A new composition theorem for S^{p}-weighted pseudo almost periodic functions and applications to semilinear differential equations

A NEW COMPOSITION THEOREM

FOR

S

_{-WEIGHTED PSEUDO ALMOST PERIODIC}

FUNCTIONS AND APPLICATIONS

TO SEMILINEAR DIFFERENTIAL EQUATIONS

Zhi-Han Zhao, Yong-Kui Chang, G.M. N’Guérékata

Abstract. In this paper, we establish a new composition theorem forSp_{-weighted pseudo} almost periodic functions under weaker conditions than the Lipschitz ones currently encoun-tered in the literatures. We apply this new composition theorem along with the Schauder’s fixed point theorem to obtain new existence theorems for weighted pseudo almost periodic mild solutions to a semilinear differential equation in a Banach space.

Keywords:Sp_{-weighted pseudo almost periodic, weighted pseudo almost periodicity,} semi-linear differential equations.

Mathematics Subject Classification:34K14, 60H10, 35B15, 34F05.

1. INTRODUCTION

In this paper, we are mainly concerned with the existence of weighted pseudo almost periodic mild solutions to the following semilinear differential equation

x′(t) =Ax(t) +f(t, Bx(t)), t∈R, (1.1)

where A generates an exponentially stable C0-semigroup {T(t)}t≥0 on X, and B is

an arbitrary bounded linear operator onX_{. Here}f is an appropriate function to be specified later.

The concept of pseudo almost periodicity was introduced by Zhang [1–3] in the early nineties. It is a natural generalization of the classical almost periodicity in the sense of Bochner. Recently, Agarwal and Diagana [4], and Diagana [5–7] introduced the concept of weighted pseudo almost periodic functions, which generalizes the one of pseudo almost periodicity. N’Guérékata et al. [8] presented stepanov-like almost automorphic functions and discussed its application to monotone differential equa-tions. Then, in [9], Diagana introduced the concept of Stepanov-like weighted pseudo

almost periodicity and applied the concept to study the existence and uniqueness of weighted pseudo almost periodic solutions to (1.1). In recent years, the existence of almost periodic, pseudo almost periodic, weighted pseudo almost periodic and

Sp-weighted pseudo almost periodic solutions of various differential equations have been considerably investigated (cf. for instance [10–23]) because of its significance and applications in physics, mechanics and mathematical biology.

Motivated by the works [9, 21, 24–26], we present here a new composition theorem for Stepanov-like weighted pseudo almost periodic functions. It is different from the existing ones which are based on Lipschitz conditions. Then we apply this new result to investigate the existence of weighted pseudo almost periodic solutions to the problem (1.1).

The rest of this paper is organized as follows. In Section 2, we present some basic definitions, lemmas, and preliminary results which will be used throughout this paper. In Section 3, we prove the existence of weighted pseudo almost periodic mild solutions to the problems (1.1).

2. PRELIMINARIES AND A NEW COMPOSITION THEOREM

In this section, we introduce some basic definitions, notations, lemmas and prelim-inary facts which will be used in the sequel. Particularly, we prove a new composition theorem forSp_{-weighted pseudo almost automorphic functions (Theorem 2.1).}

Throughout the paper, let (X_{,k · k),(}Y_{,k · kY)be two Banach spaces. The}

nota-tion L(X_,Y_{) stands for the Banach space of bounded linear operators from} Y _into X _{endowed with the uniform operator topology, and we abbreviate to L(}X₎

when-everY ₌X_{. We let C(}R,X_{) (respectively,C(}R_×Y,X_{)) denote the collection of all}

continuous functions from R _into X _{(respectively, the collection of all jointly}

con-tinuous functions f : R×Y → X_{). Furthermore, BC(}R,X_{) stands for the Banach}

space of all bounded continuous functions fromR_intoX_{equipped with the sup norm}

kxk∞:= supt∈Rkx(t)k for eachx∈BC(R,X).

Let U_{denote the set of all functions}ρ:R→(0,∞), which are locally integrable overR_{such that}ρ >0 almost everywhere. For a givenr >0 and for eachρ∈U_{, we}

setm(r, ρ) :=Rr

−rρ(t)dt.

Then we define the space of weights U_{∞ is defined by}

U_∞:={ρ∈U_{: lim}

r→∞m(r, ρ) =∞}.

