-ALMOST PERIODIC AND S

Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 22 (2021), No. 1, pp. 211–244 DOI: 10.18514/MMN.2021.3050

S

^p

-ALMOST PERIODIC AND S

^p

-ALMOST AUTOMORPHIC SOLUTIONS OF AN INTEGRAL EQUATION WITH INFINITE

DELAY

SALSABIL HAJJAJI AND FAROUK CH ´ERIF Received 26 September, 2019

Abstract. We state sufficient conditions for the existence and uniqueness of Stepanov-like pseudo almost periodic and Stepanov-like pseudo almost automorphic solutions for a class of nonlinear Volterra integral with infinite delay of the form

x(t) =f(t,x(t),x(t−r(t)))− Z +∞

t

c(t,s)g(s,x(s),x(s−r(s)))ds.

Our approach is based on Bochner’s transform, some analytic techniques, and a Banach fixed point theorem. Then we apply these results to a nonlinear differential equation when the delay is time-dependent and the force function is continuous

x⁰(t) =ax(t) +αx⁰(t−r(t))−q(t,x(t),x(t−r(t))) +h(t).

2010Mathematics Subject Classification: 34A12; 34C27; 43A60

Keywords: integral equation, almost periodic, almost automorphic, pseudo almost periodic, pseudo almost automorphic, infinite delay

1. INTRODUCTION

The existence and stability of almost periodic solutions of some models are among the most attractive topics in the qualitative theory of differential and integral equa- tions due to their applications in physical science, mathematical biology, population growth... Hence, in the literature, several studies have been conducted on Bohr’s almost periodicity and Bochner’s almost automorphic to establish sufficient condi- tions for the existence and uniqueness of various types of differential and integral equations. For instance, one can see [1–3,6,20] and the references therein. In partic- ular, it can be noted that several qualitative studies of various differential and integral equations have been carried out in recently published articles [7,17–19,25]. The no- tion of pseudo almost periodicity functions which is the central issue in this paper is a new concept introduced a few years ago by Zhang [27] as a generalization of the classical notion of Bohr’s almost periodicity. Also, the notion of almost automorphic (Stepanov) was then defined firstly by N’G u´er´ekata and Pankov [24] as an extension

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of the classical and well-known almost automorphic concept. It should be mentioned that the study of the existence of almost periodic solutions of the integral equation with a discrete delay was initiated in [16], where Fink and Gatica established the existence of a positive almost periodic solution to the following equation

x(t) = Z t

t−τf(s,x(s))ds, (1.1)

which arises in models for the spread of epidemics. Since then, many works related to the sufficient conditions on the delay and the function f in order to establish the existence of almost periodic solutions to equation (1.1).

In 1997, Ait Dads and Ezzinbi [11] studied the existence of positive almost peri- odic solutions for the following neutral integral equation

x(t) =γx(t−τ) + (1−γ) Z t

t−τ

f(s,x(s))ds. (1.2)

Later, Ait Dads et al. [10] established the existence of positive pseudo almost periodic solutions in the case of infinite delay for the equation

x(t) = Z t

−∞a(t,t−s)f(s,x(s))ds, t∈R. (1.3) Afterwards, Ding et al. [15] developed the above results to the following integral equation with neutral delay

x(t) =αx(t−β) + Z t

−∞a(t,t−s)f(s,x(s))ds+h(t,x(t)), t∈R. (1.4) Equations similar to (1.4) arise in the study of [28] where the authors established the existence and uniqueness of almost periodic and pseudo almost periodic solutions of the integral equation given by

x(t) =α(t)x(t−σ(t)) + Z t

−∞β(t,t−s)f(s,x(s),x⁰(s))ds, t∈R, (1.5) where σ(t) is almost periodic (respectively pseudo almost periodic). Recently, the authors in [12] consider two variants of Eq. (1.5), a variant where the delay σ(t) is compact almost automorphic in time and another variant where the delay is state- dependent. Also, the existence and uniqueness of periodic solutions of a more general model were established via three fixed point theorems by Islam [21].

Hence, one of the still topical subjects in the study of integral equations and/or differential equations is that if the force functions and/or the coefficients possess a specific property, are we going to find the same characteristics in the solution?

Roughly speaking, if the considered functions are Stepanov-like pseudo almost peri- odic, will the expected solutions of the differential or integral equation be of the same type? The aim of this work is to study the existence and uniqueness of Stepanov-like

S -ALMOST PERIODIC ANDS -ALMOST AUTOMORPHIC SOLUTIONS 213

pseudo almost periodic and Stepanov-like pseudo almost automorphic solutions for the following integral equation

x(t) = f(t,x(t),x(t−r(t)))− Z +∞

t

c(t,s)g(s,x(s),x(s−r(s)))ds, (1.6) where f :R×R×R−→R, g:R×R×R−→Randc:R−→R are continuous functions, r(·) is a time-dependent delay. To the best of our knowledge, there are no papers published on the S^p-pseudo almost periodic solutions and/or S^p-pseudo almost automorphic solutions of this class of Volterra equation.

Our main contributions in this paper are:

(1) The existence and uniqueness of Stepanov-like pseudo almost periodic solu- tion for system (1.6) are proved.

