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Pseudo almost periodic solutions ofinfinite class for some functionaldifferential equationsKhalil Ezzinbi a & Issa Zabsonre ba Département de Mathématiques, Faculté des Sciences Semlalia ,Université Cadi Ayyad , Marrakech, B.P. 2390 , Marocb Département de Mathématiques, Unité de Recherche et deFormation en Sciences Exactes et Appliquées , Université deOuagadougou , Ouagadougou 03, B.P. 7021 , Burkina FasoPublished online: 21 Jun 2012.

To cite this article: Khalil Ezzinbi & Issa Zabsonre (2013) Pseudo almost periodic solutions ofinfinite class for some functional differential equations, Applicable Analysis: An InternationalJournal, 92:8, 1627-1642, DOI: 10.1080/00036811.2012.698003

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Pseudo almost periodic solutions of infinite class for some

functional differential equations

Khalil Ezzinbia and Issa Zabsonreb*

aDepartement de Mathematiques, Faculte des Sciences Semlalia, Universite Cadi Ayyad,Marrakech, B.P. 2390, Maroc; bDepartement de Mathematiques, Unite de Recherche et

de Formation en Sciences Exactes et Appliquees, Universite de Ouagadougou,Ouagadougou 03, B.P. 7021, Burkina Faso

Communicated by G. N’Guerekata

(Received 30 August 2011; final version received 15 May 2012)

The aim of this work is to investigate the existence and uniqueness ofpseudo almost periodic solutions for some neutral partial functionaldifferential equations in a Banach space when the delay is distributed usingthe variation of constants formula and the spectral decomposition of thephase space developed in Adimy et al. [M. Adimy, K. Ezzinbi, andA. Ouhinou, Variation of constants formula and almost periodic solutions forsome partial functional differential equations with infinite delay, J. Math.Anal. Appl. 317(2) (2006), pp. 668–689]. Here, we assume that theundelayed part is not necessarily densely defined and satisfies the well-known Hille–Yosida condition, the delayed part is assumed to be pseudoalmost periodic with respect to the first argument and Lipschitz continuouswith respect to the second argument.

Keywords: pseudo almost periodic functions; spectral decomposition;variation of constants formula

AMS Subject Classification: 34K30

1. Introduction

The study of the existence of almost periodic, asymptotically almost periodic, almostautomorphic, asymptotically almost automorphic and pseudo almost periodicsolutions is one of the most attractive topics in the qualitative theory of differentialequations due to both its mathematical interest and the applications in physics,mathematical biology and control theory, among other areas. Most of theseproblems need to be studied in abstract spaces and the operators are defined overnon-dense domains. In this context the literature is very scarce (see [1–3] and thereferences therein).

In this work, we study the existence and uniqueness of pseudo almost periodicsolutions of infinite class with infinite delay for the following neutral partial

*Corresponding author. Email: zabsonreissa@yahoo.fr

� 2012 Taylor & Francis

http://dx.doi.org/10.1080/00036811.2012.698003

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functional differential equation

d

dtut ¼ Aut þ LðutÞ þ f ðtÞ for t2R, ð1:1Þ

where A is a linear operator on a Banach space X satisfying the Hille–Yosidacondition, that is, there exist M0� 1 and �2R such that ]�,þ1[� �(A) and

jRð�,AÞnj �M0

�� �for n2N and �4�,

where �(A) is the resolvent set of A and R(�,A)¼ (�I�A)�1 for �2 �(A). In sequel,without loss of generality, we suppose that M0¼ 1. The phase space B is a linearspace of functions mapping ]�1, 0] into X satisfying axioms which will be describedin the sequel, for every t� 0, the history ut2B is defined by

utð�Þ ¼ uðtþ �Þ for �2 � �1, 0�,

f: B!X is a continuous function. In the literature devoted to equations with finitedelay, the state space is the space of all continuous functions on [�r, 0], r� 0,endowed with the uniform norm topology.

When the delay is finite, some recent contributions concerning pseudo almostperiodic solutions for abstract differential equations similar to Equation (1.1) havebeen made. For example, in [2] the authors have shown that if the inhomogeneousterm f depends only on variable t and it is a pseudo almost periodic function, thenthe problem Equation (1.1) has a unique pseudo almost periodic solution. In [4], theauthors have proven that if f: R�X0!X is a suitable continuous function, whereX0 ¼ DðAÞ, the problem

x0ðtÞ ¼ AxðtÞ þ f ðt, xðtÞÞ, t2R

has a unique pseudo almost periodic solution, while in [1] the authors have treatedthe existence of almost periodic solutions for a class of partial neutral functionaldifferential equations defined by a linear operator of the Hille–Yosida type with non-dense domain. In [3], the authors studied the existence and uniqueness of pseudoalmost periodic solutions for a first-order abstract functional differential equationwith a linear part dominated by a Hille–Yosida type operator with a non-densedomain. In [5], the authors studied the following partial functional differentialequation when the delay r is fixed

d

dt½uðtÞ � f ðt, uðt� rÞ� ¼ A½uðtÞ þ f ðt, uðt� rÞÞ� þ gðt, uðt� rÞÞ for t � 0, ð1:2Þ

where A is a linear operator on X satisfying the Hille–Yosida condition. They provedthe existence and uniqueness of pseudo almost automorphic solutions forEquation (1.3). In [6], the authors studied the existence of Cn-almost periodicsolutions and Cn-almost automorphic solutions (n� 1) of Equation (1.1). Theyproved that the existence of a bounded integral solution on R