In addition, we defineU_{B by}

U_{B :=}{ρ∈U_:ρ is bounded and lim inf

t→∞ ρ(t)>0}. It is clear thatU_B ⊂U_∞⊂U_{, with strict inclusions.}

Definition 2.1([27,28]). A functionf ∈C(R_,X_{)is said to be (Bohr) almost periodic}

int∈R_{if for everyε >0there exists l(ε)>0such that every interval of lengthl(ε)}

contains a numberτ with the property thatkf(t+τ)−f(t)k< εfor eacht∈R_{. The}

Definition 2.2 ([27, 28]). A function f ∈C(R_×Y_,X_{)is said to be (Bohr) almost}

periodic int∈R_{uniformly iny∈}Y_{if for everyε >0and any compactK⊂}Y_there

exists l(ε)>0 such that every interval of lengthl(ε)contains a numberτ with the property thatkf(t+τ, y)−f(t, y)k< εfor eacht∈R_{andy ∈K. The collection of}

all such functions will be denoted byAP(R_×Y_,X_).

Now forρ∈U_{∞, we define}

P AP0(X, ρ) :=

f ∈BC(R,X_{) : lim}

r→∞

1 m(r, ρ)

r

Z

−r

kf(t)kρ(t)dt= 0

,

P AP0(Y,X, ρ) :=

f ∈C(R_×Y_,X_{) :f(·, y) is bounded for each y∈}Y

and lim

r→∞

1 m(r, ρ)

r

Z

−r

kf(t, y)kρ(t)dt= 0 uniformly in compact subset of Y

.

Definition 2.3 ([6, 7]). Let ρ ∈ U_{∞. A function} f ∈ BC(R,X_{) is called weighted}

pseudo almost periodic if it can be expressed as f =g+φ, where g ∈ AP(X_{) and}

φ∈P AP0(X, ρ). The collection of such functions will be denoted byW P AP(X). Definition 2.4 ([6, 7]). Letρ∈U_{∞. A function} f ∈C(R×Y,X_{)is called weighted}

pseudo almost periodic if it can be expressed asf =g+φ, whereg∈AP(R_×Y_,X_)and

φ∈P AP0(Y,X, ρ). The collection of such functions will be denoted byW P AP(R×

Y_,X_).

Lemma 2.5 ([7]). The space W P AP(X_{)is a closed subspace of (BC(}R_,X_{),k · k∞).} This yieldsW P AP(X_{) is a Banach space.}

Definition 2.6([9,20]). The Bochner transformfb(t, s), t∈R_{, s∈[0,1], of a function}

f :R_→X_{is defined by}

fb_{(t, s) :=f(t+s).}

Remark 2.7 ([9]). (i) A functionϕ(t, s), t∈R, s∈[0,1], is the Bochner transform of a certain functionf,ϕ(t, s) =fb_{(t, s), if and only ifϕ(t+τ, s−τ) =ϕ(s, t)for all}

t∈R, s∈[0,1]andτ ∈[s−1, s].

(ii) Note that if f =h+ϕ, thenfb _=hb_+ϕb_{. Moreover, (λf)}b _=λfb _{for each}

scalar λ.

Definition 2.8 ([9]). The Bochner transformfb_{(t, s, u), t∈R, s∈[0,1], u∈Xof a}

functionf :R_×X_→X_{is defined by}

fb(t, s, u) :=f(t+s, u) for each u∈X.

Definition 2.9 ([9]). Let p∈ [1,∞). The space BSp_{(X) of all Stepanov bounded}

functions, with the exponent p, consists of all measurable functionsf :R_→X_such

thatfb _∈L∞_(R;Lp_{((0,1),X)). This is a Banach space with the norm}

kfkSp=kfbkL∞₍

R;Lp₎= sup t∈R

Zt+1

t

kf(τ)kpdτ

1p

Definition 2.10 ([9]). Let ρ ∈ U_{∞. A function f ∈ BS}p_{(X) is said to be}

Stepanov-like weighted pseudo almost periodic (or Sp_{-weighted pseudo almost}

pe-riodic) if it can be decomposed as f = h+ϕ, where hb ∈ AP(Lp_{((0,1),X)) and}

ϕb ∈ P AP0(Lp((0,1),X), ρ). In other words, a function f ∈ Lploc(R,X) is said to

be Stepanov-like weighted pseudo almost periodic relatively to the weight ρ ∈U_∞,

if its Bochner transform fb _{: R→ L}p_{((0,1),X) is weighted pseudo almost periodic}

in the sense that there exist two functions h, ϕ : R _→ X _{such that} f = h+ϕ, where hb _{∈ AP(L}p_{((0,1),X)) and ϕ}b _{∈ P AP}

