
SŽ t . ␸ Ž ␪ . s

½

␸ Ž 0.
␸Žt q ␪ .

if t q ␪ G 0
if t q ␪ - 0.

We can see that Ž SŽ t .. t G 0 is a C0-semigroup on B and we set
S0 Ž t . s S Ž t . rB0 ,

where B0 [ Ä ␸ g B : ␸ Ž 0 . s 0 4 .

Let C00 be the set of continuous functions ␸ : Žyϱ, 0x ª E with compact
support and recall that a sequence of functions Ž ␸ n . ng N : Žyϱ, 0x ª E, is
uniformly bounded if sup ng N Žsupyϱ- ␪ F 0 < ␸ n Ž ␪ .<. - qϱ.
We suppose that B is a fading memory space in the sense that B
satisfies the following two axioms.
ŽC. If a uniformly bounded sequence Ž ␸ n . n in C00 converges to a
function ␸ compactly on Žyϱ, 0x, then ␸ is in B and 5 ␸ n y ␸ 5 B ª 0 as
n ª qϱ.
ŽD1. 5 S0 Ž t . ␸ 5 B ª 0 as t ª qϱ for all ␸ g B0 .
It is well known by the Banach Steinhaus theorem that sup t G 0 5 S0 Ž t .5 B
- ϱ.
Now assume that
ŽH7. Equation Ž11. has a bounded solution y on ‫ޒ‬q in the sense
that sup t g ‫ ޒ‬q < y Ž t .< - qϱ.
Let BC be the space of bounded continuous functions mapping Žyϱ, 0x
into E with the uniform norm topology. Then one has
PROPOSITION 8 w12x. Assume that B is a fading memory space. Then
BC ; B and there exists a positi¨ e constant J such that 5 ␸ 5 B F J 5 ␸ 5 B C .


FUNCTIONAL DIFFERENTIAL EQUATIONS

273

Moreo¨ er
5 x t 5 B F J sup

␴FsFt

x Ž s . q Ž 1 q JH . S0 Ž t y ␴ . 5 x␴ 5 B ,

for any function x arising in Axiom ŽA..
So, if B is a fading memory space, then the functions K Ž . . and M Ž . . in
Axiom ŽA. can be chosen bounded on w0, qϱ.. Using Žiii. in Axiom ŽA. we
obtain the existence of a positive constant N1 such that
sup 5 yt 5 B F N1 .

Ž 12 .

tG0

Assume that ŽH1., ŽH2., ŽH7., ŽC., and ŽD1. are satis-

THEOREM 5.
fied. If
inf

0- ␴ - ␻

M Ž ␻ y ␴ . HK Ž ␴ . sup

0FtF ␴

T0 Ž t . q M Ž ␴ . - 1,

Ž 13 .

then Eq. Ž11. has an ␻-periodic solution.
Proof. Set D [ coÄ yn ␻ : n g ‫ގ‬4 where co denotes the closure of the
convex hull. Then D is a nonempty closed convex subset of X . D is
bounded and x␻ Ž . , ␺ . g D for all ␺ g D. In fact let ␺ g D. Then there
exists a sequence Ž ␺ k . k ; Ä yn ␻ : n g ‫ގ‬4 such that
nk

␺ks

nk

Ý ␣ ik yn ␻ ,
k
i

␣ ik G 0,

is1

Ý ␣ ik s 1, and 5 ␺ y ␺ k 5 B ª 0 as k ª ϱ.
is1

Therefore
5␺ 5 B F 5␺ k 5 B q 5␺ k y ␺ 5 B
nk

F

Ý ␣ ik 5 yn ␻ 5 B q 5 ␺ k y ␺ 5 B ,
k

i

is1

and Ž12. yields 5 ␺ 5 B - N1. On the other hand, one has
nk

x␻ Ž . , ␺ k . s

Ý ␣ ik x␻ Ž . , yn ␻ .
k
i

is1
nk

s

Ý ␣ ik yn ␻q ␻ ,
k
i

is1

so x␻ Ž . , ␺ k . g D. Since 5 x␻ Ž . , ␺ . y x␻ Ž . , ␺ k .5 B ª 0 as k ª qϱ, and
D is closed we deduce that x␻ Ž . , ␺ . g D for all ␺ g D. Hence assumption