þ implies the existenceof Cn-almost periodic and Cn-almost automorphic strict solutions. When theexponential dichotomy holds for the homogeneous linear equation, they showed theuniqueness of Cn-almost periodic and Cn-almost automorphic strict solutions. In [7],the authors studied some basic results related to pseudo almost periodic functions ofclass r. Upon making some suitable assumptions, they obtained the existence anduniqueness of pseudo almost periodic solutions of following first-order partial

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neutral functional differential equations of class r

d

dt½uðtÞ þ f ðt, ut� ¼ A½uðtÞ þ f ðt, utÞ� þ gðt, utÞ for t � 0 ð1:3Þ

d

dt½uðtÞ þ f ðt, uðt� �1ðtÞÞ� ¼ A½uðtÞ þ f ðt, uðt� rÞÞ� þ gðt, uðt� �2ðtÞÞÞ for t � 0,

ð1:4Þ

where f, g, �1 and �2 are some suitable functions.The aim of this work is to prove the existence of pseudo almost periodic solutions

of Equation (1.1) when the delay is distributed on ]�1, 0]. Our approach is based on

the variation of constants formula and the spectral decomposition of the phase space

developed in [3].This work is organized as follows: in Section 2 we recall some prelimary results

on variation of constants formula and spectral decomposition. In Section 3, we recall

some preliminary results on pseudo almost periodic functions and neutral partial

functional differential equations that will be used in this work. In Section 4, we

discuss the main result of this article. Using the strict contraction principle we show

the existence and uniqueness of pseudo almost periodic solution of infinite class with

infinite delay for Equation (1.1). Finally, for illustration, we propose to study the

existence and uniqueness of pseudo almost periodic solution for some model arising

in the population dynamics.

2. Variation of constants formula and spectral decomposition

In this work, we assume that the state space (B, j�jB) is a normed linear

space of functions mapping ]�1, 0] into X satisfying the following fundamental

axioms.

(A1) There exist a positive constant H and functions K(�), M(�): Rþ!R

þ, with K

continuous and M locally bounded, such that for any � 2R and a> 0, if u:

]�1, a]!X, u� 2B, and u(�) is continuous on [�, �þ a], then for every t2 [�, �þ a]

the following conditions hold:

(i) ut2B,(ii) ju(t)j �HjutjB, which is equivalent to j’(0)j �Hj’jB for every ’2B,(iii) jutjB � Kðt� �Þ sup��s�t juðsÞj þMðt� �Þju�jB:

(A2) For the function u(�) in (A1), t � ut is a B-valued continuous function for

t2 [�, �þ a].(B) The space B is a Banach space.We assume that:(C1) If (’n)n�0 is a sequence in B such that ’n! 0 in B as n!þ1, then (’n(�))n�0converges to 0 in X.Let C( ]�1, 0],X) be the space of continuous functions from ]�1,0] into X.

We make the following assumptions:(C2) B�C(]�1, 0],X ).

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(C3) There exists �02R such that, for all �2C with Re �>�0 and x2X we have

e�.x2B and

K0 ¼ supRe�4 �0,x2X

x 6¼0

je�:xjBjxj

51,

where (e�.x)(�)¼ e��x for � 2 ]�1, 0] and x2X.

We associate the following initial value problem to Equation (1.1):

d

dtut ¼ Aut þ LðutÞ þ f ðtÞ for t � 0,

u0 ¼ ’2B,

8<: ð2:1Þ

where f: Rþ!X is a continuous function.

Definition 2.1 [6] We say that a continuous function u from ]�1,þ1[ into X is an

integral solution of Equation (2.1), if the following conditions hold:

(i)R t0 usds2DðAÞ for t� 0,

(ii) ut ¼ ’þ AR t0 usdsþ

R t0ðLðusÞ þ f ðsÞÞds for t� 0,

(iii) u0¼’.

If DðAÞ ¼ X, the integral solutions coincide with the known mild solutions. One

can see that if ut is an integral solution of Equation (2.1), then ut 2DðAÞ for all t� 0,

in particular ’2DðAÞ. Let us introduce the part A0 of the operator A in DðAÞ which,

defined by

DðA0Þ ¼ fx2DðAÞ : Ax2DðAÞg,

A0x ¼ Ax for x2DðA0Þ:

(We make the following assumption:

(H0) A satisfies the Hille–Yosida condition.

LEMMA 2.2 [1] A0 generates a strongly continuous semigroup (T0(t))t�0 on DðAÞ.

PROPOSITION 2.3 [2] Assume that B satisfies A and B and (H0) holds. Then for all

’2B such that ’ð0Þ 2DðAÞ, Equation (2.1) has a unique integral solution u on R

given by

ut ¼ T0ðtÞ’þ lim�!þ1

Z t

0

T0ðt� sÞB�ðLðusÞ þ f ðsÞÞds, for t � 0,

where B�¼ �R(�,A), for �>!.

