Pseudo-almost periodic solutions for some classes of ...

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Pseudo-almost periodic solutions for some classes ofnonautonomous partial evolution equations

Toka Diagana

Department of Mathematics, Howard University, 2441 6th Street NW, Washington, DC 20059, USA

Received 7 July 2009; received in revised form 29 May 2011; accepted 2 June 2011

Abstract

In this paper upon making some suitable assumptions such as the Acquistapace–Terreni

conditions and exponential dichotomy we obtain the existence of pseudo-almost periodic solutions to

some classes of nonautonomous evolution equations of Sobolev-type. An example is given at the end

of the paper to illustrate our abstract result.

& 2011 The Franklin Institute Published by Elsevier Ltd. All rights reserved.

1. Introduction

Let ðX,J � JÞ be a Banach space. In Diagana et al. [15], the existence pseudo-almostperiodic solutions to the class of nonautonomous first-order partial neutral functionaldifferential with unbounded delay given by

d

dt½uðtÞ þ F ðt,utÞ� ¼AðtÞuðtÞ þ Gðt,utÞ, ð1:1Þ

where AðtÞ : DðAðtÞÞ � X-X is a family of densely defined linear operators on a commondomain D¼DðAðtÞÞ, independent of t 2 R, the history ut : ð�1,0�-X defined byutðyÞ ¼ uðtþ yÞ for each y 2 ð�1,0� belongs to some phase space B, and F ,G : R� B-X

are some appropriate functions, was established. For that, Diagana et al. assumed thatthere exists an evolution family T ¼ fV ðt,sÞgtZs associated with the family of linearoperators A(t), which in addition is uniformly asymptotically stable.

In this paper, we consider a more general setting and use different techniques to study theexistence of pseudo-almost periodic solutions to the class of nonautonomous Sobolev-type

2.00 & 2011 The Franklin Institute Published by Elsevier Ltd. All rights reserved.

.jfranklin.2011.06.001

dress: [email protected]

this article as: T. Diagana, Pseudo-almost periodic solutions for some classes of nonautonomous

lution equations, J. Franklin Inst. (2011), doi:10.1016/j.jfranklin.2011.06.001

dx.doi.org/10.1016/j.jfranklin.2011.06.001

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T. Diagana / Journal of the Franklin Institute ] (]]]]) ]]]–]]]2

differential equations:

d

dt½uðtÞ þ f ðt,BðtÞuðtÞÞ� ¼AðtÞuðtÞ þ gðt,CðtÞuðtÞÞ, t 2 R, ð1:2Þ

where A(t) for t 2 R is a family of closed linear operators on DðAðtÞÞ satisfying Acquistapace–Terreni conditions, BðtÞ,CðtÞ (t 2 R) are families of (possibly unbounded) linear operators,and f : R�X-Xt

b ð0oaobo1Þ and g : R�X-X are pseudo-almost periodic in t 2 R

uniformly in the second variable. It is well-known that in that event, there exists an evolutionfamily U ¼ fUðt,sÞgtZs associated with the family of linear operators A(t). Assuming that theevolution family U ¼ fUðt,sÞgtZs is exponentially dichotomic (hyperbolic) and under someadditional assumptions it will be shown that Eq. (1.2) has a unique pseudo-almost periodicsolution. The main result of this paper (Theorem 3.7) generalizes most of the known results onpseudo-almost periodic solutions to autonomous and nonautonomous differential equations,especially those in Refs. [6,12,10,15–22,24,33,47–49].Eqs. (1.1) and (1.2) are of a great interest as both equations are differential equations of

Sobolev-type. It is well-known that Sobolev-type differential equations have variousapplications notably in wave propagations or in dynamic of fluids [25].Various treatments of these equations can be easily found in literature. For different

formulations of these equations, we refer the readers to Refs. [9,34].The existence of almost periodic, almost automorphic, pseudo-almost periodic, and

pseudo-almost automorphic constitutes one of the most attractive topics in qualitativetheory of differential equations due to their applications. Some contributions on pseudo-almost periodic solutions to abstract differential and Sobolev-type differential equationshave recently been made, among them are Refs. [6,12,15,17,18,20–22,33,47–49]. However,the existence of pseudo-almost periodic solutions to evolution equations of the formEq. (1.2) is an untreated original topic, which in fact is the main motivation of thepresent paper.The paper is organized as follows: Section 2 is devoted to preliminaries facts related to

the existence of an evolution family. Some preliminary results on intermediate spaces arealso stated there. In addition, basic definitions and results on the concept of pseudo-almostperiodic functions are given. In Section 3, we first state and prove a key technical lemma(Lemma 3.2) and next we prove the main result (Theorem 3.7). An example is given at theend of the paper to illustrate our main result.

2. Preliminaries

This section is devoted to some preliminary results needed in the sequel. We basically usethe same setting as in Ref. [7] with slight adjustments.Throughout this paper ðX,J � JÞ stands for a Banach space, A(t) for t 2 R is a family of

closed linear operators on DðAðtÞÞ satisfying Acquistapace–Terreni conditions (Assump-tion (H.1)). Moreover, the linear operators A(t) are not necessarily densely defined. The(possibly unbounded) linear operators AðtÞ,BðtÞ,CðtÞ are defined on X such that theiralgebraic sum AðtÞ þ BðtÞ þ CðtÞ is not trivial for each t 2 R as every solution u to Eq. (1.2)belongs to DðAðtÞ þ BðtÞ þ CðtÞÞ ¼DðAðtÞÞ \DðBðtÞÞ \DðCðtÞÞ.The functions, f : R�X-Xt

b (0oaobo1), g : R�X-X are respectively jointlycontinuous satisfying some additional assumptions.

Please cite this article as: T. Diagana, Pseudo-almost periodic solutions for some classes of nonautonomous

partial evolution equations, J. Franklin Inst. (2011), doi:10.1016/j.jfranklin.2011.06.001


T. Diagana / Journal of the Franklin Institute ] (]]]]) ]]]–]]] 3

If L is a linear operator on X, then rðLÞ, sðLÞ, D(L), N(L), R(L) stand for the resolvent,spectrum, domain, kernel, and range of L. Moreover, one sets Rðl,LÞ :¼ ðlI�LÞ�1 for alll 2 rðLÞ.

Throughout the paper, we set QðsÞ ¼ I�PðsÞ for a family of projections P(s) with s 2 R.The space BðY,ZÞ denotes the collection of all bounded linear operators from Y into Z

equipped with its natural topology. When Y¼ Z, this is simply denoted by BðYÞ.