274

BENKHALTI, BOUZAHIR, AND EZZINBI

ŽH6. is true and by Theorem 3, we conclude that Eq. Ž11. has an
␻-periodic solution.
COROLLARY 1. Assume that B s C␥ , ␥ ) 0. Under the conditions ŽH1.,
ŽH2., and ŽH7., Eq. Ž11. has a m ␻-periodic solution, for a certain m g ‫ގ‬U .
Proof. It is known from w12x that if ␥ ) 0, the phase space C␥ satisfies
ŽC. and ŽD1.. Moreover K Ž ␴ . s 1 and M Ž ␴ . s ey␥ ␴ . Hence
inf

0- ␴ - ␻

s

½

M Ž ␻ y ␴ . HK Ž ␴ . sup

0FtF ␴

inf

0- ␴ - ␻

½ H sup

0FtF ␴

T0 Ž t . q M Ž ␴ .

5

T0 Ž t . ey␥ Ž ␻ y ␴ . q ey ␥␻ .

5

Furthermore, for a certain m g ‫ގ‬U , Ž13. in Theorem 5 is satisfied with
m ␻ instead of ␻ . Consequently Theorem 5 implies that Eq. Ž11. has a
m ␻-periodic solution.
Next we suppose that B is a normed space and a uniform fading
memory space; that means that B satisfies the extra axioms ŽC. and
ŽD2.

5 S0 Ž t .5 ª 0 as t ª qϱ.

We will show the existence of a periodic solution without condition Ž13..
Before doing so let us recall the following results.
Let H : Z ª Z be a closed linear operator with a dense domain DŽ H . in
a Banach space Z. We denote by ␴ Ž H . the spectrum of H .
We define the essential spectrum of H , denoted by ␴e s s Ž H ., as the set of
all ␭ in ␴ Ž H . for which at least one of the following holds:
Ži.
Žii.
is infinite
Žiii.

ImŽ ␭ I y H . s ÄŽ ␭ I y H . z : z g Z 4 is not closed.
the generalized eigenspace M␭Ž H . s Dk G 1 kerŽ ␭ I y H . k of ␭
dimensional,
␭ g ␴ Ž H . _ ␭.

For a bounded linear operator H , the Kuratowski measure of noncompactness ␣ Ž H . of H is defined by

␣ Ž H . s inf Ä k g ‫ޒ‬q : ␣ Ž H Ž B . . F k ␣ Ž B .
for every bounded subset B of Z 4 .
The essential radius of ␴e s s Ž H . is given by
r e s s Ž H . s sup Ä < ␭ < : ␭ g ␴e s s Ž H . 4 .
In w16x, Nussbaum has proved the formula
r e s s Ž H . s lim ␣ Ž H n .
nªϱ

1rn

.

Ž 14 .


275

FUNCTIONAL DIFFERENTIAL EQUATIONS

Let V [ Ž V Ž t .. t G 0 be a C0 semigroup on a Banach space Z, the essential
growth bound of V is defined by

␻ e s s Ž V . [ lim

tªϱ

1

t

log ␣ Ž V Ž t . . s inf

t)0

1
t

log ␣ Ž V Ž t . . ,

and according to w24x, we have
r e s s Ž V Ž t . . s exp Ž t ␻ e s s Ž V . . , t ) 0.

Ž 15 .