In the sequel of this work, for simplicity, integral solutions are called solutions.The phase space BA of Equation (2.1) is defined by

BA ¼ f’2B : ’ð0Þ 2DðAÞg:

For each t� 0, we define the linear operator U(t) on BA by

UðtÞ ¼ vtð�, ’Þ,

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where v(�, ’) is the solution of the following homogeneous equation:

d

dtvt ¼ Avt þ LðvtÞ for t � 0,

v0 ¼ ’2B:

8<:PROPOSITION 2.4 [3] (U(t))t�0 is a strongly continuous semigroup of linear operators onBA. Moreover, (U(t))t�0 satisfies, for t� 0 and � 2 ]�1, 0], the following translationproperty:

ðUðtÞ’Þð�Þ ¼ðUðtþ �Þ’Þð0Þ for tþ � � 0,

’ðtþ �Þ for tþ � � 0:

(THEOREM 2.5 [3] Assume that B satisfies A, B, (C1) and (C2). Then AU definedon BA by

DðAUÞ¼n’2C1ð��1,0�;XÞ\BA; ’0 2BA, ’ð0Þ2DðAÞ and ’

0ð0Þ ¼A’ð0ÞþLð’Þo,

AU’¼ ’0 for ’2DðAUÞ

(is the infinitesimal generator of the semigroup (U(t))t�0 on BA.

Let hX0i be the space defined by

hX0i ¼ fX0x : x2Xg,

where the function X0x is defined by

ðX0xÞð�Þ ¼0 if �2 � �1, 0½,

x if � ¼ 0:

�The space BAhX0i equipped with the norm j�þX0cjB¼ j�jBþ jcj for(�, c)2BA�X is a Banach space and consider the extension AU defined onBAhX0i by

DðfAUÞ ¼ n’2C1ð� �1, 0�;XÞ : ’2DðAÞ and ’0 2DðAÞo,fAU’ ¼ ’0 þ X0ðA’þ Lð’Þ � ’0Þ:

8<:LEMMA 2.6 [3] Assume that B satisfies A, B, (C1), (C2) and (C3). Then, fAU satisfiesthe Hille-Yosida condition on BAhX0i.

Now, we can state the variation of constants formula associated withEquation (2.1).

Let C00 be the space of X-valued continuous function on ]�1, 0] with compactsupport. We assume that:

(D) If (’n)n�0 is a Cauchy sequence in B and converges compactly to ’ on]�1, 0], then ’2B and j’n�’j! 0.

THEOREM 2.7 [3] Assume that (C1), (C2) and (C3) hold. Then the integral solution x ofEquation (2.1) is given by the following variation of constants formula:

ut ¼ UðtÞ’þ lim�!þ1

Z t

0

Uðt� sÞeB�ðX0f ðsÞÞds for t � 0,

where eB� ¼ �ð�I� eAUÞ�1.

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Let (S0 (t))t�0 be the strongly continuous semigroup defined on the subspace

B0 ¼ f’2B : ’ð0Þ ¼ 0g,

by

ðS0ðtÞ�Þð�Þ ¼�ðtþ �Þ if tþ � � 0,

0 if tþ � � 0:

�Definition 2.8 Assume that the space B satisfies axioms (B) and (D), B is said to bea fading memory space, if for all ’2B0,

jS0ðtÞj ! 0 as t!þ1 in B0:

Moreover, B is said to be a uniform fading memory space, if

jS0ðtÞj ! 0 as t!þ1:

LEMMA 2.9 If B is a uniform fading memory space, then we can choose the functionK constant and the function M such that M(t)! 0 as t!þ1.

PROPOSITION 2.10 If the phase space B is a fading memory space, then the space BC(]�1, 0],X) of bounded continuous X-valued functions on ]�1, 0] endowed with theuniform norm topology is continuous embedding in B. In particular B satisfies (C3), for�0> 0.

For the sequel, we make the following assumption:

(H1) T0(t) is compact on DðAÞ for every t> 0.(H2) B is a uniform fading memory space.

THEOREM 2.11 [3] Assume that B satisfies A, B, (C1) and (H0), (H1), (H2) hold. Thenthe semigroup (U(t))t�0 is decomposed on BA as follows:

UðtÞ ¼ U1ðtÞ þ U2ðtÞ for t � 0,

where (U1(t))t�0 is an exponentially stable semigroup onBA, which means that there arepositive constants 0 and N0 such that

jU1ðtÞj �Me�!tj’j for t � 0 and ’2BA

and (U2(t))t�0 is compact for for every t> 0.

We introduce the Kuratowski measure of noncompactness (�) of bounded sets Kin a Banach space Y by

ðKÞ ¼ inff"4 0 : K has a finite cover of balls of diameter5 "g:

For a bounded linear operator B on Y, jBj is defined by

jBj ¼ inff"4 0 : BððKÞÞ � "ðKÞ for any bounded set K of Yg:

The essential growth bound !ess(U) of the semigroup (U(t))t�0 is defined by

!essðUÞ ¼ limt!þ1

1

tlog jUðtÞj ¼ inf

t40

1

tlog jUðtÞj:

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THEOREM 2.12 [3] Assume that B satisfies A, B, (C1) and (H1), (H2) hold. Then thesemigroup (U(t))t�0 is quasicompact, namely !ess(U)< 0.

We also get the following result on the spectral decomposition of the phasespace BA.