(2) A new proof for the composition theorem in the spacePAPS^pis given based mainly on the Banach’s transform.

(3) The existence and uniqueness of Stepanov-like pseudo almost automorphic solution for system (1.6) are proved.

(4) The existence and uniqueness of Stepanov-like pseudo almost periodic solu- tions and Stepanov-like pseudo almost automorphic of a class of logistic dif- ferential equation are established.

The organization of this work is as follows. In Section2, we present some defin- itions and lemmas that will be used later. In Section 3, we state our main results.

More precisely, we give sufficient conditions for the existence and the uniqueness ofS^p-pseudo almost automorphic andS^p-pseudo almost periodic solutions of the in- tegral equation (1.6). Our approach is based mainly on Bochner’s transform, using analytic techniques and Banach’s fixed point theorem. Finally, in Section4, we study the validity of our theoretical result, therefore we give an illustrating application. It should be mentioned that the main results of this paper include Theorems1,2,3and 4.

2. PRELIMINARIES: SPACES OF FUNCTIONS

Throughout this article(E,k · k_E)and(F,k · k_F)denote Banach spaces and

C

⁽E,F) the Banach space of continuous functions fromEtoF. We denote byBC(R,E)the Banach space of bounded and continuous defined functions onRwith the sup norm defined by

kfk=sup

t∈R

kf(t)k. (2.1)

Definition 1([4]). A setDof real numbers is said to be relatively dense if there exists a number` >0 such that any interval of length`contains at least one number ofD.

Definition 2([4]). A function f ∈C(R,E)is called (Bohr) almost periodic if for eachε>0 the setT(f,ε) ={τ:f(t+τ)−f(t)}is relatively dense, i.e. for anyε>0

there existsl=l(ε)>0 such that every interval of lengthlcontains a numberτwith the property that

kf(t+τ)−f(t)k<ε, t∈R. The collection of all such functions will be denoted byAP(R,E).

Definition 3([26]). A function f ∈C(R×E,F)is called (Bohr) almost periodic int∈Runiformly iny∈K where K⊂Eis any compact subset if for each ε>0 there existsl=l(ε)>0 such that every interval of lengthlcontains a numberτwith the property that

kf(t+τ,y)−f(t,y)k<ε, t∈R,y∈K.

The collection of such functions will be denoted byAP(R×E,F).

Lemma 1 ([5]). Let f ∈ AP(R×E,F) and φ∈ AP(R,E) then the function [t7−→F(t,φ(t))]∈AP(R,F).

Definition 4([27]). A continuous function f :R−→E is called pseudo almost periodic if it can be written as f =h+φwhereh∈AP(R,E)andφ∈PAP₀(R,E) where the spacePAP₀(R,E)is defined by

PAP₀(R,E) =

f∈BC(R,E),

M

^{(kfk) = lim}

T−→∞

1 2T

Z T

−T

kf(t)kdt=0

. The functionshandφin above definition are respectively called the almost periodic components and the ergodic perturbation of the pseudo-almost periodic function f. The collection of all pseudo almost periodic functions which map fromRtoEwill be denoted byPAP(R,E).

Definition 5([13]). The Bochner transform f^b(t,s)witht∈R,s∈[0,1]of a func- tion f :R7→Eis defined by f^b(t,s):= f(t+s).

Definition 6([13]). The Bochner transformF^b:R×[0,1]×E7→Eof a function F :R×E7→Eis defined byF^b(t,s,u):=F(t+s,u) for eacht∈R, s∈[0,1],and u∈E.

Definition 7 ([13]). Let p∈[1,∞[. The space BS^p(R,E) of all Stepanov-like bounded functions, with exponent p, consists of all measurable functions f :R7→E such that f^b∈L^∞(R,L^p((0,1),E)). This is a Banach space with the norm

kfk_BSp(R,E):=kf^bk_L∞(R,L^p)=sup

t∈R

^Z t+1 t

kf(τ)k^pdτ 1/p

.

Definition 8([14]). A function f∈BS^p(R,E)is called Stepanov-like almost peri- odic if f^b∈AP(R,L^p((0,1),E)). The collection of these functions will be denoted byAPS^p(R,E).

S -ALMOST PERIODIC ANDS -ALMOST AUTOMORPHIC SOLUTIONS 215

Definition 9 ([14]). A function f :R×E→ F,(t,u) 7→ f(t,u) with f(·,u) ∈ BS^p(R,), for eachu∈E, is called Stepanov almost periodic function int∈Runi- formly foru∈Eif for eachε>0 and each compact setK⊂Ethere exists a relatively dense setP=P(ε,f,K)⊂Rsuch that

sup

t∈R

^Z 1 0

kf(t+s+τ,u)−f(t+s,u)kds1/p

<ε, (2.2) for eachτ∈P,u∈K. We denote byAPS^p(R×E,F)the set of such functions.