2.1. Evolution families

(H.1): The family of closed linear operators A(t) for t 2 R on X with domain DðAðtÞÞ

(possibly not densely defined) satisfy the so-called Acquistapace–Terreni conditions,that is, there exist constants o 2 R, y 2 ðp=2,pÞ, L40 and m,n 2 ð0,1� with mþ n41such that

Sy [ f0g � rðAðtÞ�oÞ 3 l, JRðl,AðtÞ�oÞJrK

1þ jljfor all t 2 R ð2:1Þ

and

JðAðtÞ�oÞRðl,AðtÞ�oÞ½Rðo,AðtÞÞ�Rðo,AðsÞÞ�JrLjt�sjm

jljnð2:2Þ

for t,s 2 R, l 2 Sy :¼ fl 2 C\f0g : jargljryg.Among other things, Acquistapace–Terreni Conditions ensure that there exists a unique

evolution family:

U ¼ fUðt,sÞ : t,s 2 R such that tZsg

on X associated with A(t) such that Uðt,sÞXDDðAðtÞÞ for all t,s 2 R with tZs, and

(a)

Ple

pa

Uðt,sÞUðs,rÞ ¼Uðt,rÞ for t,s 2 R such that tZsZs;
(b) Uðt,tÞ ¼ I for t 2 R where I is the identity operator of X; (c) ðt,sÞ-Uðt,sÞ 2 BðXÞ is continuous for t4s;
Let us indicate that the above-mentioned properties were mainly established in Refs.
[1, Theorem 2.3,45, Theorem 2.1], see also [3,44]. In that case we say that Að�Þ generates theevolution family Uð�,�Þ.
Definition 2.1 (Baroun et al. [7]). An evolution family U is said to have an exponential

dichotomy (or is hyperbolic) if there are projections P(t) (t 2 R) that are uniformly boundedand strongly continuous in t and constants d40 and NZ1 such that

(e)
Uðt,sÞPðsÞ ¼PðtÞUðt,sÞ; (f) the restriction UQðt,sÞ : QðsÞX-QðtÞX of Uðt,sÞ is invertible (we then set
~U Qðs,tÞ :¼ UQðt,sÞ�1); and

(g)
JUðt,sÞPðsÞJrNe�dðt�sÞ and J ~U Qðs,tÞQðtÞJrNe�dðt�sÞ for tZs and t,s 2 R.
This setting requires some estimates related to U ¼ fUðt,sÞgtZs. For that, we introduce
the interpolation spaces for A(t). We refer the reader to the following excellent books[4,26,36] for proofs and further information on theses interpolation spaces.
ase cite this article as: T. Diagana, Pseudo-almost periodic solutions for some classes of nonautonomous

rtial evolution equations, J. Franklin Inst. (2011), doi:10.1016/j.jfranklin.2011.06.001



Let A be a sectorial operator on X (Assumption (H.1) holds when A(t) is replacedwith A) and let a 2 ð0,1Þ. Define the real interpolation space:

XAa :¼ fx 2 X : JxJA

a :¼ supr40

JraðA�oÞRðr,A�oÞxJo1g,

which, by the way, is a Banach space when endowed with the norm J � JAa . For convenience

we further write

XA0 :¼ X, JxJA

0 :¼ JxJ, XA1 :¼ DðAÞ

and JxJA1 :¼ Jðo�AÞxJ. Moreover, let X

A:¼ DðAÞ of X. In particular, we will frequently

be using the following continuous embedding

DðAÞ+XAb+Dððo�AÞaÞ+XA

a+XA� X, ð2:3Þ

for all 0oaobo1, where the fractional powers are defined in the usual way.In general, D(A) is not dense in the spaces XA

a and X. However, we have the followingcontinuous injection:

XAb+DðAÞ

J�JAa

ð2:4Þ

for 0oaobo1.

Definition 2.2. Given the family of linear operators A(t) for t 2 R, satisfying (H.1), we set

Xta :¼ XAðtÞ

a , Xt:¼ X

AðtÞ

for 0rar1 and t 2 R, with the corresponding norms. Then the embedding in Eq. (2.3)holds with constants independent of t 2 R.These interpolation spaces are of class J a [36, Definition 1.1.1 ] and it can be shown that

JyJtarKaL1�aJyJ1�aJAðtÞyJa, y 2 DðAðtÞÞ, ð2:5Þ

where K ,L are the constants appearing in (H.1).We have the following fundamental estimates for the evolution family U. Its proof was

given in Ref. [7] though for the sake of clarity, we reproduce it here.

Proposition 2.3. For x 2 X, 0rar1, the following hold:

(i)

Ple

par

There is a constant cðaÞ, such that

JUðt,sÞPðsÞxJtarcðaÞe�ðd=2Þðt�sÞðt�sÞ�aJxJ, t4s: ð2:6Þ

(ii)
There is a constant mðaÞ, such that
J ~U Qðs,tÞQðsÞxJsarmðaÞe�dðt�sÞJxJ, trs: ð2:7Þ

Proof. (i) Using Eq. (2.5) we obtain

JUðt,sÞPðsÞxJtarrðaÞJUðt,sÞPðsÞxJ1�aJAðtÞUðt,sÞPðsÞxJa

rrðaÞJUðt,sÞPðsÞxJ1�aJAðtÞUðt,t�1ÞUðt�1,sÞPðsÞxJa

rrðaÞJUðt,sÞPðsÞxJ1�aJAðtÞUðt,t�1ÞJaJUðt�1,sÞPðsÞxJa

rc0ðaÞðt�sÞ�ae�ðd=2Þðt�sÞðt�sÞae�ðd=2Þðt�sÞJxJ

for t�sZ1 and x 2 X.

ase cite this article as: T. Diagana, Pseudo-almost periodic solutions for some classes of nonautonomous

tial evolution equations, J. Franklin Inst. (2011), doi:10.1016/j.jfranklin.2011.06.001



Since ðt�sÞae�ðd=2Þðt�sÞ-0 as t-þ1 it easily follows that there exists c1ðaÞ40 such that

JUðt,sÞPðsÞxJtarc1ðaÞðt�sÞ�ae�ðd=2Þðt�sÞJxJ:

If 0ot�sr1, we have

JUðt,sÞPðsÞxJtarrðaÞJUðt,sÞPðsÞxJ1�aJAðtÞUðt,sÞPðsÞxJa

rrðaÞJUðt,sÞPðsÞxJ1�a AðtÞU t,tþ s

2

� �U

tþ s

2,s

� �PðsÞx

�� a

rrðaÞJUðt,sÞPðsÞxJ1�a AðtÞU t,tþ s

2

� �� a Utþ s

2,s

� �PðsÞx

�� arc2ðaÞe�ðd=2Þðt�sÞðt�sÞ�aJxJ,

and hence

JUðt,sÞPðsÞxJtarcðaÞðt�sÞ�ae�ðd=2Þðt�sÞJxJ for t4s:

(ii)

J ~U Qðs,tÞQðtÞxJsarrðaÞJ ~U Qðs,tÞQðtÞxJ

1�aJAðsÞ ~U Qðs,tÞQðtÞxJa

rrðaÞJ ~U Qðs,tÞQðtÞxJ1�aJAðsÞQðsÞ ~U Qðs,tÞQðtÞxJ

a

rrðaÞJ ~U Qðs,tÞQðtÞxJ1�aJAðsÞQðsÞJaJ ~U Qðs,tÞQðtÞxJ

a

rrðaÞNe�dðt�sÞð1�aÞJAðsÞQðsÞJae�dðt�sÞaJxJrmðaÞe�dðt�sÞJxJ:

In the last inequality we have used that JAðsÞQðsÞJrc for some constant cZ0, see e.g.[43, Proposition 3.18]. &

In addition to above, we also need the following assumptions:(H.2). The evolution family U ¼ fUðt,sÞgtZs generated by Að�Þ has an exponential

dichotomy with constants N,d40 and dichotomy projections P(t) for t 2 R. Moreover,0 2 rðAðtÞÞ for each t 2 R and the following holds

supt,s2R

JAðsÞA�1ðtÞJBðX,XbÞoc0: ð2:8Þ

Remark 2.4. Note that Eq. (2.8) is satisfied in many cases in the literature. In particular, itholds when AðtÞ ¼ dðtÞA where A : DðAÞ � X-X is any closed linear operator such that0 2 rðAÞ and d : R-R with inf t2RjdðtÞj40 and supt2RjdðtÞjo1.

(H.3). There exists 0raobo1 such that

Xta ¼Xa and Xt

b ¼Xb

for all t 2 R, with uniform equivalent norms.If 0raobo1, then we let kðaÞ,k1,k2 be the bounds of the continuous injections

Xb+Xa, Xa+X, Xb+X.

2.2. Pseudo-almost periodic functions

Let BCðR,XÞ (respectively, BCðR�Y,XÞ) denote the collection of all X-valued boundedcontinuous functions (respectively, the class of jointly bounded continuous functionsF : R�Y-X). The space BCðR,XÞ equipped with its natural norm, that is, the sup norm





defined by

JuJ1 ¼ supt2R

JuðtÞJ

is a Banach space. Similarly, BðR,XaÞ for a 2 ð0,1Þ stands for the Banach space of all boundedcontinuous functions j : R-Xa when equipped with the a-sup norm:

JjJa,1 :¼ supt2R

JjðtÞJa

for j 2 BCðR,XaÞ. Furthermore, CðR,YÞ (respectively, CðR�Y,XÞ) denotes the class ofcontinuous functions from R into Y (respectively, the class of jointly continuous functionsF : R�Y-X).

Definition 2.5. A function f 2 CðR,XÞ is called (Bohr) almost periodic if for each e40there exists lðeÞ40 such that every interval of length lðeÞ contains a number t with theproperty that

Jf ðtþ tÞ�f ðtÞJoe for each t 2 R:

The number t above is called an e-translation number of f, and the collection of all suchfunctions will be denoted as APðXÞ.

Definition 2.6. A function f 2 CðR,XÞ is called (Bochner) almost periodic if for anysequence of real numbers ðs0nÞn2N there exists a subsequence ðtnÞn2N � ðs

0nÞn2N such that the

sequence of functions ff ðtþ tnÞgn2N converges uniformly in t 2 R.

It is well-known that Bohr and Bochner’s definitions of almost periodicity are equivalent.

Definition 2.7. A function F 2 CðR�Y,XÞ is called (Bohr) almost periodic in t 2 R

uniformly in y 2 Y if for each e40 and any compact K � Y there exists lðeÞ such that everyinterval of length lðeÞ contains a number t with the property that

JF ðtþ t,yÞ�F ðt,yÞJoe for each t 2 R, y 2 K :

The collection of those functions is denoted by APðY,XÞ.

Define

PAP0ðXÞ :¼ f 2 BCðR,XÞ : limT-1

1

2 T

Z T

�T

Jf ðsÞJ ds¼ 0

� �:

In the same way, we define PAP0ðY,XÞ as the collection of jointly continuous functionsF : R�Y-X such that F ð�,yÞ is bounded for each y 2 Y and

limT-1

1

2 T

Z T

�T

JF ðs,yÞJ ds¼ 0

uniformly in compact subset of Y.

Definition 2.8. A function f 2 BCðR,XÞ is said to be a pseudo-almost periodic if it can beexpressed as f ¼ gþ f, where g 2 APðXÞ and f 2 PAP0ðXÞ. The collection of suchfunctions will be denoted by PAPðXÞ.





Lemma 2.9. The space ðPAPðXÞ,J � J1Þ is a Banach space.

Theorem 2.10. The decomposition of a pseudo-almost periodic function f ¼ gþ f, where

g 2 APðXÞ and f 2 PAP0ðXÞ, is unique.

Definition 2.11. A function F 2 BCðR�X,YÞ is called pseudo-almost periodic in t 2 R

uniformly in y 2 Y if F ¼G þ F, where G 2 APðX,YÞ and F 2 PAP0ðX,YÞ. The class ofsuch functions will be denoted by PAPðX,YÞ.