THEOREM 2.13 [3] Assume that B satisfies A, B, (C1), and (H0), (H1), (H2) hold. Thenthe space BA is decomposed as a direct sum

BA ¼ SU

of two U(t) invariant closed subspaces S and U such that the restricted semigroup on Uis a group and there exist positive constants M and ! such that

jUðtÞ’j �Me�!tj’j for t � 0 and ’2S,

jUðtÞ’j �Me!tj’j for t � 0 and ’2U,

where S and U are called the stable and unstable space, respectively, �s and �u denotethe projection operator on S and U, respectively.

3. Pseudo almost periodic functions

In this section, we recall some properties about pseudo almost periodic functions.Let BC(R;X) be the space of all bounded and continuous function from R to Xequipped with the uniform norm topology.

Definition 3.1 A bounded continuous function �: R!X is called almost periodic iffor each "> 0, there exists a relatively dense subset of R denoted by K(",�,X) suchthat j�(tþ )��(t)j<" for all (t, )2R�K(", �,X).

We denote by AP(R; X), the space of all such functions.

Definition 3.2 Let X1 and X2 be two Banach spaces. A bounded continuousfunction �: R�X1!X2 is called almost periodic in t2R uniformly in x2X1 if foreach "> 0 and all compact K�X1, there exists a relatively dense subset of R denotedby K(",�,K) such that j�(tþ , x)��(t, x)j<" for all t2R, x2K, 2K(",�,K).

We denote by AP(R�X1;X2), the space of all such functions.The next lemma is also a characterization of almost periodic functions.

LEMMA 3.3 A function �2C(R,X) is almost periodic if and only if the space offunctions {�: 2R}, where (�)(t)¼� (tþ ), is relatively compact in BC(R;X).

PAP0(R; X) stands for the space of functions

PAP0ðR;XÞ ¼

(u2BCðR;XÞ : lim

!þ1

1

2

Z þ�

juðtÞjdt ¼ 0

):

To study delayed differential equations for which the history belong toB, we need tointroduce the space

PAP0ðR;X,1Þ ¼

(u2BCðR;XÞ : lim

!þ1

1

2

Z þ�

�sup

� 2 ��1,t�juð�Þj

�dt ¼ 0

):

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In addition to above-mentioned space, we consider the following spaces:

PAP0ðR� X1,X2Þ ¼

(u2BCðR� X1;X2Þ : lim

!þ1

1

2

Z þ�

juðt, xÞjX2dt ¼ 0

),

PAP0ðR� X1;X2,1Þ

¼

(u2BCðR� X1;X2Þ : lim

!þ1

1

2

Z þ�

�sup

� 2 ��1,t�juð�, xÞjX2

�dt ¼ 0

),

where in both cases the limit (as !þ1) is uniform in compact subset of X1.In view of previous definitions, it is clear that the spaces PAP0(R;X,1)

and PAP0(R�X1; X2,1) are continuously embedded in PAP0(R;X) and

PAP0(R�X1,X2), respectively.

Definition 3.4 A bounded continuous function �: R!X is called pseudo almost

periodic if �¼�1þ�2, where �12AP(R,X) and �22PAP0(R;X) .

We denote by PAP(R;X) the space of all such functions.

Definition 3.5 Let X1 and X2 be two Banach spaces. A bounded continuous

function �: R�X1!X2 is called uniformly pseudo almost periodic if �¼�1þ�2,where �12AP(R�X1;X2) and �22PAP0(R�X1,X2).

We denote by PAP(R�X1;X2), the space of all such functions.We consider some concepts of pseudo almost periodic function introduced in [7].

Definition 3.6 A bounded continuous function �:R!X is called pseudo almost

periodic of infinite class if �¼�1þ�2, where �12AP(R; X) and �22PAP0(R;X,1) .We denote by PAP(R;X,1), the space of all such functions.

Definition 3.7 Let X1 and X2 be two Banach spaces. A bounded continuous

function �: R�X1!X2 is called uniformly pseudo almost periodic of infinite class if

�¼�1þ�2, where �12AP(R�X1;X2) and �22PAP0(R�X1;X2,1).

We denote by PAP(R�X1;X2,1), the space of all such functions.

LEMMA 3.8 PAP(R;X,1) is a closed subspace of PAP(R;X).

Proof Let (xn)n be a sequence in PAP(R;X,1) such that limn!1 xn ¼ x in

PAP(R;X). For each n, let xn¼ ynþ zn with yn2AP(R;X) and zn2PAP0(R;X,1).

From [8, Lemma 1.2], (yn)n converges to some y2AP(R;X). Consequently, (zn)n also

converges to some z2BC(R;X). Note that

1

2

Z þ�

�sup

�2 ��1,t�jzð�Þj

�dt

�1

2

Z þ�

�sup

� 2 ��1,t�jznð�Þ � zð�Þj

�dtþ

1

2

Z þ�

�sup

� 2 ��1,t�jznð�Þj

�dt

�1

2

Z þ�

�supt2R

jznðtÞ � zðtÞj

�dtþ

1

2

Z þ�

�sup

� 2 ��1,t�jznð�Þj

�dt

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� kzn � zk þ1

2

Z þ�

�sup

� 2 ��1,t�jznð�Þj

�dt:

Then we get z2PAP0(R;X,1), hence x2PAP(R;X,1). g

Consequently, using [8, Lemma 1.2], one obtains the following:

LEMMA 3.9 The space PAP(R;X,1) endowed with the uniform convergence

topology is a Banach space.