THEOREM 6. Suppose that B is a uniform fading memory space. Under
conditions ŽH1., ŽH2., and ŽH7., Eq. Ž11. has an ␻-periodic solution.
The proof is based on the following two results.
LEMMA 2. Let conditions ŽH1. and ŽH2. hold. Then, for any ␧ ) 0,
there exists a constant C␧ ) 0 such that ␣ ŽU1Ž t .. F C␧ M Ž t y ␧ . for t ) ␧ .
Proof. Let ⍀ be a bounded set in X such that ␣ Ž ⍀ . s ␤ . For ␦ ) 0,
there exists a finite cover Ž ⍀ i .1 F i F n of ⍀ such that diamŽ ⍀ i . - ␤ q ␦ .
Using Axiom ŽA.Žiii., we have for t ) ␧ and ␸ , ␾ g ⍀,
U1 Ž t . ␸ y U1 Ž t . ␾

BFKŽty␧.

sup T0 Ž s . ␸ Ž 0 . y T0 Ž s . ␾ Ž 0 .

␧FsFt

q M Ž t y ␧ . U1 Ž ␧ . ␸ y U1 Ž ␧ . ␾

B.

Ž 16 .

On the other hand,
U1 Ž ␧ . ␸ y U1 Ž ␧ . ␾

BFKŽ␧.

sup
0FsF ␧

T0 Ž s . ␸ Ž 0 . y T0 Ž s . ␾ Ž 0 .

q MŽ ␧ . 5␸ y ␾5 B,
which implies that
U1 Ž ␧ . ␸ y U1 Ž ␧ . ␾

5
5
B F C␧ ␸ y ␾ B ,

Ž 17 .

where C␧ s K Ž ␧ . H sup 0 F s F ␧ 5 T0 Ž s .5 q M Ž ␧ .. Consequently we have
U1 Ž t . ␸ y U1 Ž t . ␾

BFKŽty␧.

sup T0 Ž s . ␸ Ž 0 . y T0 Ž s . ␾ Ž 0 .

␧FsFt

q M Ž t y ␧ . C␧ 5 ␸ y ␾ 5 B .

Ž 18 .

Now, ⍀ is bounded in B and it follows from Axiom ŽA.Žii.X that Ä ␺ Ž0. : ␺
g ⍀ 4 is bounded in E. Using the compactness of the semigroup ŽT0 Ž t .. t G 0 ,
we can prove that ÄT0 Ž . . ␺ Ž0. : ␺ g ⍀ 4 is relatively compact in C Žw ␧ , t x, E ..
Then there exists a finite cover Ž ⌫i .1 F iF m of ÄT0 Ž . . ␺ Ž0. : ␺ g ⍀ 4 in
C Žw ␧ , t x, E . such that diamŽ ⌫i . - ␦ . Let ⍀ i, j s Ä ␺ g ⍀ i : T0 Ž . . ␺ Ž0. g ⌫j4 .


276

BENKHALTI, BOUZAHIR, AND EZZINBI

Then ⍀ ; Di, j ⍀ i, j . For ␸ , ␾ g ⍀ i, j by Ž18. we have
U1 Ž t . ␸ y U1 Ž t . ␾

B F K Ž t y ␧ . ␦ q M Ž t y ␧ . C␧ Ž

␤ q ␦ . 5␸ y ␾5 B,

and letting ␦ go to zero we obtain

␣ Ž U1 Ž t . . F C␧ M Ž t y ␧ . ,

for t ) ␧ .

This completes the proof of the lemma.
THEOREM 7 w6x. 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 ImŽ I y L. is closed and there exists an x 0 g Z such that Ž P n x 0 . nG 0
is bounded, then P has at least a fixed point.
Proof of Theorem 6. Let x Ž . , ␸ , f . be the solution of Eq. Ž11.. Then
x Ž . , ␸ , f . s x Ž . , ␸ , 0. q x Ž . , 0, f ., where x Ž . , ␸ , 0. and x Ž . , 0, f . are respectively the solutions of Eq. Ž11. with f s 0 and ␸ s 0. It follows that
the Poincare
´ map P␻ is given by P␻ ␸ s L ␸ q ␺ , with L ␸ s x␻ Ž . , ␸ , 0.
and ␺ s x␻ Ž . , 0, f .. According to Proposition 5, L is decomposed as
follows L s U1Ž ␻ . q U2 Ž ␻ ., where ŽU1Ž t .. t G 0 is the solution semigroup of
Eq. Ž8. and U2 Ž ␻ . is compact. By Proposition 8, K and M can be chosen
such that K is bounded and M Ž t . ª 0 as t ª qϱ. Let ␧ ) 0 and t 0 ) 0
such that C␧ M Ž t 0 y ␧ . - 1. Then, by Lemma 6, we have

␻ e s s Ž U1 . F

log Ž ␣ Ž U1 Ž t 0 . . .
t0

- 0.