3. Main results

Throughout the rest of this paper we suppose that there exists three real numbers a,bsuch that 0oaobo1 with

2b4aþ 1:

Moreover, we denote by G1,G2,G3, and G4, the nonlinear integral operators defined by

ðG1uÞðtÞ :¼

Z t

�1

AðsÞUðt,sÞPðsÞf ðs,BðsÞuðsÞÞ ds,

ðG2uÞðtÞ :¼

Z 1t

AðsÞUQðt,sÞQðsÞf ðs,BðsÞuðsÞÞ ds,

ðG3uÞðtÞ :¼

Z t

�1

Uðt,sÞPðsÞgðs,CðsÞuðsÞÞ ds, and

ðG4uÞðtÞ :¼

Z 1t

UQðt,sÞQðsÞgðs,CðsÞuðsÞÞ ds:

Additionally, we suppose that the linear operators BðtÞ,CðtÞ : Xa-X are boundeduniformly in t 2 R. Moreover, both t-BðtÞ and t-CðtÞ belong to APðBðXa,XÞÞ. Wethen set

$ :¼ max supt2R

JBðtÞJBðXa,XÞ, supt2R

JCðtÞJBðXa,XÞ

� �:

To study Eq. (1.2), in addition to the previous assumptions, we require the followingadditional assumptions:

(H.4)

Please

partia

Rðo,Að�ÞÞ 2 APðBðXaÞÞ: Moreover, there exists a function H : ½0,1Þ-½0,1Þ withH 2 L1½0,1Þ such that for every e40 there exists lðeÞ such that every interval oflength lðeÞ contains a t with the property

JAðtþ tÞUðtþ t,sþ tÞ�AðtÞUðt,sÞJBðXa,XÞreHðt�sÞ

for all t,s 2 R with t4s.
(H.5) The function f : R�X-Xb belongs to PAPðX,XbÞ while g : R�X-X belongs to
PAPðX,XÞ. Moreover, the functions f ,g are uniformly Lipschitz with respect to thesecond argument in the following sense: there exists K40 such that

Jf ðt,uÞ�f ðt,vÞJbrKJu�vJ

cite this article as: T. Diagana, Pseudo-almost periodic solutions for some classes of nonautonomous

l evolution equations, J. Franklin Inst. (2011), doi:10.1016/j.jfranklin.2011.06.001



Please

partia

andJgðt,uÞ�gðt,vÞJrKJu�vJ

for all u,v 2 X and t 2 R.

To study the existence and uniqueness of pseudo-almost periodic solutions to Eq. (1.2)we first introduce the notion of mild solution, which has been adapted from Diagana et al.[15, Defintion 3.1].

Definition 3.1. A continuous function u : R-Xa is said to be a mild solution to Eq. (1.2)provided that the function s-AðsÞUðt,sÞPðsÞf ðs,BðsÞuðsÞÞ is integrable on (s,t), s-AðsÞ

Uðt,sÞQðsÞf ðs,BðsÞuðsÞÞ is integrable on ðt,sÞ and

uðtÞ ¼�f ðt,BðtÞuðtÞÞ þUðt,sÞðuðsÞ þ f ðs,BðsÞuðsÞÞÞ

�

Z t

s

AðsÞUðt,sÞPðsÞf ðs,BðsÞuðsÞÞ dsþ

Z s

t

AðsÞUðt,sÞQðsÞf ðs,BðsÞuðsÞÞ ds

þ

Z t

s

Uðt,sÞPðsÞgðs,CðsÞuðsÞÞ ds�

Z s

t

Uðt,sÞQðsÞgðs,CðsÞuðsÞÞds

for tZs and for all t,s 2 R.

Under Assumptions (H.1)–(H.3) and (H.5), it can be easily shown that Eq. (1.2) has aunique mild solution given by

uðtÞ ¼�f ðt,BðtÞuðtÞÞ�

Z t

�1

AðsÞUðt,sÞPðsÞf ðs,BðsÞuðsÞÞ ds

þ

Z 1t

AðsÞUQðt,sÞQðsÞf ðs,BðsÞuðsÞÞ dsþ

Z t

�1

Uðt,sÞPðsÞgðs,CðsÞuðsÞÞ ds

�

Z 1t

UQðt,sÞQðsÞgðs,CðsÞuðsÞÞ ds

for each t 2 R.The proof of our main result requires the next technical lemmas.

Lemma 3.2. For each x 2 X, suppose that Assumptions (H.1)–(H.3) hold and let a,b be real

numbers such that 0oaobo1 with 2b4aþ 1, Then there are two constants r0ða,bÞ,dðbÞ40such that

JAðtÞUðt,sÞPðsÞxJbrr0ða,bÞe�d=4ðt�sÞðt�sÞ�bJxJ, t4s ð3:1Þand

JAðsÞ ~U Qðs,tÞQðtÞxJbrdðbÞe�dðs�tÞJxJ, trs: ð3:2Þ

Proof. Let x 2 X. First of all, note that JAðtÞUðt,sÞJBðX,XbÞrKðt�sÞ�ð1�bÞ for all t,s suchthat 0ot�sr1 and b 2 ½0,1�.Letting t�sZ1 and using the above-mentioned approximate, we obtain

JAðtÞUðt,sÞPðsÞxJb ¼ JAðtÞUðt,t�1ÞUðt�1,sÞPðsÞxJb

rJAðtÞUðt,t�1ÞJBðX,XbÞJUðt�1,sÞPðsÞxJrNKede�dðt�sÞJxJ

¼K1e�dðt�sÞJxJ¼K1e�ð3d=4Þðt�sÞðt�sÞbðt�sÞ�be�ðd=4Þðt�sÞJxJ:





Now since e�ð3d=4Þðt�sÞðt�sÞb-0 as t-1 it follows that there exists c4ðbÞ40 such that

JAðtÞUðt,sÞPðsÞxJbrc4ðbÞðt�sÞ�be�ðd=4Þðt�sÞJxJ:

Now, let 0ot�sr1. Using Eq. (2.6) and the fact 2b4aþ 1, we obtain

JAðtÞUðt,sÞPðsÞxJb ¼ AðtÞU t,tþ s

2

� �U

tþ s

2,s

� �PðsÞx

�� b

r AðtÞU t,tþ s

2

� �� BðX,XbÞ

Utþ s

2,s

� �PðsÞx

�� rk1

AðtÞU t,tþ s

2

� �BðX,XbÞ

U tþ s

2,s

� �PðsÞx

a

rk1Kt�s

2

� �b�1cðaÞ

t�s

2

� ��ae�ðd=4Þðt�sÞJxJ

¼ c5ða,bÞðt�sÞb�1�ae�ðd=4Þðt�sÞJxJrc5ða,bÞðt�sÞ�be�ðd=4Þðt�sÞJxJ:

In summary, there exists r0ðb,aÞ40 such that

JAðtÞUðt,sÞPðsÞxJbrr0ða,bÞðt�sÞ�be�ðd=4Þðt�sÞJxJ

for all t,s 2 R with t4s.Let x 2 X. Since the restriction of A(s) to RðQðsÞÞ is a bounded linear operator it follows

that

JAðsÞ ~U Qðs,tÞQðtÞxJb ¼r~cJ ~U Qðs,tÞQðsÞxJbr~cmðbÞe�dðs�tÞJxJ¼ dðbÞe�dðs�tÞJxJ

for trs by using Eq. (2.7). &

A straightforward consequence of Lemma 3.2 is the following:

Corollary 3.3. For each x 2 X, suppose that Assumptions (H.1)–(H.3) hold and let a,b be

real numbers such that 0oaobo1 with 2b4aþ 1. Then there is a constant rða,bÞ such that

JAðsÞUðt,sÞPðsÞxJbrrða,bÞe�ðd=4Þðt�sÞðt�sÞ�bJxJ, t4s: ð3:3Þ

Proof. We make use of (H.2) and Lemma 3.2. Indeed, for each x 2 X:

JAðsÞUðt,sÞPðsÞxJb ¼ JAðsÞA�1ðtÞAðtÞUðt,sÞPðsÞxJb

rJAðsÞA�1ðtÞJBðX,XbÞJAðtÞUðt,sÞPðsÞxJ

rc0k2JAðtÞUðt,sÞPðsÞxJbrc0k2r0ða,bÞe�ðd=4Þðt�sÞðt�sÞ�bJxJ

¼ rða,bÞe�ðd=4Þðt�sÞðt�sÞ�bJxJ, t4s: &

Lemma 3.4. Under previous assumptions, if u 2 PAPðXaÞ, then Cð�Þuð�Þ 2 PAPðXÞ.Similarly, Bð�Þuð�Þ 2 PAPðXÞ.

Proof. Let u 2 PAPðXaÞ and suppose u¼ u1 þ u2 where u1 2 APðXaÞ and u2 2 PAP0ðXaÞ.Then, CðtÞuðtÞ ¼CðtÞu1ðtÞ þ CðtÞu2ðtÞ for all t 2 R. Since u1 2 APðXaÞ, for every e40 thereexists T0ðeÞ such that every interval of length T0ðeÞ contains a t such

Ju1ðtþ tÞ�u1ðtÞJaoe, t 2 R:





Similarly, since CðtÞ 2 APðBðXa,XÞÞ, we have

JCðtþ tÞ�CðtÞJBðXa,XÞoe, t 2 R:

Now

JCðtþ tÞu1ðtþ tÞ�CðtÞu1ðtÞJ

¼ JCðtþ tÞu1ðtþ tÞ�CðtÞu1ðtþ tÞþCðtÞu1ðtþ tÞ�CðtÞu1ðtÞJrJ½Cðtþ tÞ�CðtÞ�u1ðtþ tÞJþ JCðtÞ½u1ðtþ tÞ�u1ðtÞ�J

rJCðtþ tÞ�CðtÞJBðXa,XÞJu1ðtþ tÞJa þ JCðtÞJBðXa,XÞJu1ðtþ tÞ�u1ðtÞJa

r supt2R

Ju1ðtÞJa þ$

� �e,

and hence t-CðtÞu1ðtÞ belongs to APðXÞ.To complete the proof, it suffices to notice that

1

2T

Z T

�T

JCðtÞu2ðtÞJ dtr$

2T

Z T

�T

Ju2ðtÞJa dt

and hence

limT-1

1

2T

Z T

�T

JCðtÞu2ðtÞJ dt¼ 0: &

Lemma 3.5. Under Assumptions (H.1)–(H.5), the integral operators G3 and G4 defined above

map PAPðXaÞ into itself.

Proof. Let u 2 PAPðXaÞ. From Lemma 3.4 it follows that Cð�Þuð�Þ 2 PAPðXÞ. SettinghðtÞ ¼ gðt,CðtÞuðtÞÞ and using the theorem of composition of pseudo-almost periodicfunctions [17] it follows that h 2 PAPðXÞ. Now write h¼ fþ z where f 2 APðXÞ andz 2 PAP0ðXÞ. Thus G3u can be rewritten as

ðG3uÞðtÞ ¼

Z t

�1

Uðt,sÞPðsÞfðsÞ dsþ

Z t

�1

Uðt,sÞPðsÞzðsÞ ds:

Set

FðtÞ ¼Z t

�1

Uðt,sÞPðsÞfðsÞ ds and CðtÞ ¼Z t

�1

Uðt,sÞPðsÞzðsÞ ds

for each t 2 R.The next step consists of showing that F 2 APðXaÞ and C 2 PAP0ðXaÞ. Obviously,

F 2 APðXaÞ. Indeed, since f 2 APðXÞ, for every e40 there exists lðeÞ40 such that forevery interval of length lðeÞ contains a t with the property:

Jfðtþ tÞ�fðtÞJoem for each t 2 R,

where m¼ d1�a=cðaÞ21�aGð1�aÞ with G being the classical G function.Now

Fðtþ tÞ�FðtÞ ¼Z tþt

�1

Uðtþ t,sÞPðsÞfðsÞ ds�

Z t

�1

Uðt,sÞPðsÞfðsÞ ds

¼

Z t

�1

Uðtþ t,sþ tÞPðsþ tÞfðsþ tÞ ds�

Z t

�1






¼

Z t

�1

Uðtþ t,sþ tÞPðsþ tÞfðsþ tÞ ds

�

Z t

�1

Uðtþ t,sþ tÞPðsþ tÞfðsÞ ds

þ

Z t

�1

Uðtþ t,sþ tÞPðsþ tÞfðsÞ ds�

Z t

�1


¼

Z t

�1

Uðtþ t,sþ tÞPðsþ tÞðfðsþ tÞ�fðsÞÞ ds

þ

Z t

�1

ðUðtþ t,sþ tÞPðsþ tÞ�Uðt,sÞPðsÞÞfðsÞ ds:

Using [8,37] it follows thatZ t

�1

ðUðtþ t,sþ tÞPðsþ tÞ�Uðt,sÞPðsÞÞfðsÞ ds

��ar

2JfJ1d

e:

Similarly, using Eq. (2.6), it follows thatZ t

�1

Uðtþ t,sþ tÞPðsþ tÞðfðsþ tÞ�fðsÞÞ ds

��are:

Therefore,

JFðtþ tÞ�FðtÞJao 1þ2JfJ1

d

� �e for each t 2 R

and hence, F 2 APðXaÞ.To complete the proof for G3, we have to show that C 2 PAP0ðXaÞ. First, note that

s-CðsÞ is a bounded continuous function. It remains to show that

limT-1

1

2T

Z T

�T

JCðtÞJa dt¼ 0:

Again using Eq. (2.6) it follows that

limT-1

1

2T

Z T

�T

JCðtÞJa dtr limT-1

cðaÞ2 T

Z T

�T

Z þ10

s�ae�ðd=2ÞsJzðt�sÞJ ds dt

r limT-1

cðaÞZ þ10

s�ae�ðd=2Þs1

2T

Z T

�T

Jzðt�sÞJ dt ds:

Set

GsðTÞ ¼1

2T

Z T

�T

Jzðt�sÞJ dt:

Since PAP0ðXÞ is translation invariant it follows that t-zðt�sÞ belongs to PAP0ðXÞ foreach s 2 R, and hence

limT-1

1

2T

Z T

�T

Jzðt�sÞJ dt¼ 0

for each s 2 R.One completes the proof by using the well-known Lebesgue dominated convergence

theorem and the fact GsðTÞ-0 as T-1 for each s 2 R.





The proof for G4uð�Þ is similar to that of G3uð�Þ. However one makes use of Eq. (2.7)rather than Eq. (2.6). &

Lemma 3.6. Under Assumptions (H.1)–(H.5), the integral operators G1 and G2 defined above

map PAPðXaÞ into itself.

Proof. Let u 2 PAPðXaÞ. From Lemma 3.4 it follows that the function t-BðtÞuðtÞ belongsto PAPðXÞ. Again, using the composition of pseudo-almost periodic functions [17] itfollows that cð�Þ ¼ f ð�,BðtÞuð�ÞÞ is in PAPðXbÞ whenever u 2 PAPðXaÞ. In particular,

JcJ1,b ¼ supt2R

Jf ðt,BðtÞuðtÞÞJbo1:

Now write c¼fþ z, where f 2 APðXbÞ and z 2 PAP0ðXbÞ, that is, G1c¼XðfÞ þ XðzÞwhere

XfðtÞ :¼Z t

�1

AðsÞUðt,sÞPðsÞfðsÞ ds and

XzðtÞ :¼

Z t

�1

AðsÞUðt,sÞPðsÞzðsÞ ds:

Clearly, XðfÞ 2 APðXaÞ. Indeed, since f 2 APðXbÞ, for every e40 there exists lðeÞ40such that every interval of length lðeÞ contains a t with the property

Jfðtþ tÞ�fðtÞJboen for each t 2 R

where n¼ d1�b=k2kðaÞrða,bÞ41�bGð1�bÞ.

Xfðtþ tÞ�XfðtÞ ¼Z tþt

�1

AðsÞUðtþ t,sÞPðsÞfðsÞ ds�

Z t

�1

AðsÞUðt,sÞPðsÞfðsÞ ds

¼

Z t

�1

Aðsþ tÞUðtþ t,sþ tÞPðsþ tÞfðsþ tÞ ds

�

Z t

�1


¼

Z t

�1

Aðsþ tÞUðtþ t,sþ tÞPðsþ tÞfðsþ tÞ ds

�

Z t

�1

Aðsþ tÞUðtþ t,sþ tÞPðsþ tÞfðsÞ ds

þ

Z t

�1

Aðsþ tÞUðtþ t,sþ tÞPðsþ tÞfðsÞ ds

�

Z t

�1


¼

Z t

�1

Aðsþ tÞUðtþ t,sþ tÞPðsþ tÞðfðsþ tÞ�fðsÞÞ ds

þ

Z t

�1

ðAðsþ tÞUðtþ t,sþ tÞPðsþ tÞ�AðsÞUðt,sÞPðsÞÞfðsÞ ds:





Using Eq. (3.3) it follows thatZ t

�1

Aðsþ tÞUðtþ t,sþ tÞPðsþ tÞðfðsþ tÞ�fðsÞÞ ds

��are:

Similarly, using Assumption (H.4), it follows thatZ t

�1

ðAðsþ tÞUðtþ t,sþ tÞPðsþ tÞ�AðsÞUðt,sÞPðsÞÞfðsÞ ds

��arek0JHJL1JfJ1,b

where JHJL1 ¼R10 HðsÞ dso1.

Therefore,

JXðfÞðtþ tÞ�XðfÞðtÞJaoð1þ k0JHJL1JfJ1,bÞe

for each t 2 R, and hence XðfÞ 2 APðXaÞ.Now, let T40. Again from Eq. (3.3), we have

1

2T

Z T

�T

JðXzÞðtÞJa dtr1

2T

Z T

�T

Z þ10

JAðsÞUðt,sÞPðsÞzðt�sÞJa ds dt

rk2kðaÞrða,bÞ

2 T

Z T

�T

Z þ10

e�ðd=4ÞsJzðt�sÞJbds dt

rk2kðaÞrða,bÞ:Z þ10

e�d=4s 1

2T

Z T

�T

Jzðt�sÞJbdt

� �ds:

Now

limT-1

1

2T

Z T

�T

Jzðt�sÞJb dt¼ 0,

as t-zðt�sÞ 2 PAP0ðXbÞ for every s 2 R. One completes the proof by using the Lebesgue’sdominated convergence theorem.

The proof for G2uð�Þ is similar to that of G1uð�Þ except that one makes use of Eq. (3.2)instead of Eq. (3.3). &

Theorem 3.7. Under Assumptions (H.1)–(H.5), the evolution equation (1.2) has a unique

pseudo-almost periodic mild solution whenever K is small enough.

Proof. Consider the nonlinear operator M defined on PAPðXaÞ by

MuðtÞ ¼ �f ðt,BðtÞuðtÞÞ�

Z t

�1

AðsÞUðt,sÞPðsÞf ðs,BðsÞuðsÞÞ ds

þ

Z 1t

AðsÞUQðt,sÞQðsÞf ðs,BðsÞuðsÞÞ dsþ

Z t

�1

Uðt,sÞPðsÞgðs,CðsÞuðsÞÞ ds

�

Z 1t

UQðt,sÞQðsÞgðs,CðsÞuðsÞÞ ds

for each t 2 R.As we have previously seen, for every u 2 PAPðXaÞ, f ð�,Bð�Þuð�ÞÞ 2 PAPðXbÞ �

PAPðXaÞ. In view of Lemmas 3.5 and 3.6, it follows that M maps PAPðXaÞ into itself.To complete the proof one has to show that M has a unique fixed-point.