4. Pseudo almost periodic solutions of infinite class

In what follows, we will be looking at the existence of bounded integral solutions of

infinite class of Equation (1.1).

THEOREM 4.1 [3] Assume that B satisfies A, B, (C1), (C2) and (H1), holds. If

f2BC(R;X), then there exists a unique bounded solution u of Eq. (1.1) on R, given by

ut¼ lim�!þ1

Z t

�1

Usðt� sÞ�sðeB�X0f ðsÞÞdsþ lim�!þ1

Z t

þ1

Uuðt� sÞ�uðeB�X0fðsÞÞds for t2R,

where �s and �u are the projections of BA onto the stable and unstable subspaces,

respectively.

THEOREM 4.2 Let h2AP(R; X) and � be the mapping defined for t2R by

�hðtÞ ¼

"lim

�!þ1

Z t

�1

Usðt� sÞ�sðeB�X0hðsÞÞdsþ lim�!þ1

Z t

þ1

Uuðt� sÞ�uðeB�X0hðsÞÞds

#ð0Þ:

Then � h2AP(R,X).

Proof We can see that �h2BC(R;X). Since h is a almost periodic function, then the

set of functions {h: �2R}, h(t)¼ h(tþ ) is precompact in BC(R;X). On the other

hand, we have

ð�hÞðtÞ ¼ ð�hÞðtþ Þ

¼

"lim

�!þ1

Z tþ

�1

Usðtþ � sÞ�sðeB�X0hðsÞÞds

þ lim�!þ1

Z tþ

þ1

Uuðtþ � sÞ�uðeB�X0hðsÞÞds

#ð0Þ

¼

"lim

�!þ1

Z t

�1

Usðt� sÞ�sðeB�X0hðsþ ÞÞds

þ lim�!þ1

Z t

þ1

Uuðt� sÞ�uðeB�X0hðsþ ÞÞds

#ð0Þ

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¼

"lim

�!þ1

Z t

�1

Usðt� sÞ�sðeB�X0hðsÞÞds

þ lim�!þ1

Z t

þ1

Uuðt� sÞ�uðeB�X0hðsÞÞds

#ð0Þ

¼ ð�hÞðtÞ for all t2R:

Thus (�h)¼ (�h), which implies that {(�h)�: �2R} is relatively compact

in BC(R;X) since � is continuous from BC(R;X) into BC(R;X). Thus, �

h2AP(R,X). g

THEOREM 4.3 Let g2PAP0(R;X,1), then �g2PAP0(R;X,1) .

Proof In fact, for > 0 we get�����Z

�

sup

� 2 ��1,t�lim

�!þ1

Z �

�1

Usð� � sÞ�sðeB�X0gðsÞÞds

!dt

þ

Z

�

sup

� 2 ��1,t�lim

�!þ1

Z �

þ1

Uuð� � sÞ�uðeB�X0gðsÞÞds

!dt

������M eM Z

�

sup

� 2 ��1,t�

Z �

�1

e�!ð��sÞj�sj j gðsÞjds

!dt

þM eM Z

�

sup

� 2 ��1,t�

Z þ1�

e!ð��sÞj�uj j gðsÞjds

!dt

�M eMj�sj

Z

�

sups2 ��1,t�

j gðsÞj

sup

� 2 ��1,t�

Z �

�1

e�!ð��sÞds

!dt

þM eMj�uj

Z

�

sups2 ��1,t�

j gðsÞj

sup

� 2 ��1,t�

Z 1�

e!ð��sÞds

!dt

�M eMj�sj þM eMj�uj

!

Z

�

�sup

s2 ��1,t�j gðsÞj

�dt:

Consequently

1

2

Z

�

�sup

� 2 ��1,t�ð�gÞð�Þ

�dt �

M eMj�sj þM eMj�uj

!

"1

2

Z

�

�sup

s2 ��1,t�j gðsÞj

�dt

#,

which converges to zero as !þ1. Thus, we obtain the desired result. g

For the existence of pseudo almost periodic solution of infinite class, we make the

following assertion.

(H3) f: R!X is pseudo almost periodic of infinite class.

COROLLARY 4.4 Assume that B satisfies A, B, (C1), (C2) and (H0), (H1), (H3) hold.

Then Equation (1.1) has a unique pseudo almost periodic solution of infinite class.

Proof Since f is a pseudo almost periodic function, f has a decomposition f¼ f1þ f2where f12AP(R; X) and f22PAP0(R;X,1) . Using Theorems 4.1–4.3, we get the

desired result. g

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Our next objective is to show the existence of pseudo almost periodic solutions of

infinite class for the following problem:

d

dtut ¼ Aut þ LðutÞ þ f ðt, utÞ for t2R ð4:1Þ

where f: R�C!X is a continuous.To obtain our result, we prove the following results concerning the composition

of pseudo almost periodic functions of infinite class.

THEOREM 4.5 Let �2PAP(R�X1;X2,1) and h2PAP(R;X1,1). Assume thatthere exists a function L�: R! [0,þ1[ sastisfying

j�ðt, x1Þ � �ðt, x2Þj � l�ðtÞjx1 � x2j for t2R and for x1, x2 2X1: ð4:2Þ

If

lim!þ1

1

2

Z þ�

�sup

�2��1,t�L�ð�Þ

�dt51 and lim

!þ1

1

2

Z þ�

�sup

�2��1,t�L�ð�Þ

��ðtÞdt¼ 0

ð4:3Þ

for each �2PAP0(R), then the function t!�(t, h(t)) belongs to PAP(R;X2,1).