Hence Ž15. gives
r e s s Ž U1 Ž ␻ . . s e ␻␻ e s sŽU1 . - 1.
On the other hand, we have

Ln s U1 Ž n ␻ . q Vn ,

for all n ) 1,

where Vn is a compact operator. Using formula Ž14. we get
r e s s Ž L . s r e s s Ž U1 Ž ␻ . . - 1.
Consequently ImŽ I y L. is closed. Let y Ž . , ␾ , f . be the bounded solution.
Then Ž P␻n␾ . nG 0 s Ž yn ␻ Ž . , ␾ , f .. nG 0 is bounded. Hence, by Theorem 7,
the map P␻ has a fixed point in B, which gives an ␻-periodic solution of
Eq. Ž11..


FUNCTIONAL DIFFERENTIAL EQUATIONS

277

5. APPLICATION
In order to apply the abstract result of the previous section, we consider
the following partial functional differential equation with infinite delay

¡Ѩ
Ѩt

wŽ t, ␰ . s

Ѩ2
Ѩ␰ 2

~

wŽ t, ␰ .

q bŽ t .

0

Hyϱ G Ž ␪ . w Ž t q ␪ , ␰ . d␪ q g Ž t , ␰ . ,

w Ž t , 0 . s w Ž t , ␲ . s 0,

¢w Ž ␪ , ␰ . s w Ž ␪ , ␰ . ,
0

Ž 19 .

t G 0, 0 F ␰ F ␲ ,
t G 0,
y ϱ - ␪ F 0, 0 F ␰ F ␲ ,

where b : ‫ޒ‬qª ‫ ޒ‬is a continuous ␻-periodic function, g : ‫ޒ‬q=w0, ␲ x ª ‫ޒ‬
is a continuous function, ␻-periodic with respect to time, G is a positive
function on Žyϱ, 0x, and w 0 : Žyϱ, 0x = w0, ␲ x ª ‫ ޒ‬is a continuous function.
Note that Eq. Ž19. can be written as Eq. Ž11.. To do this choose
E s C Žw0, ␲ x : ‫ޒ‬. endowed with the uniform norm topology and consider
the operator A : DŽ A. ; E ª E defined by

½

D Ž A . s Ä y g C 2 Ž w 0, ␲ x : ‫ ޒ‬. : y Ž 0 . s y Ž ␲ . s 0 4 ,
Ay s yY .

From w7x, we know that

Ž 0, qϱ. ; ␳ Ž A .

and

Ž ␭ I y A.

y1

F

1

␭

, for ␭ ) 0.

This implies that assumption ŽH1. is satisfied. Moreover one has
D Ž A . s Ä y g E : y Ž 0. s y Ž ␲ . s 04 .
It is known from w12x that the space C␥ , ␥ ) 0, introduced in Proposition 6,
is a normed uniform fading memory space.
We suppose that:
Ži. G Ž . . ey␥ is integrable on Žyϱ, 0x,
Žii. w 0 g C ŽŽyϱ, 0x = w0, ␲ x : ‫ޒ‬., with lim ␪ ªyϱŽ e␥␪ sup 0 F ␰ F ␲ < w 0 Ž ␪ ,
␰ .<. exists, and w 0 Ž0, 0. s w 0 Ž0, ␲ . s 0.


278

BENKHALTI, BOUZAHIR, AND EZZINBI

Let

¡LŽ t , ␾ . Ž ␰ . s b Ž t . H

0

~

yϱ

G Ž ␪ . ␾ Ž ␪ . Ž ␰ . d␪ ,

t G 0, ␰ g w 0, ␲ x ,

␾ g C␥ ,

¢f Ž t . Ž ␰ . s g Ž t , ␰ . ,

t G 0, ␰ g w 0, ␲ x .