Let v,w 2 PAPðXaÞ

JG1ðvÞðtÞ�G1ðwÞðtÞJa

rZ t

�1

JAðsÞUðt,sÞPðsÞ½f ðs,BðsÞvðsÞÞ�f ðs,BðsÞwðsÞÞ�Ja ds

rk2kðaÞrða,bÞZ t

�1

ðt�sÞ�ae�ðd=4Þðt�sÞJf ðs,BðsÞvðsÞÞ�f ðs,BðsÞwðsÞÞJb ds

rk2kðaÞrða,bÞKZ t

�1

ðt�sÞ�ae�ðd=4Þðt�sÞJBðsÞvðsÞ�BðsÞwðsÞJ ds

rk2kðaÞrða,bÞK$Z t

�1

ðt�sÞ�ae�ðd=4Þðt�sÞJvðsÞ�wðsÞJa ds

¼ k2kðaÞrða,bÞK$ð4d�1Þ1�aGð1�aÞ$Jv�wJa,1:

Now


rZ 1

t

JAðsÞUQðt,sÞQðsÞ½f ðs,BðsÞvðsÞÞ�f ðs,BðsÞwðsÞÞ�Ja ds

rkðaÞk2dðbÞZ 1

t

e�dðs�tÞJf ðs,BðsÞvðsÞÞ�f ðs,BðsÞwðsÞÞJb ds

rkðaÞk2dðbÞKZ þ1

t

e�dðs�tÞJBðsÞvðsÞ�BðsÞwðsÞJ ds

rkðaÞk2dðbÞK$Z þ1

t

edðt�sÞJvðsÞ�wðsÞJa ds

rkðaÞk2dðbÞK$d�1Jv�wJa,1:

Now for G3 and G4, we have the following approximations:


rZ t

�1

JUðt,sÞPðsÞ½gðs,CðsÞvðsÞÞ�gðs,CðsÞwðsÞÞ�Ja ds

rZ t

�1

cðaÞðt�sÞ�ae�ðd=2Þðt�sÞJgðs,CðsÞvðsÞÞ�gðs,CðsÞwðsÞÞJ ds

rKcðaÞZ t

�1

ðt�sÞ�ae�ðd=2Þðt�sÞJCðsÞvðsÞ�CðsÞwðsÞJ ds

r$KcðaÞZ t

�1

ðt�sÞ�ae�ðd=2Þðt�sÞJvðsÞ�wðsÞJa ds

rK$cðaÞð2d�1Þ1�a Gð1�aÞJv�wJa,1

and


rZ 1

t

JUQðt,sÞQðsÞ½gðs,CðsÞvðsÞÞ�gðs,CðsÞwðsÞÞ�Ja ds

rZ 1

t

mðaÞedðt�sÞJgðs,CðsÞvðsÞÞ�gðs,CðsÞwðsÞÞJ ds





rZ 1

t

mðaÞKedðt�sÞJCðsÞvðsÞ�CðsÞwðsÞJ ds

r$mðaÞKZ 1

t

edðt�sÞJvðsÞ�wðsÞJa ds

rK$mðaÞd�1Jv�wJa,1:

Combining previous inequalities it follows that

JMv�MwJa,1rKYJv�wJa,1,

where

Y :¼ $½mðaÞd�1 þ cðaÞð2d�1Þ1�aGð1�aÞ þ kðaÞk2dðbÞd�1

þk2kðaÞrða,bÞð4d�1Þ1�aGð1�aÞ�:

Therefore, if K is small enough, that is, KoY�1, then Eq. (1.2) has a unique solution,which obviously is its only mild pseudo-almost periodic solution. &

Example 3.8. Let O � RN (NZ1) be an open bounded subset with C2 boundary G¼ @Oand let X¼ L2ðOÞ equipped with its natural topology J � JL2ðOÞ.

Let 0oaobo1 with 2b4aþ 1. We study the existence of pseudo-almost periodicsolutions to the following boundary-value problem:

@

@t½jþ F ðt,bðt,xÞjÞ� ¼ aðt,xÞD2jþ Gðt,cðt,xÞjÞ in R� O

j¼ Dj¼ 0 on R� G,

8<: : ð3:4Þ

where the coefficients a,b,c : R� O-R are almost periodic, and F : R� L2ðOÞ-Xb, G :R� L2ðOÞ-L2ðOÞ are pseudo-almost periodic functions with

Xa ¼ ðL2ðOÞ,H2

0 ðOÞ \H4ðOÞÞa,1:

Define the linear operators A(t) appearing in Eq. (3.4) as follows:

AðtÞu¼ aðt,xÞD2u for all u 2 DðAðtÞÞ ¼H20 ðOÞ \H4ðOÞ,

where a : R� O-R, in addition of being almost periodic satisfies the followingassumptions:

(H.6)

Please

partia

inf t2R,x2Oaðt,xÞ ¼m040, and
(H.7) there exists L40 and 0omr1 such that
jaðt,xÞ�aðs,xÞjrLjs�tjm

for all t,s 2 R uniformly in x 2 O.

Clearly, A(t) is sectorial and invertible. Moreover it can be shown that the analyticsemigroup ðe�sAðtÞÞsZ0 associated with �AðtÞ is exponentially stable and hence hyperbolic.

Let I denote the identity operator of L2ðOÞ. Here we take BðtÞ ¼ bðt,xÞI andCðtÞ ¼ cðt,xÞI . Under previous assumptions, it is clear that the operators A(t) definedabove are invertible and satisfy Acquistapace–Terreni conditions. Moreover, Assumptions(H.2)–(H.4) are fulfilled.





In addition to the above, we require the following assumption:

(H.8)

Please

partia

Let F : R-X-Xb, G : R�X-X be pseudo-almost periodic in t 2 R uniformly inu 2 Xa. Moreover, the functions F ,G are globally Lipschitz with respect to thesecond argument in the following sense: there exists K 040 such that

JF ðt,jÞ�F ðt,cÞJbrK 0Jj�cJL2ðOÞ,

and

JGðt,jÞ�Gðt,cÞJL2ðOÞrK 0Jj�cJL2ðOÞ

for all j,c 2 L2ðOÞ and t 2 R.