Proof Assume that �¼�1þ�2, h¼ h1þ h2 where �1, h12AP(R; X1) and�2,h22PAP0(R;X2,1). Consider the following decomposition:

�ðt, hðtÞÞ ¼ �1ðt, h1ðtÞÞ þ ½�ðt, hðtÞÞ � �ðt, h1ðtÞÞ� þ �2ðt, h1ðtÞÞ:

From [7], �1(�, h1(�))2AP(R; X). It remains to prove that both �(�, h(�))��(�, h1(�))and �2(�, h1(�)) belong to PAP0(R;X2,1). Using Equation (4.2), it follows thatZ þ

�

�sup

� 2 ��1,t�j�ð�, hð�ÞÞ � �ð�, h1ð�ÞÞj

�dt �

Z þ�

�sup

� 2 ��1,t�L�ð�Þjh2ð�Þj

�dt

and Z þ�

�sup

� 2 ��1,t�L�ð�Þjh2ð�Þj

�dt �

Z þ�

�sup

�2 ��1,t�L�ð�Þ

��sup

� 2 ��1,t�jh2ð�Þj

�dt,

which implies that �(�, h(�))��(�, h1(�))2PAP0(R;X2,1) by Equation (4.3).Since h1(R) is relatively compact in X1 and �1 is uniformly continuous on sets of

the form R�K where K�X1 is a compact subset, for "> 0, there exists �2 ]0, "[such that

j�1ðt, zÞ � �1ðt, z0Þj � " for z, z0 2 h1ðRÞ, with jz� z0j5 �:

Now, fix z1, . . . , zn2 h1(R) such that h1ðRÞ �Sn

i¼1 B�ðziÞ. Obviously, the setsEi¼ h�1(B�(zi)) form an open covering of R, and therefore using the sets B1¼E1,

B2¼E2 nE1 and Bi ¼ Ei nSi�1

j¼1 Ej one obtains a covering of R by disjoint open sets.For t2Bi, h1(t)2B�(zi), we have

j�2ðt, h1ðtÞÞj � j�ðt, h1ðtÞÞ � �ðt, ziÞj þ j � �1ðt, h1ðtÞÞ þ �1ðt, ziÞj þ j�2ðt, ziÞj

� L�ðtÞjh1ðtÞ � zij þ "þ j�2ðt, ziÞj

� L�ðtÞ"þ "þ j�2ðt, ziÞj:

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Now using the previous inequality it follows that

1

2

Z þ�

�sup

� 2 ��1,t�j�2ð�, h1ð�ÞÞj

�dt �

1

2

Xni¼1

ZBi\½�,�

�sup

� 2 ��1,t�j�2ð�, h1ð�ÞÞj

�dt

�1

2

Xni¼1

ZBi\½�,�

�sup

j¼1,...,,n

�sup

� 2 ��1,t�\Bj

j�2ð�, h1ð�ÞÞj

��dt

�1

2

Xni¼1

ZBi\½�,�

�sup

j¼1,...,,n

�sup

� 2 ��1,t�\Bj

j�ð�, h1ð�ÞÞ � �ð�, zj Þj

��dt

þ1

2

Xni¼1

ZBi\½�,�

�sup

j¼1,...,,n

�sup

� 2 ��1,t�\Bj

j�1ð�, h1ð�ÞÞ � �1ð�, zj Þj

��dt

þ1

2

Xni¼1

ZBi\½�,�

�sup

j¼1,...,,n

�sup

� 2 ��1,t�\Bj

j�2ð�, zj Þj

��dt

�1

2

Z

�

�sup

� 2 ��1,t�L�ð�Þ"þ "

�dtþ

Xni¼1

1

2

Z

�

�sup

�2 ��1,t�j�2ð�, zj Þj

�dt:

In view of the above it follows that �2(�, h1(�)) belongs to PAP0(R;X2,1). g

We have the following result.

THEOREM 4.6 Let u2PAP(R;X,1), then the function t! ut belongs to PAP(B,1).

Proof Assume that u¼ hþ g where h2AP(R; X) and g2PAP0(R;X,1). From [7],

ut¼ htþ gt and ht is almost periodic. On the other hand, we have

1

2

Z þ�

�sup

� 2 ��1,t�

�sup

�2 ��1,0�j gð� þ �Þj

���

1

2

Z þ�

�sup

� 2 ��1,t�j gð�Þj

�,

which shows that ut belongs to PAP(B,1). Thus, we obtain the desired result. g

For the sequel, we make the following assertion.(H4) f : R�C(]�1, 0]; X))!X is uniformly pseudo almost periodic of infinite

class such that there exists a continuous function Lf: R! [0, þ1[ such that

j f ðt,’1Þ � f ðt, ’2Þj � Lf ðtÞj’1 � ’2j for all t2R and ’1, ’2 2B

and Lf satisfies (4.2) and (4.3).