0(Lp((0,1),X), ρ). We denoted by W P AP Sp_{(X)the set of all such functions.}

Definition 2.11 ([9]). Let ρ ∈ U_{∞. A function f :} R_×Y _→ X_{, (t, x) → f(t, x)}

withf(·, x)∈Lp_loc(R,X_{)for each}x∈X_{, is said to be Stepanov-like weighted pseudo}

almost periodic (or Sp_{-weighted pseudo almost periodic) if it can be decomposed as}

f =h+ϕ, wherehb_{∈AP(R×Y, L}p_{((0,1),X))and ϕ}b _{∈P AP}

0(Y, Lp((0,1),X), ρ).

We denote byW P AP Sp_{(R×Y,X)the set of all such functions.}

Lemma 2.12([9]). The spaceW P AP Sp(X_{)is a closed subspace ofBC(}R_{, L}p_((0,1), X₎₎, relatively to the norm k · kSp, and hence is a Banach space.

Lemma 2.13. Let ρ∈U_{∞ and letf ∈BS}p_{(X). Then f}b _{∈P AP}

0(Lp((0,1),X), ρ)

if and only if for anyε >0,

lim

r→∞

1 m(r, ρ)

Z

Mr,ε(f)

ρ(t)dt= 0, (2.1)

whereMr,ε(f) =

t∈[−r, r] :

Rt+1

t kf(s)kpds

1p ≥ε

.

Proof. “Necessity”. Suppose the contrary, that there exists ε0 > 0 such that 1

m(r,ρ)

R

Mr,ε(f)ρ(t)dt does not converge to 0 as r → ∞. Then there exists δ > 0 such that for eachn,

1 m(rn, ρ)

Z

Mrn,ε0(f)

for somern> n. Then

1 m(rn, ρ)

rn Z

−rn Zt+1

t

kf(s)kp_ds

1p

ρ(t)dt=

= 1 m(rn, ρ)

Z

Mrn,ε0(f)

Zt+1

t

kf(s)kp_ds

_p1

ρ(t)dt+

+

Z

[−rn,rn]\Mrn,ε0(f)

Z t+1

t

kf(s)kp_ds

1p

ρ(t)dt

≥

≥ 1

m(rn, ρ)

Z

Mrn,ε0(f)

Zt+1

t

kf(s)kp_ds

1p

ρ(t)dt≥

≥ ε0

m(rn, ρ)

Z

Mrn,ε0(f)

ρ(t)dt≥

≥ε0δ,

which contradicts the fact thatfb_{∈P AP}

0(Lp((0,1),X), ρ), that is,

lim

r→∞

1 m(r, ρ)

r

Z

−r

Zt+1

t

kf(s)kp_ds

1p

ρ(t)dt= 0.

Thus (2.1) holds.

“Sufficiency”. From the statement of the lemma it is clear thatkfkSp<∞and for anyε >0, there existsr0>0such that forr > r0,

1 m(r, ρ)

Z

Mr,ε(f)

ρ(t)dt < ε

kfkSp

Then

1 m(r, ρ)

r

Z

−r

Zt+1

t

kf(s)kp_ds

1p

ρ(t)dt=

= 1 m(r, ρ)

Z

Mr,ε(f) tZ+1

t

kf(s)kp_ds

1p

ρ(t)dt+

+

Z

[−r,r]\Mr,ε(f) Zt+1

t

kf(s)kp_ds

1p

ρ(t)dt

≤

≤ 1

m(r, ρ)

kfkSp Z

Mr,ε(f)

ρ(t)dt+ε

Z

[−r,r]\Mr,ε(f)

ρ(t)dt

≤

≤ε+ε= 2ε

forr > r0. Solimr→∞_m(1_r,ρ)

r

R

−r

_t+1

R

t

kf(s)kp_ds

1p

ρ(t)dt= 0. That is,

fb ∈P AP0(Lp((0,1),X), ρ).