We have

Theorem 3.9. Under Assumptions (H.6)–(H.8), then the boundary-value problem Eq. (3.4)has a unique solution j 2 PAPðL2ðOÞ,H2

0ðOÞ \H4ðOÞÞa,1Þ whenever K 0 is small enough.

Acknowledgments

The author would like thank Profs. Baroun and Maniar for their careful reading of themanuscript and insightful comments. Moreover, the author is thankful the referees fortheir careful reading of the manuscript and insightful comments.

References

[1] P. Acquistapace, Evolution operators and strong solutions of abstract linear parabolic equations, Differential

Integr. Equations 1 (1988) 433–457.

[3] P. Acquistapace, B. Terreni, A Unified approach to abstract linear nonautonomous parabolic equations,

Rend. Sem. Mat. Univ. Padova 78 (1987) 47–107.

[4] H. Amann, Linear and Quasilinear Parabolic Problems, Birkhauser, Berlin, 1995.

[6] B. Amir, L. Maniar, Composition of pseudo-almost periodic functions and Cauchy problems with operator

of nondense domain, Ann. Math. Blaise Pascal 6 (1) (1999) 1–11.

[7] M. Baroun, S. Boulite, T. Diagana, L. Maniar, Almost periodic solutions to some semilinear nonautonomous

thermoelastic plate equations, J. Math. Anal. Appl. 349 (1) (2009) 74–84.

[8] M. Baroun, S. Boulite, G.M. N’Gu�er�ekata, L. Maniar, Almost automorphy of semilinear parabolic

evolution equations, Electron. J. Differential Equation 2008 (60) (2008) 1–9.

[9] H. Brill, A semilinear Sobolev evolution equation in a Banach space, J. Differential Equation 24 (3) (1977)

412–425.

[10] A. Caicedo, C. Cuevas, G.M. Mophou, G.M. N’Gu�er�ekata, Asymptotic behavior of solutions of some

semilinear functional differential and integro-differential equations with infinite delay in Banach spaces,

J. Franklin Inst., in press, doi:10.1016/j.jfranklin.2011.02.001.

[12] C. Cuevas, M. Pinto, Existence and uniqueness of pseudo almost periodic solutions of semilinear Cauchy

problems with nondense domain, Nonlinear Anal. 45 (1) (2001) 73–83.

[15] T. Diagana, E. Hern�andez, M. Rabello, Pseudo almost periodic solutions to some nonautonomous

neutral functional differential equations with unbounded delay, Math. Comput. Model. 45 (9–10) (2007)

1241–1252.

[16] T. Diagana, Pseudo almost periodic solutions to some differential equations, Nonlinear Anal. 60 (7) (2005)

1277–1286.

[17] T. Diagana, Pseudo Almost Periodic Functions in Banach Spaces, Nova Science Publishers Inc., New York,

2007.



10.1016/j.jfranklin.2011.02.001



[18] T. Diagana, E. Hern�andez, Existence and uniqueness of pseudo almost periodic solutions to some abstract

partial neutral functional-differential equations and applications, J. Math. Anal. Appl. 327 (2) (2007)

776–791.

[19] T. Diagana, C.M. Mahop, G.M. N’Gu�er�ekata, Pseudo almost periodic solution to some semilinear

differential equations, Math. Comp. Model. 43 (1–2) (2006) 89–96.

[20] T. Diagana, C.M. Mahop, G.M. N’Gu�er�ekata, B. Toni, Existence and uniqueness of pseudo almost periodic

solutions to some classes of semilinear differential equations and applications, Nonlinear Anal. 64 (11) (2006)

2442–2453.

[21] T. Diagana, Existence and uniqueness of pseudo almost periodic solutions to some classes of partial

evolution equations, Nonlinear Anal. 66 (2) (2007) 384–395.

[22] T. Diagana, G.M. N’Gu�er�ekata, Pseudo almost periodic mild solutions to hyperbolic evolution equations in

abstract intermediate Banach spaces, Appl. Anal. 85 (6–7) (2006) 769–780.

[24] T. Diagana, An introduction to classical and p-adic theory of linear operators and applications, Nova

Science Publishers, New York, 2006.

[25] S.A. Dubey, Numerical solution for nonlocal Sobolev-type differential equations, Electron. J. Differential

Equations 19 (2010) 75–83 (Conference).

[26] K.J. Engel, R. Nagel, One parameter semigroups for linear evolution equations, Graduate Texts in

Mathematics, Springer Verlag, 1999.

[33] H.X. Li, F.L. Huang, J.Y. Li, Composition of pseudo almost-periodic functions and semilinear differential

equations, J. Math. Anal. Appl. 255 (2) (2001) 436–446.

[34] J.H. Lightbourne III, S.M. Rankin III, A Partial functional differential equation of Sobolev-type, J. Math.

Anal. Appl. 93 (1983) 328–337.

[36] A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems, PNLDE, vol. 16, Birkhauser

Verlag, Basel, 1995.

[37] L. Maniar, R. Schnaubelt, Almost periodicity of inhomogeneous parabolic evolution equations, Lecture

Notes in Pure and Applied Mathematics, vol. 234, Dekker, New York, 2003, pp. 299–318.

[43] R. Schnaubelt, Asymptotic behavior of parabolic nonautonomous evolution equations, in: M. Iannelli,

R. Nagel, S. Piazzera (Eds.), Functional Analytic Methods for Evolution Equations, Lecture Notes in

Mathematics, vol. 1855, Springer-Verlag, Berlin, 2004, pp. 401–472.

[44] A. Yagi, Parabolic equations in which the coefficients are generators of infinitely differentiable semigroups II,

Funkcial. Ekvac. 33 (1990) 139–150.

[45] A. Yagi, Abstract quasilinear evolution equations of parabolic type in Banach spaces, Boll. Un. Mat. Ital. B

5 (7) (1991) 341–368.

[47] C.Y. Zhang, Pseudo almost periodic solutions of some differential equations, J. Math. Anal. Appl. 181 (1)

(1994) 62–76.

[48] C.Y. Zhang, Pseudo almost periodic solutions of some differential equations. II, J. Math. Anal. Appl. 192 (2)

(1995) 543–561.

[49] C.Y. Zhang, Integration of vector-valued pseudo almost periodic functions, Proc. Amer. Math. Soc. 121 (1)

(1994) 167–174.




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