THEOREM 4.7 Assume that B is a uniform fading memory space and (C1), (C2), (H0),

(H1), (H2), (H4) hold. If

2M eMC supt2R

�j�sj

Z t

�1

e�!ðt�sÞLf ðsÞdsþ j�uj

Z þ1t

e!ðt�sÞLf ðsÞds

�5 1,

where C ¼ max�supt2R jMðtÞj, supt2R jKðtÞj

, then Equation (4.1) has a unique

pseudo almost periodic solution of infinite class.

Proof Let x be a function in PAP(R;X,1), from Theorem 4.6 the function t!xtbelongs to PAP(C( ]�1, 0];X),1). Hence Theorem 4.5 implies that the function

g(�) :¼ f (�, x.) is in PAP(R;X,1). Consider the mapping

H : PAPðR;X,1Þ ! PAPðR;X,1Þ

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defined for t2R by

ðHxÞðtÞ ¼

�lim

�!þ1

Z t

�1

Usðt� sÞ�sðeB�X0f ðs, xsÞÞds

þ lim�!þ1

Z t

þ1

Uuðt� sÞ�uðeB�X0f ðs,xsÞÞds

�ð0Þ:

From Theorem 4.1, Theorem 4.2 and taking into account Theorem 4.3, it suffices

now to show that the operator H has a unique fixed point in PAP(R;X,1). Let x1,

x22PAP(R; X,1). Since B is a uniform fading memory space, by the Lemma 2.9,

we can choose the function K constant and the function M such that M(t)! 0

as t!þ1. Let C ¼ max�supt2R jMðtÞj, supt2R jKðtÞj

, then we have

jHx1ðtÞ � Hx2ðtÞj �

���� lim�!þ1

Z t

�1

Usðt� sÞ�sðeB�X0½ f ððs,x1sÞÞ � f ððs,x1sÞÞ�ds

����þ

���� lim�!þ1

Z t

þ1

Usðt� sÞ�uðeB�X0½ f ððs, x2sÞÞ � f ððs,x2sÞÞ�ds

�����M eM�

j�sj

Z t

�1

e�!ðt�sÞLf ðsÞjx1s � x2sjBds

þ j�uj

Z þ1t

e!ðt�sÞLf ðsÞjx1s � x2sjBds

��M eM�

j�sj

Z t

�1

e�!ðt�sÞLf ðsÞðKðtÞ sup0�s�tjx1ðsÞ � x2ðsÞj

þMðtÞjx10 � x20 jB

�ds

þ j�uj

Z þ1t

e!ðt�sÞLf ðsÞ

�KðtÞ sup

0�s�tjx1ðsÞ � x2ðsÞj

þMðtÞjx10 � x20 jBÞds

�

j Hx1ðtÞ � Hx2ðtÞj � 2M eMC supt2R

j�sj

Z t

�1

e�!ðt�sÞLf ðsÞds

þ j�uj

Z þ1t

e!ðt�sÞLf ðsÞds

!jx1 � x2j:

This means that H is a strict contraction. Thus by a fixed point theorem, H has a

unique fixed point u in PAP(R; X,1). We conclude that Equation (1.1), has one and

only one pseudo almost periodic solution of infinite class. g

COROLLARY 4.8 Assume that B is a uniform fading memory space and (C1), (H0),

(H1), (H2) and f is lipschitzian with respect the second argument. If Lip(f) is small

enough, then Eq. (1.1) has a unique pseudo almost periodic solution of infinite class,

where Lip( f ) is the Lipschitz constant of f.

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5. Application

For illustration, we propose to study the existence of solutions for the following

model:

@

@tzðt, xÞ ¼

@2

@x2zðt, xÞ þ

Z 0

�1

Gð�Þðt, zðtþ �, xÞÞd�

þ ðsin tþ sinðffiffiffi2p

tÞ þ gðtÞÞ

Z 0

�1

hð�, zðtþ �, xÞÞd� for t2R and x2 ½0, �

zðt, 0Þ ¼ zðt, Þ ¼ 0 for t2R,

8>>>>><>>>>>:ð5:1Þ

where G: ]�1, 0]!R is a continuous function, h: ]�1, 0]�R!R is continuous

and g: R� [0, ]!R is a bounded continuous function defined by

gðtÞ ¼0 for t � 0,

te�t for t � 0:

(

To rewrite Equation (5.1) in the abstract form, we introduce the space

X¼C([0, ];R) of continuous function from [0, ] to Rþ equipped with the uniform

norm topology. Let A: D(A)!X be defined by

DðAÞ ¼ fy2X \ C2ð½0, �,RÞ : y00 2Xg,

Ay ¼ y00:

�Then A satisfies the Hille–Yosida condition in X. Moreover the part A0 of A in DðAÞ

is the generator of strongly continuous compact semigroup (T0(t))t�0 on DðAÞ.