Definition 10([13]). Let p≥1. A function f ∈BS^p(R,E) is called S^p-pseudo almost periodic (or Stepanov-like pseudo almost periodic) if it can be expressed as

f =h+φ, (2.3)

whereh^b∈AP(L^p((0,1),E))andφ^b∈PAP₀(L^p((0,1),E)). In other words, a func- tion f ∈L^p(R,E) is said to beS^p-pseudo almost periodic if its Bochner transform f^b :R−→L^p((0,1),E) is pseudo almost periodic in the sense that there exist two functionsh,φ:R→Esuch that f =h+φ, where h^b∈AP(L^p((0,1),E))andφ^b∈ PAP₀(L^p((0,1),E))that is,

T→∞lim 1 2T

Z T

−T

_{Z t+1}

t

kϕ(σ)k^pdσ ¹_p

dt=0. (2.4)

The collection of such functions will be denoted byPAPS^p(R,E).

Definition 11([23]). A continuous function f :R−→Eis almost automorphic if for every sequence of real numbers(sn)n∈Nthere exists a subsequence(sn)n∈Nsuch that

g(t) = lim

n→+∞f(t+sn) (2.5)

is well defined for eacht∈R, and

n→+∞lim g(t−s_n) = f(t) (2.6) for eacht∈R. The collection of all almost automorphic functions which map from RtoEis denoted byAA(R,E).

Definition 12 ([5]). A function f :R×E−→E (t,x)7−→ f(t,x) is said to be almost automorphic int∈Rfor eachu∈Ewhen it satisfies the two following con- ditions:

(1) For allx∈E,the function f(·,x)∈AA(R,E).

(2) For all subset compactKofE, for allε>0 there existsδ=δ(k,ε)>0 such that, for allx,z∈k, ifd(x,z)≤δthen we haved(f(x,t),f(z,t))≤εfor all t∈R.

The collection of such functions will be denoted byAA(E×R,F).

Lemma 2([5]). Let f ∈AA(R×E,F)and u∈AA(R,E), then we have [t7−→ f(t,u(t))]∈AA(R,F).

Lemma 3([9]). If the functions x(·)∈PAPS^p(R,R)and r(·)∈APS^p(R,R)then we have x(· −r(·))∈PAPS^p(R,R).

Definition 13([14]). A functionf ∈BS^p(R,E)is calledS^p-almost automorphic if f^b ∈ AA(L^p((0,1),E)). The collection of such functions will be denoted by AAS^p(R,E).

Definition 14([14]). A function f∈BS^p(R×E,F),(t,u)7→F(t,u)whereF(·,u)

∈L^p(R,E) for each u∈E, is called S^p-pseudo almost automorphic int∈Runi- formly inu∈Eift7→F(t,u)isS^p-pseudo automorphic for eachu∈KwhereK⊂E is a bounded subset. The collection of such functions will be denoted by PAAS^p (R×E,F).

3. MAIN RESULTS

3.1. Stepanov-like pseudo almost periodic solutions In this section, we consider the following integral equation

x(t) = f(t,x(t),x(t−r(t)))− Z +∞

t

c(t,s)g(s,x(s),x(s−r(s)))ds, (3.1) where f :R×R×R→R,g:R×R×R→R,c,r:R→Rare continuous functions.

We give sufficient conditions which guarantee the existence ofS^p-pseudo almost peri- odic solutions for equation (3.1).

(H1) f :R×R²−→R is S^p-pseudo almost periodic, i.e. f^b =h^b+φ^b, where h^b ∈AP(R×R²,L^p((0,1),R)) and φ^b ∈PAP₀(R×R²,L^p((0,1),R)) such that

T→+∞lim 1 2T

Z T

−T

_{Z t+1}

t

|φ(σ,u)|^pdσ ¹_p

dt=0, (3.2)

uniformly inu∈R².

(H2) f is Lipschitz i.e.∃L¹_f,L²_f >0 such that∀x₁,x₂,y₁,y₂∈R,

|f(t,x₁,x₂)−f(t,y₁,y₂)| ≤L¹_f|x₁−y₁|+L²_f|x₂−y₂|. (3.3) (H3) g:R×R²−→ R is S^p-pseudo almost periodic, i.e. g^b =g^b₁+g^b₂, where g^b₁ ∈AP(R×R²,L^p((0,1),R)) and g^b₂ ∈PAP₀(R×R²,L^p((0,1),R)) such that

Tlim→+∞

1 2T

Z T

−T

_{Z t+1}

t

|g₂(σ,u)|^pdσ ¹_p

dt=0, (3.4)

uniformly for allu∈R².

S -ALMOST PERIODIC ANDS -ALMOST AUTOMORPHIC SOLUTIONS 217

(H4) gis Lipschitz i.e.∃L¹_g,L²_g>0 such that∀x₁,x₂,y₁,y₂∈R

|g(t,x₁,x2)−g(t,y1,y2)| ≤L_g¹|x₁−y₁|+L²_g|x₂−y₂|. (3.5) (H5) There exists a constantλ>0 such thatc(t,s)≤e^λ(t−s),for alls≥t.

(H6) The functiont7→r(t)∈APS^p(R,R)∩C¹(R,R)with

0≤r(t)≤r,r(t)≤r^∗<1, for all t∈R. (3.6) Lemma 4. Assume that (H1)-(H3) hold, if x(·)∈PAPS^p(R,R), then the function β:R−→Rdefined byβ(·) = f(·,x(·),x(· −r(·)))belongs to PAPS^p(R,R).