The proof is complete.

Remark 2.14. Whenρ(t) = 1for eacht∈R_{, we obtain Li’s}et al.result [25, Lemma 2.7] as corollary of previous lemma.

We now present a new composition result for weighted pseudo almost periodic functions.

Theorem 2.15. Let ρ∈U_∞ and letf :R×Y→X be aSp_{-weighted pseudo almost}

periodic functions. Suppose thatf satisfies the following conditions:

(i) f(t, ψ) is uniformly continuous in every bounded subset K ⊂ Y _{uniformly in}

t∈R_.

(ii) For every bounded subsetK⊂Y,{f(·, ψ) :ψ∈K}is bounded inW P AP Sp_(X).

If x∈W P AP(Y_{), thenf(·, x(·))∈W P AP S}p_(X).

Proof. Since f ∈ W P AP Sp_{(R ×Y,X) and x ∈ W P AP(Y), we have by}

defi-nition that f = g +φ and x = α+β, where gb _{∈ AP(R×Y, L}p_((0,1),X)),

φb _{∈ P AP}

0(Y, Lp((0,1),X), ρ), α ∈ AP(Y) and β ∈ P AP0(Y, ρ). So the function f can be decomposed as

f(t, x(t)) =g(t, α(t)) +f(t, x(t))−g(t, α(t)) =

where G(t) = g(t, α(t)), F(t) = f(t, x(t))−f(t, α(t)) and Φ(t) = φ(t, α(t)). By a standard argument, it is easy to prove that Gb _{∈ AP(L}p_{((0,1),X)) (see, e.g.,}

[27, Theorem 2.11]). To show that f(·, x(·)) ∈ W P AP Sp(X_{), it is enough to show}

thatFb,Φb_{∈P AP}

0(Lp((0,1),X), ρ).

We first prove thatFb∈P AP0(Lp((0,1),X), ρ). Sincex(·)andα(·)are bounded,

we can choose a bounded subsetK⊂Y_{such that x(}R_{), α(}R_)⊂K. It is easy to get from (ii) thatF ∈BSp_{(X). Under assumption (i),f is uniformly continuous on the}

bounded subsetK ⊂Y_{uniformly for} t∈R_{. So, given}ε >0, there existsδ >0 such thatu, v ∈Kandku−vk_Y< δ imply thatkf(t, u)−f(t, v)k< εfor allt∈R_{. Then}

we have

Zt+1

t

kf(s, u)−f(s, v)kp_ds

1p

< ε.

Hence, for eacht∈R_,kβ(s)kY< δ,s∈[t, t+ 1]implies that for allt∈R_,

tZ+1

t

kF(s)kp_ds

1p

=

Zt+1

t

kf(s, x(s))−f(s, α(s))kp_ds

1p

< ε,

whereβ(s) =x(s)−α(s). LetM(r, δ, β) :={t∈[−r, r+ 1] :kβ(t)kY≥δ}. So we get

Mr,ε(F) =Mr,ε(f(·, x(·))−f(·, α(·)))⊂M(r, δ, β).

Sinceβ∈P AP0(Y, ρ), by an argument similar to the proof of Proposition 3.1 in [24],

we can obtain that

lim

r→∞

1 m(r, ρ)

Z

M(r,δ,β)

ρ(t)dt= 0.

Thus

lim

r→∞

1 m(r, ρ)

Z

Mr,ε(F)

ρ(t)dt= 0.

This show thatFb_{∈P AP}

0(Lp((0,1),X), ρ)by Lemma 2.13.

Next, we show that Φb _{∈ P AP}

0(Lp((0,1),X), ρ). Since α ∈ AP(Y) and gb ∈ AP(R_×Y_{, L}p_{((0,1),X)), α(R) is compact andg}b _{is uniformly continuous in R×}

α(R_{). Then it follows from (i) thatφ}b_=fb_−gb _{is uniformly continuous inu∈α(R)}

uniformly in t∈R_{. That is, for anyε >0, there existsδ >0 such thatu, v ∈α(}R₎

andku−vk_Y< δ imply that

Zt+1

t

kφ(s, u)−φ(s, v)kp_ds

p1

<ε

8 for all t∈R. (2.2)

Meanwhile, one can find in α(R_{) a finite δ-net of α(}R_{). Namely, there exist finite}

number of points uk ∈ α(R), k = 1,2,· · ·, m, such that for any v ∈ α(R), we have kv−ukkY< δ for some 1≤k≤m. Let

ThenR_=∪m

k=1Ok. Let

B_1=O1, B_{k=Ok\ ∪}k−1

i=1Oi

, k= 2,3,· · ·, m.