Then L is continuous form ‫ޒ‬q= C␥ into E, linear with the second
argument and ␻-periodic in t and f : ‫ޒ‬qª E is continuous ␻-periodic. If
we put

½

t G 0, ␰ g w 0, ␲ x ,

␪ F 0, ␰ g w 0, ␲ x ,

xŽ t. Ž ␰ . s wŽ t, ␰ . ,
␸ Ž ␪ . Ž ␰ . s w0 Ž ␪ , ␰ . ,

then Eq. Ž19. is written in E as

¡d
¢x s ␸ .

~ dt x Ž t . s Ax Ž t . q L Ž t , x . q f Ž t . ,
t

t G 0,

Ž 20 .

0

We can see that assumption Žii. implies that ␸ g C␥ , and ␸ Ž0. g D Ž A . .
Let A 0 be the part of the operator A in D Ž A . given by

½

D Ž A 0 . s Ä y g C 2 Ž w 0, ␲ x : ‫ ޒ‬. : y Ž 0 . s yY Ž 0 . s y Ž ␲ . s yY Ž ␲ . s 0 4 ,
A 0 y s yY .

Then A 0 generates a strongly continuous compact ŽT0 Ž t .. t G 0 in D Ž A . .
Moreover 5 T0 Ž t .5 F eyt , t G 0.
Assume further that

0
Žiii. there exists d g Ž0, 1. such that 0 - lHyϱ
G Ž ␪ . d ␪ - 1 y d,
where l s sup 0 F t F ␻ < bŽ t .<.

Let ␳ s 1 q

m
d

, where m s sup 0 F t F ␻ < f Ž t .<. Then we have

LEMMA 3. Under the abo¨ e assumptions, let ␸ g C␥ such that < ␸ Ž ␪ .< - ␳
for all ␪ g Žyϱ, 0x. Then the integral solution of Eq. Ž20. is bounded by ␳
on ‫ޒ‬q.
Proof. Let x Ž . , ␸ . be the integral solution of Eq. Ž20.. Then < x Ž t, ␸ .< ␳ for t G 0. We proceed by contradiction. Suppose that there exists t 0 ) 0
such that < x Ž t 0 , ␸ .< ) ␳ . Let
t 1 s inf Ä t ) 0 : x Ž t , ␸ . ) ␳ 4 .


279

FUNCTIONAL DIFFERENTIAL EQUATIONS

By continuity, it follows that
x Ž t1 , ␸ . s ␳ ,

Ž 21 .

and there exists ␦ ) 0 such that < x Ž t, ␸ .< ) ␳ for t g Ž t 0 , t 0 q ␦ .. Since

x Ž t , ␸ . s T0 Ž t . ␸ Ž 0 . q lim

t

H T Ž t y s . B Ž L Ž s, x Ž . , ␸ . . q f Ž s . . s,
␭ªqϱ 0
␭

0

s

for t G 0,
we have
x Ž t 1 , ␸ . F eyt 1␳ q

t1 yŽ t ys.
1

H0

L Ž s, x s Ž . , ␸ . . ds q

e

t1 yŽ t ys.
1

H0

e

m ds.

Since yϱ - s q ␪ F t 1 q ␪ F t 1 for all ␪ g Žyϱ, 0x and s g w0, t 1 x, we
have
L Ž s, x s Ž . , ␸ . . s b Ž s .
F ␳l

0

Hyϱ G Ž ␪ .

x Ž s q ␪ , ␸ . d␪

0

Hyϱ G Ž ␪ . d␪ .

Hence, by assumption Žiii.
x Ž t 1 , ␸ . F eyt 1␳ q Ž ␳ y ␳ d q m . Ž 1 y eyt 1 . .
Therefore
x Ž t 1 , ␸ . F ␳ y d Ž 1 y eyt 1 .
- 1,
which contradicts Ž21. by definition of t 1. This completes the proof of the
lemma.
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.


280

BENKHALTI, BOUZAHIR, AND EZZINBI

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