Proof. Let f =h+φwhereh^b∈AP(R×R²,L^p((0,1),R))and the functionφ^b∈ PAP₀(R×R²,L^p((0,1),R)).Similarly, let x^b(·) =x^b₁(·) +x^b₂(·) where the function x^b₁∈AP(R,L^p((0,1),R))andx^b₂∈PAP₀(R,L^p((0,1),R))that is

T→+∞lim 1 2T

Z T

−T

_{Z t+1}

t

|x₂(σ)|^pdσ ¹_p

dt=0, (3.7)

for allt∈R. By Lemma3we getx(· −r(·))∈PAPS^p(R,R), then

x^b(· −r(·)) =x^b₁(· −r^b(·)) +x^b₂(· −r^b(·)), (3.8) wherex^b₁(· −r^b(·))∈AP(R,L^p((0,1),R))andx^b₂(· −r^b(·))∈PAP₀(R,L^p((0,1),R)) that is

T→+∞lim 1 2T

Z T

−T

_{Z t+1}

t

|x₂(σ−r(σ))|^pdσ ¹_p

dt=0. (3.9)

Since f^b:R−→L^p((0,1),R)we decompose f^bas follows f^b(·,x^b(·),x^b(· −r(t)))

=h^b(·,x^b₁(·),x^b₁(· −r(t))) +f^b(·,x^b(·),x^b(· −r(t)))−h^b(·,x^b₁(·),x^b₁(· −r(t)))

=h^b(·,x^b₁(·),x^b₁(· −r(t))) +f^b(·,x^b(·),x^b(· −r(t)))−f^b(·,x^b₁(·),x^b₁(· −r(t))) +φ^b(·,x^b₁(·),x^b₁(· −r(t))).

Let us prove thath^b(·,x^b₁(·),x₁^b(· −r(·)))∈AP(R,L^p((0,1),R)). First, the function x^b₁(·)∈AP(R,L^p((0,1),R))andx^b₁(· −r^b(·))∈AP(R,L^p((0,1),R)).Then the func- tion u^b₁(·) = (x^b₁(·),x^b₁(· −r(t)))∈AP(R,L^p((0,1),R²)). Indeed, ∀ε>0,∃` >0,

∀a∈R,∃τ∈[a,a+`]such that sup

t∈R

_{Z 1}

0

u^b₁(t+τ)−u^b₁(t)

p

∞ds ¹_p

=sup

t∈R

_{Z 1}

0

max

|x^b₁(t+τ)−x^b₁(t)|,|x^b₁(t+τ−r^b(t))−x₁^b(t−r^b(t))|p

ds ¹_p

≤ε.

Since the functionh^b∈AP(R×R²,L^p((0,1),R))andu^b₁(·)∈AP(R,L^p((0,1),R²)) then, we can apply the composition theorem of almost periodic functions 1, thus h^b(·,u^b₁(·))∈AP(R,L^p((0,1),R)).Now, set

G^b(·) =f^b(·,x^b(·),x^b(· −r^b(·)))−f^b(·,x^b₁(·),x^b₁(· −r^b(·))). (3.10) G^b(·)∈PAP₀(R,(L^p((0,1),R)). Indeed, letT >0,we have

T→+∞lim 1 2T

Z T

−T

_{Z 1}

0

G^b(σ)

pdσ ¹_p

dt

≤ lim

T→+∞

L¹_f 2T

Z T

−T

_{Z 1}

0

x^b(σ)−x^b₁(σ)

pdσ ¹_p

dt + lim

T→+∞

L²_f 2T

Z T

−T

_{Z 1}

0

x^b(σ−r^b(σ))−x^b₁(σ−r^b(σ))

pdσ ¹_p

dt

≤ lim

T→+∞

L¹_f 2T

Z T

−T

_{Z 1}

0

x^b₂(σ)

pdσ ¹_p

dt + lim

T→+∞

L²_f 2T

Z T

−T

_{Z 1}

0

x^b₂(σ−r^b(σ))

pdσ ¹_p

dt.

Using (3.7) and (3.9) we get 1 2T

Z T

−T

_{Z 1}

0

G^b₂(σ)

pdσ ¹_p

dt=0.

Moreover, using the composition theorem of ergodic functions (cf. [22]) we have φ^b(·,u^b₁(·))∈PAP₀(R,L^p((0,1),R))such that

T→∞lim 1 2T

Z T

−T

_{Z 1}

0

φ^b(σ,u^b₁(σ))

pdσ ¹_p

dt=0. (3.11)

Lemma 5. Assume that (H3)-(H5) hold. If x(·)∈PAPS^p(R,R), then the function Θ:t7−→

Z +∞

t

c(t,s)g(s,x(s),x(s−r(s)))ds∈PAPS^p(R,R)for all s∈R.

Proof. Using Lemma 4 and the hypothesis (H5), we obtain that the integral is convergent and consequentlyt7−→

Z +∞

t

c(t,s)g(s,x(s),x(s−r(s)))dsis well defined.