Thus

R_=∪m

k=1Bk andBi∩Bj =∅, i6=j, 1≤i, j≤m. (2.3)

Define a function h : R _→ Y _{by h(t) = uk for t ∈} B_{k, k = 1,2,· · · , m. Then}

kα(t)−h(t)k_Y< δ fort∈R_{, and it is easy to get from (2.2) and (2.3) that}

m X

k=1

Z

Bk∩[t,t+1]

kφ(s, α(s))−φ(s, uk)kpds

1p

= (2.4)

=

Zt+1

t

φ s, α(s)

−φ s, h(s)

p

ds

1p

< ε

8. (2.5)

Sinceφb_{∈P AP}

0(Y, Lp((0,1),X), ρ), there existsr0>0such that

1 m(r, ρ)

r

Z

−r

Zt+1

t

kφ(s, uk)kpds

1p

ρ(t)dt < ε

8m2 (2.6)

for allr > r0and1≤k≤m. Now by (2.3)–(2.6), for allr > r0, we have

1 m(r, ρ)

r

Z

−r

Zt+1

t

kΦ(s)kp_ds

1p

ρ(t)dt=

= 1 m(r, ρ)

r

Z

−r

m X

k=1

Z

B_k∩[t,t+1]

kφ(s, α(s))−φ(s, uk) +φ(s, uk)kpds

1p

≤ 1

m(r, ρ)

r

Z

−r

2p m

X

k=1

Z

B_k∩[t,t+1]

kφ(s, α(s))−φ(s, uk)kpds+

+

Z

B_k∩[t,t+1]

kφ(s, uk)kpds

1p

ρ(t)dt≤

≤ 2

1+1 p

m(r, ρ)

r

Z

−r

m X

k=1

Z

B_k∩[t,t+1]

kφ(s, α(s))−φ(s, uk)kpds

1p

ρ(t)dt+

+ 2 1+1

p

m(r, ρ)

r

Z

−r

m X

k=1

Z

Bk∩[t,t+1]

kφ(s, uk)kpds

1p

ρ(t)dt <

< 4 m(r, ρ)

r

Z

−r

ε

8ρ(t)dt+ 4m1p

m(r, ρ)

m

X

k=1

r

Z

−r

Z

B_k∩[t,t+1]

kφ(s, uk)kpds

1p

ρ(t)dt <

< ε 2+m

1 p ε

2m < ε,

which implies thatΦb_{∈P AP}

0(Lp((0,1),X), ρ). This completes the proof.

Remark 2.16. We obtain Li et al.’s result [25, Theorem 2.8] as an immediate con-sequence of previous theorem whenρ= 1.

Definition 2.17. A continuous functionuis called a weighted pseudo almost periodic mild solution of Eq. (1.1) onR_{ifu∈W P AP(}X_{)andu(t)satisfies}

u(t) =T(t−a)u(a) +

t

Z

a

T(t−s)f(s, Bu(s))ds

fort≥a.

3. EXISTENCE OF WEIGHTED PSEUDO ALMOST PERIODIC SOLUTIONS

In this section, we investigate the existence of weighted pseudo almost periodic mild solutions for the problem (1.1). We first list the following basic assumptions of this paper:

(H1) The operator A is the infinitesimal generator of an exponentially stable

C0-semigroup {T(t)}t≥0 on X; that is, there exist constants M > 0, δ > 0 such

thatkT(t)k ≤M e−δt _{for allt≥0. Moreover,T(t)is compact fort >0.}

(H2) The operatorB :X_→X_{is bounded linear operator andBK⊂K, whereK is}

(H3) Letp >1andq≥1 such that 1

p+

1

q = 1.