It follows that (H0) and (H1) are satisfied.The phase space B¼C�, � > 0 where

C� ¼n’2Cð� �1, 0�;XÞ : lim

�!�1e��’ð�Þ exist in X

owith the following norm

j’j� ¼ sup��0je��’ð�Þj:

This space is a uniform fading memory space satisfying (C1), (C2).We define f: R�C!X and L: C!X as follows:

f ðt, ’ÞðxÞ ¼ ðsin tþ sinðffiffiffi2p

tÞ þ gðtÞÞ

Z 0

�1

hð�,’ð�ÞðxÞÞd� for x2 ½0, � and t2R,

Lð’ÞðxÞ ¼

Z 0

�1

Gð�Þ’ð�ÞðxÞÞd� for �1 � � � 0 and x2 ½0, �:

Let us pose v(t)¼ z(t, x). Then Equation (5.1) takes the following abstract form:

d

dtvt ¼ Avt þ LðvtÞ þ f ðt, vtÞ for t2R: ð5:2Þ

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From [7] ðsin tþ sinðffiffiffi2p

tÞÞ is almost periodic. Let �> 0 be given, then we have

1

2

Z þ�

j gsjds ¼1

2

Z þ0

j gsjds

¼1

2

Z �þ1

0

j gsjdsþ1

2

Z þ�þ1

j gsjds:

Since g is a decreasing function on [1,þ1[, then jgsj ¼ jg(s� �)j on [�þ 1,þ1[.

Hence

1

2

Z þ�

j gsjds ¼1

2

Z �þ1

0

j gsjdsþ1

2

Z þ�þ1

j gðs� �Þjds:

Since � is given, thenZ þ1�þ1

j gðs� �Þjds ¼

Z þ1�þ1

ðs� �Þe�ðs��Þds ¼2

e:

Consequently

lim!þ1

1

2

Z þ�

j gsjds ¼ 0:

It follows that g is ergodic of infinite class, consequently, f is uniformly

pseudo almost periodic of infinite class. Moreover, L is a bounded linear operator

from C to X.We suppose that there exists a function k1(�)2L

1(]�1, 0]; Rþ ) such that

jhð�, x1Þ � hð�, x2Þj � kð�Þjx1 � x2j for � � 0 and x1, x2 2R, ð5:3Þ

hð�, 0Þ ¼ 0: ð5:4Þ

Assumptions (5.3) and (5.4) imply that f(’)2X. In fact, ’2B, then

f ð’ÞðxÞ ¼

Z 0

�1

hð�, ’ð�ÞðxÞÞd� for x2 ½0, �

and

j f ð’ÞðxÞj �

Z 0

�1

k1ð�Þj ð�ÞðxÞÞjd�:

Consequently

j f ð Þj �

�Z 0

�1

k1ð�Þd�

�j jB:

Moreover assumption (5.4) implies that

f ð Þð0Þ ¼ f ð Þð Þ ¼ 0:

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Using the dominated convergence theorem, one can show that f(’) is a continuousfunction on [0, ]. Moreover, for every ’1, ’22B, we have

j f ð’1Þ � f ð’2Þj ¼ sup0�x�

j f ð’1ÞðxÞ � f ð’2ÞðxÞj

� sup0�x�

Z 0

�1

j gð�,’1ð�ÞðxÞÞ � gð�, ’2ð�ÞðxÞÞjd�

� sup0�x�

Z 0

�1

k1ð�Þj’1ð�ÞðxÞ � ’2ð�ÞðxÞjd�

�

�Z 0

�1

k1ð�Þd�

�sup�15��00�x�

j’1ð�ÞðxÞ � ’2ð�ÞðxÞj:

Consequently, we conclude that f is Lipschitz continuous and uniformly pseudoalmost periodic of infinite class.

Then by Corollary 4.8 we deduce the following result.

PROPOSITION 5.1 Under the above assumptions, ifR 0�1

k1ð�Þd� is small enough, thenEquation (5.2) has a unique pseudo almost periodic solution v of infinite class.

References

[1] M. Adimy, A. Elazzouzi, and K. Ezzimbi, Bohr-Neugebauer type theorem for some partialneutral functional differential equations, Nonlinear Anal. Theory, Methods Appl. 66(5)

(2007), pp. 1145–1160.[2] M. Adimy and K. Ezzinbi, Existence and linearized stability for partial neutral functional

differential equations, Differ. Eqns Dyn. Syst. 7 (1999), pp. 371–417.

[3] M. Adimy, K. Ezzinbi, and A. Ouhinou, Variation of constants formula and almost periodicsolutions for some partial functional differential equations with infinite delay, J. Math. Anal.Appl. 317(2) (2006), pp. 668–689.

[4] B. Amir and L. Maniar, Composition of pseudo almost periodic functions and Cauchyproblems with operator of non dense domain, Ann. Math. Blaise Pascal 6(1) (1999), pp. 1–11.

[5] K. Ezzinbi, S. Fatajou, and G.M. N0Guerekata, Pseudo-almost-automorphic solutions tosome neutral partial functional differential equations in Banach spaces, Nonlinear Anal.

Theory Method Appl. 70(4) (2009), pp. 1641–1647.[6] K. Ezzinbi, S. Fatajou, and G.M. NGuerekata, Cn-almost automorphic solutions for partial

neutral functional differential equations, Appl. Anal. 86(9) (2007), pp. 1127–1146.

[7] T. Diagana and E. Hernandez, Existence and uniqueness of pseudo almost periodic solutionsto some abstract partial neutral functional-differential equations and applications, J. Math.Anal. Appl. 327(2) (2007), pp. 776–791.

[8] H.X. Li, F.L. Huang, and J.Y. Li, Composition of pseudo almost-periodic functions andsemilinear differential equations, J. Math. Anal. Appl. 255(2) (2001), pp. 436–446.

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