Otherwise, since[s7−→g(s,x(s),x(s−r(s))]∈PAPS^p(R,R), one can write

g=g₁+g₂ (3.12)

with g₁∈APS^p(R,R) i.e. for each ε⁰>0 , there exists` >0 such that every in- terval of length`contains aτ such thatkg₁(t+τ)−g₁(t)k_S^p <ε⁰ and the function

S -ALMOST PERIODIC ANDS -ALMOST AUTOMORPHIC SOLUTIONS 219

g₂∈PAP₀(R,L^p((t,t+1),R))such that

T→+∞lim 1 2T

Z T

−T

_{Z t+1}

t

|g₂(σ)|^pdσ ¹_p

dt=0. (3.13)

Then

Θ(t) = Z +∞

t

c(t,s)g₁(s)ds+ Z +∞

t

c(t,s)g₂(s)ds=Θ1(t) +Θ2(t).

Now, we shall study theS^p-almost periodicity ofΘ₁(·). We have

|Θ₁(t+τ)−Θ₁(t)| ≤

Z +∞

t+τ

e^λ(t+τ−s)g₁(s)ds− Z +∞

t

e^λ(t−s)g₁(s)ds

=

Z +∞

t

e^λ(t−ξ)g₁(ξ+τ)dξ− Z +∞

t

e^λ(t−s)g₂(s)ds

≤ Z +∞

t

e^λ(t−s)|g₁(s+τ)−g₁(s)|ds.

According to H¨older inequality 1

p+1 q =1

one has for allτ∈R,

|Θ₁(t+τ)−Θ₁(t)| ≤ Z +∞

0

e^−λs|g₁(s+t+τ)−g₁(s+t)|ds

≤ 2

λq

¹_{qZ +∞}

0

e^−λps2 |g₁(s+t+τ)−g₁(s+t)|^pds ¹_p

.

Using Fubini’s theorem, we get for allτ∈R sup

x∈R

_{Z x+1}

x

|Θ₁(t+τ)−Θ1(t)|^pdt ¹_p

≤ 2

λq ¹_q

sup

x∈R

_{Z x+1}

x

Z +∞

0

e^−λps2 |g₁(s+t+τ)−g₁(s+t)|^pdsdt ¹_p

≤ 2

λq

¹_{qZ +∞}

0

e^−λps2 sup

x∈R

Z x+1 x

|g₁(s+t+τ)−g₁(s+t)|^pdtds ¹_p

.

Asg₁isS^palmost periodic, forε⁰=Cε>0 we have sup

x∈R

_{Z x+1}

x

|g₁(t+ +s+τ)−g₁(t+s)|^pdt ¹_p

<ε⁰=Cε, (3.14)

whereC= λp

2 ¹_p

λq 2

¹_q

. Then sup

x∈R

_{Z x+1}

x

|Θ₁(t+τ)−Θ1(t)|^pdt ¹_p

≤ε. This proves theS^p-almost periodicity ofΘ₁. Now let’s show the ergodicity ofΘ₂(·). Since

Θ₂(·)∈BC(R,R),it remains to show that

T→+∞lim 1 2T

Z T

−T

_{Z t+1}

t

|Θ₂(σ)|^pdσ ¹_p

dt=0. (3.15)

Letq∈[1,∞[such that 1 p+1

q=1 then, by H¨older’s inequality and Fubini’s theorem we obtain

Z T

−T

_{Z t+1}

t

|Θ₂(σ)|^pdσ ¹_p

dt

≤(2T)^1q _{Z T}

−T

_{Z t+1}

t

|Θ₂(σ)|^pdσ

dt ¹_p

≤ |Θ₂|^1q

∞(2T) 1

2T Z T

−T

_{Z t+1}

t

Z +∞

σ

e^λ(σ−s)g₂(s)ds

dσ

dt ¹_p

,

which gives

T→+∞lim 1 2T

Z T

−T

_{Z t+1}

t

|Θ2(σ)|^pdσ ¹_p

dt

≤ lim

T→+∞|Θ₂|^1q

∞

1 2T

Z T

−T

_{Z t+1}

t

Z +∞

σ

e^λ(σ−s)g₂(s)ds

dσ

dt ¹_p

. On the other hand, we get

T→+∞lim 1 2T

Z T

−T

_{Z t+1}

t

Z +∞

σ

e^λ(σ−s)g₂(s)ds

dσ

dt≤I+J, (3.16) with

I= lim

T→+∞

1 2T

Z T

−T

_{Z t+1}

t

Z T σ

e^λ(σ−s)g₂(s)ds

dσ

dt (3.17)

and

J= lim

T→+∞

1 2T

Z T

−T

_{Z t+1}

t

Z +∞

T

e^λ(σ−s)g₂(s)ds

dσ

dt. (3.18)

Further

I≤ lim

T→+∞

1 2T

Z T

−T

_{Z t+1}

t

e^λσ Z T

σ

e^−λs|g₂(s)|dsdσ

dt.

Now, we have[s7−→e^−λs]and[s7−→ |g₂(s)|]are two continuous functions on[σ,T], furthermore[s7−→ |g₂(s)|]keep a constant sign, so∃ξ∈]σ,T[such that,

I= lim

T→+∞

1 2T

Z T

−T

_{Z t+1}

t

e^λσ|g₂(ξ)|

Z T σ

e^−λsds

dσ

dt

= lim

T→+∞

1 2Tλ

Z T

−T

_{Z t+1}

t

|g₂(ξ)|h

1−e^{−λ(T−σ)} i

dσ

dt.

S -ALMOST PERIODIC ANDS -ALMOST AUTOMORPHIC SOLUTIONS 221

Now as 1−e^{−λ(T−σ)}≤1,one hasI ≤ lim

T→+∞

1 2Tλ

Z T

−T

_{Z t+1}

t

|g₂(ξ)|dσ

dt.Since ξ∈]σ,T[, then ξ= (1−α)σ+αT, whereα∈]0,1[.Hence,

I= lim

T→+∞

1 2Tλ

Z T

−T

_{Z t+1}

t

(|g₂((1−α)σ+αT)|)dσ

dt. (3.19) On the other hand, sincet≤σ≤t+1, we get

(1−α)t+αT ≤(1−α)σ+αT ≤(1−α)t+ (1−α) +αT.

Besides, 1−α<1,thus

(1−α)t+αT ≤(1−α)σ+αT ≤(1−α)t+αT+1.

Setz= (1−α)σ+αT andu= (1−α)σ+αT.We obtain, I≤ lim

T→+∞

1 2Tλ

Z T

−T

_{Z z+1}

z

|g₂(u)|du

dt. (3.20)

According to the hypothesisg₂∈PAP₀(R,L^p((t,t+1),R))we conclude thatI=0.

Meanwhile, by applying Fubini’s theorem we obtain J= lim

T→+∞

1 2T

Z T

−T

_{Z t+1}

t

Z +∞

T

e^λ(α−s)g₂(s)ds

dα

dt

= lim

T→+∞

1 2Tλ

Z T

−T

_{Z +∞}

T

|g₂(s)|e^−sλh

e^λ(t+1)−e^λt i

ds

dt.

Thus,J≤J₁+J₂with J₁= lim

T→+∞

e^λ 2Tλ

Z T

−T

Z +∞

T

|g₂(s)|e^λ(t−s)dsdt, (3.21) and

J₂= lim

T→+∞

1 2Tλ

Z T

−T

Z +∞

T

|g₂(s)|e^λ(t−s)dsdt. (3.22) Letξ=s−t,then

J₁≤ lim

T→+∞

e^λ|g₂|_∞ 2Tλ

Z T

−T

Z +∞

T−t e^−λξdξdt= lim

T→+∞

e^λ|g₂|_∞ 2Tλ³

h

1−e^−2λT i

=0.

Similarly, it is easy to see thatJ₂=0 which implies thatJ=0. Then we have

T→+∞lim |Θ₂|^1q

∞

1 2T

Z T

−T

_{Z t+1}

t

Z +∞

σ

e^λ(σ−s)g₂(s)ds

dσ

dt ¹_p

=0. (3.23) Consequently,

T→+∞lim 1 2T

Z T

−T

_{Z t+1}

t

|Θ2(σ)|^pdσ ¹_p

dt=0. (3.24)

Therefore the function Θ : t 7−→

Z +∞

t

c(t,s)g(s,x(s),x(s−r(s)))ds belongs to

PAPS^p(R,R).

Now, we are able to establish the existence and uniqueness of the Stepanov-like pseudo almost periodic solutions of (3.1).

Theorem 1. We assume (H1)-(H5) hold. If m<1then, (3.1) has a unique S^p- pseudo almost periodic solution with

m=max L¹_f,L²_f(1−r^∗)^−1p,L¹_g 2

λq ¹_q

2 λp

¹_p ,L²_g

2 λq

¹_q 2 λp

¹_p

(1−r^∗)^−1p

! .

Proof. Define the operator onPAPS^p(R,R)by Γ(x)(t) =f(t,x(t),x(t−r(t)))−

Z +∞

t

c(t,s)g(s,x(s),x(s−r(s)))ds, t∈R. (3.25) Using Lemma3and (H1) we get that the functiont7→ f(t,x(t),x(t−r(t)))is con- tinuous. Furthermore, by Lemma4and the hypothesis (H5) we get that the integral defined byt7−→

Z +∞

t

c(t,s)g(s,x(s),x(s−r(s)))dsexists. Thus,Γxis well defined.

Moreover, from Lemmas4and5we deduce that

Γ:PAPS^p(R,R)−→PAPS^p(R,R).

Letx,y∈PAPS^p(R,R), according to H¨older’s inequality 1

p+1 q=1

, we get

|Γx(t)−Γy(t)| ≤ |f(t,x(t),x(t−r(t)))−f(t,y(t),y(t−r(t)))|

+

Z +∞

t

c(t,s)

g(s,y(s),y(s−r(s)))ds−g(s,x(s),x(s−r(s))) ds

=|f(t,x(t),x(t−r(t)))−f(t,y(t),y(t−r(t)))|

+ 2

qλ

¹_{qZ +∞}

0

e^−λps2 |g(s+t,y(s+t),y(s+t−r(t)))

−g(s+t,x(s+t),x(s+t−r(t)))|^pds ¹_p

.

Then using Fubini’s theorem and Minkowski’s inequality, we get sup

ξ∈R

_{Z ξ+1}

ξ

|Γx(t)−Γy(t)|^pdt ¹_p

≤sup

ξ∈R

_{Z ξ+1}

ξ

L¹_f|x(t)−y(t)|+L²_f|x(t−r(t))−y(t−r(t))|^p dt

¹_p

S -ALMOST PERIODIC ANDS -ALMOST AUTOMORPHIC SOLUTIONS 223

+ 2

λq

¹_{qZ ∞}

0

e^−λps2 sup

ξ∈R

Z ξ+1 ξ

L¹_g|y(t+s)−x(t+s)|

+L²_g|y(t+s−r(s))−x(t+s−r(s))|p

dtds ¹_p

≤sup

ξ∈R

_{Z ξ+1}

ξ

L¹_f|x(t)−y(t)|+L²_f|x(t−r(t))−y(t−r(t))|p

dt ¹_p

+ 2

λq ¹_q

× _{Z ∞}

0

e^−λps2 sup

ξ⁰∈R

Z ξ⁰+1 ξ⁰

L¹_g|y(t⁰)−x(t⁰)|

+L²_g|y(t⁰−r(t))−x(t⁰−r(t))|p

dt⁰ds ¹_p

≤sup

ξ∈R

_{Z ξ+1}

ξ

L¹_f|x(t)−y(t)|+L²_f|x(t−r(t))−y(t−r(t))|p

dt ¹_p

+ 2

λq ¹_q

2 λp

¹_p sup

ξ⁰∈R

Z ξ⁰+1 ξ⁰

L¹_g|y(t⁰)−x(t⁰)|

+L²_g|y(t⁰−r(t⁰))−x(t⁰−r(t⁰))|^p dt⁰

¹_p

≤L¹_fkx−yk_S^p+L¹_g 2

λq ¹_q

2 λp

¹_p

kx−yk_S^p

+L²_f 1−r⁰(t)−¹_p

sup

ξ∈R

_{Z ξ+1−r(ξ+1)}

ξ−r(ξ)

|x(ρ)−y(ρ)|^pdρ ¹_p

+L²_g 2

λq ¹_q

2 λp

¹_p

1−r⁰(t)−¹_p

×sup

ξ∈R

_{Z ξ+1−r(ξ+1)}

ξ−r(ξ)

|x(ρ)−y(ρ)|^pdρ ¹_p

≤L¹_fkx−yk_S^p+L²_f(1−r^∗)^−1psup

ξ∈R

_{Z ξ+1−r}

ξ−r

|x(ρ)−y(ρ)|^pdρ ¹_p

+L¹_g 2

λq ¹_q

2 λp

¹_p

kx−yk_S^p

+L²_g 2

λq ¹_q

2 λp

¹_p

(1−r^∗)^−1psup

ξ∈R

_{Z ξ+1−r}

ξ−r

|x(ρ)−y(ρ)|^pdρ ¹_p

≤mkx−yk_S^p,

where

m=max L¹_f,L²_f(1−r^∗)^−1p,L¹_g 2

λq ¹_q

2 λp

¹_p ,L²_g

2 λq

¹_q 2 λp

¹_p

(1−r^∗)^−1p

! . Since m<1, the operator Γ:(PAPS^p(R,R),k · k_S^p)−→(PAPS^p(R,R),k · k_S^p) is a contraction. Therefore, by applying the Banach fixed point theorem, there is a unique x_∗∈PAPS^p(R,R)such thatΓ(x∗) =x_∗, which corresponds to the uniqueS^p-almost

periodic pseudo solution of equation (3.1).

3.2. Stepanov like (pseudo) almost automorphic solutions

In this section, we establish the existence of pseudo almost automorphic solutions of equation (3.1). For this study, we make the following assumptions:

(H1) f :R×R²−→RisS^p-pseudo almost automorphic, i.e. f^b=h^b+φ^b,where the function h^b ∈ AA(R×R²,L^p((0,1),R)) and the function φ^b ∈PAP₀ ((R×R²,L^p((0,1),R))such that

T→+∞lim 1 2T

Z T

−T

_{Z t+1}

t

|φ(σ,u)|^pdσ ¹_p

dt=0, (3.26)

uniformly for allu∈R².

(H2) f is a Lipschitz function, i.e.∃L¹_f,L²_f >0 such that∀x₁,x₂,y₁,y₂∈R,

|f(t,x₁,x₂)−f(t,y₁,y₂)| ≤L¹_f|x₁−y₁|+L²_f|x₂−y₂|. (3.27) (H3) g:R×R²−→RisS^p-pseudo almost automorphic, i.e. g^b=g^b₁+g^b₂,where g^b₁ ∈AA(R×R²,L^p((0,1),R)) and g^b₂ ∈PAP₀(R×R²,L^p((0,1),R)) such that

Tlim→+∞

1 2T

Z T

−T

_{Z t+1}

t

|g₂(σ,u)|^pdσ ¹_p

dt=0, (3.28)

uniformly for allu∈R².

(H4) gis Lipschitz , i.e.∃L¹_g,L_g²>0 such that∀x₁,x₂,y₁,y₂∈R,

|g(t,x₁,x₂)−g(t,y₁,y₂)| ≤L_g¹|x₁−y₁|+L²_g|x₂−y₂|. (3.29) (H5) There exists a constantλ>0 such thatc(t,s)≤e^λ(t−s),for alls≥t.

(H6) The functiont7→r(t)∈C¹(R,R)with

0≤r(t)≤r, r(t)≤r^∗<1. (3.30) Lemma 6. Assume that (H6) holds. If x(·) ∈PAAS^p(R,R) then x(· −r(·))∈ PAAS^p(R).

S -ALMOST PERIODIC ANDS -ALMOST AUTOMORPHIC SOLUTIONS 225

Proof. Sincex(·)∈PAAS^p(R,R), thenx(·)can be written as x=x₁+x₂,where x^b₁(·)∈AA(R,L^p([0,1],R))andx^b₂(·)∈PAP₀(R,L^p([0,1],R)),such that

T→+∞lim 1 2T

Z T

−T

_{Z t+1}

t

|x₂(σ)|^pdσ ¹_p

dt=0. (3.31)

Let

x(· −r(·)) =x₁(· −r(·)) +x(· −r(·))−x₁(· −r(·)) =Ψ₁(·) +Ψ₂(·), (3.32) whereΨ₁(·) =x₁(· −r(·))andΨ₂(·) =x(· −r(·))−x₁(· −r(·)).Note that the func- tion Ψ^b₂(·)∈PAP₀(R,L^p((0,1),R))(cf. [9]). Hence, it only remains to show that x^b₁(· −r^b(·))∈AA(R,L^p((0,1),R)). Sincex^b₁(·)∈AA(R,L^p((0,1),R))then for any sequence(s⁰_n)n∈R there exists a subsequence(sn)n∈R and a functiong∈L_loc^p (R,R) such that

_{Z t+1}

t

|x₁(s+sn)−g(s)|^pds ¹_p

−→n→∞0 (3.33)

and

_{Z t+1}

t

|g(s−sn)−x₁(s)|^pds ¹_p

−→n→∞0. (3.34)

Thus we could find _{Z t+1}

t

|x₁(s+sn−r(s+sn))−g(s−r(s))|^pds ¹_p

≤ _{Z t+1}

t

|x₁(s+sn−r(s+sn))−g(s−r(s+sn)|^pds ¹_p

+ _{Z t+1}

t

|g(s−r(s+sn)−g(s−r(s))|^pds ¹_p

.

Let(gn)n∈Nbe a sequence such thatg_n→gasn→∞inBS^p(R,R)which is domin- ated by some integrable functionw, then

_{Z t+1}

t

|g(s−r(s+sn)−g(s−r(s))|^pds ¹_p

≤I+J+K, where

I= _{Z t+1}

t

|g(s−r(s+s_n)−g_n(s−r(s+s_n)|^pds ¹_p

, (3.35)

J= _{Z t+1}

t

|g_n(s−r(s+sn)−gn(s−r(s))|^pds ¹_p

(3.36) and

K= _{Z t+1}

t

|g_n(s−r(s))−g(s−r(s))|^pds ¹_p

. (3.37)

Let us show thatI=0. For that, lettings⁰=s−r(s+s_n)one obtains I= 1−r⁰(s+sn)−¹_pZ t+1−r(t+1−s_n)

t−r(t+s_n)

g(s⁰)−gn(s⁰)

pds^{0 1}_p

≤(1−r^∗))^−1p

_{Z t+1−r}

t−r

g(s⁰)−g_n(s⁰)

pds^{0 1}_p

≤(1−r^∗))^−1p

_{Z t+1}

t

g(s⁰)−gn(s⁰)

pds^{0 1}_p

−→n→∞0.

In addition, by applying the dominated convergence theorem we getJ=0. Moreover, lets⁰=s−r(s), then

K= 1−r⁰(s)−¹_{pZ t+1−r(t+1)}

t−r(t)

g(s⁰)−g_n(s⁰)

pds^{0 1}_p

≤(1−r^∗))^−1p

_{Z t+1−r}

t−r

g(s⁰)−g_n(s⁰)

pds^{0 1}_p

≤(1−r^∗))^−1p

_{Z t+1}

t

g(s⁰)−g_n(s⁰)

pds^{0 1}_p

−→n→∞0.

What is left to show that _{Z t+1}

t

|x₁(s+s_n−r(s+s_n))−g(s−r(s+s_n)|^pds ¹_p

−→n→∞0. (3.38) For this purpose, we sets−r(s+sn) =s⁰, then

_{Z t+1}

t

|x₁(s+sn−r(s+sn))−g(s−r(s+sn)|^pds ¹_p

= 1−r⁰(s+s_n)−¹_{pZ t+1−r(t+1−sn)}

t−r(t+sn)

x₁(s⁰+s_n)−g(s⁰)

pds^{0 1}_p

≤(1−r^∗))^−1p

_{Z t+1−r}

t−r

x₁(s⁰+sn)−g(s⁰)

pds^{0 1}_p

≤(1−r^∗))^−1p

_{Z t+1}

t

x₁(s⁰+s_n)−g(s⁰)

pds^{0 1}_p

−→n→∞0.

Similarly, we can get _{Z t+1}

t

|g(s−sn+r(s−sn))−x₁(s−r(s))|^pds ¹_p

−→n→∞0. (3.39) Consequently,

x^b₁(· −r^b(·))∈AA(R,L^p([0,1],R)).