(H4) The functionf :R_×X_→X_{satisfies the following conditions:}

(i)f isSp-weighted pseudo almost periodic andf(t,·)is uniformly continuous in every bounded subsetK⊂X_{uniformly int∈}R_.

(ii) There exists L >0such that

Mf := sup t∈R,kBxk≤L





t+1

Z

t

kf(s, Bx(s))kp_ds





1 p

≤ L

∆(M, q, δ),

where∆(M, q, δ) =Mqq eqδ−1 qδ

P∞

k=1e−δk.

(iii) Let{xn} ⊂W P AP(X)be uniformly bounded inRand uniformly convergent

in each compact subset ofR_{. Then{f(·, xn(·))} is relatively compact inBS}p_(X).

(H5) The space W P AP(X_{) is translation invariant, that is, the weighted function}

ρ∈U_{∞ and satisfies:}

lim sup

t→∞

ρ(t+τ)

ρ(t) <∞ and lim supr→∞

m(r+τ, ρ) m(r, ρ) <∞

for everyτ ∈R_.

Remark 3.1. Note that condition (H5) was introduced by Diagana in [5–7].

To establish our first main result, we need the following technical lemma.

Lemma 3.2. Let ρ ∈ U_{∞ and suppose that assumptions (H1)–(H3), (H4)(i) and}

(H5) hold. Letx:R_→X_{be a weighted pseudo almost periodic function and let Λ be} the nonlinear operator defined by

(Λx)(t) :=

t

Z

−∞

T(t−s)f(s, Bx(s))ds for each t∈R.

ThenΛ is continuous and mapsW P AP(X₎into itself.

Proof. We first prove thatΛ mapsW P AP(X_{)into itself. Let} x∈W P AP(X_{). Since}

B∈ L(X_{), then}Bx∈W P AP(X_{). SettingF(t) =}f(t, Bx(t))and using Theorem 2.15 it follows that F ∈W P AP Sp_{(X). Write F = ϕ+ψ where ϕ}b _∈AP(Lp_((0,1),X))

andψb _{∈P AP}

0(Lp((0,1),X), ρ). HenceΛxcan be rewritten as

(Λx)(t) = Φ(t) + Ψ(t),

where Φ(t) =

t

R

−∞

T(t−s)ϕ(s)ds and Ψ(t) =

t

R

−∞

T(t−s)ψ(s)ds for each t ∈ R_{. It}

Now, we show thatΨ(t)∈P AP0(X, ρ). For this, we consider

Ψk(t) := t−k+1

Z

t−k

T(t−s)ψ(s)ds=

k

Z

k−1

T(s)ψ(t−s)ds

for each t ∈ R _{and k = 1,2,· · ·. From assumption (H1) and Hölder’s inequality, it}

follows that

kΨk(t)k ≤ M t−k+1

Z

t−k

e−δ(t−s)kψ(s)kds≤

≤ M





t−k+1

Z

t−k

e−qδ(t−s)ds





1 q 



t−k+1

Z

t−k

kψ(s)kp_ds

  1 p ≤ ≤ M   k Z

k−1

e−qδsds





1 q 



t−k+1

Z

t−k

kψ(s)kp_ds





1 p

≤

≤ M q

s

eqδ₋₁

qδ e

−δk





t−k+1

Z

t−k

kψ(s)kp_ds





1 p

.

Then, forr >0, we see that

1 m(r, ρ)

r

Z

−r

kΨk(t)kρ(t)dt≤

≤ M q

s

eqδ₋₁

qδ e

−δk 1

m(r, ρ) r Z −r  

t−k+1

Z

t−k

kψ(s)kp_ds





1 p

ρ(t)dt.

Sinceψb_{∈P AP}

0(Lp((0,1),X), ρ), the above inequality leads toΨk ∈P AP0(X, ρ)for

eachk= 1,2,· · ·. The above inequality leads also to

kΨk(t)k ≤M q

s

eqδ₋₁

qδ e

−δk_kψkSp.

SinceMqq eqδ₋₁ qδ

P∞

k=1e−δk<∞, we deduce from the Weierstrass test that the series

P∞

k=1Ψk(t)is uniformly convergent onR. Furthermore,

Ψ(t) =

t

Z

−∞

T(t−s)ψ(s)ds=

∞ X

and clearlyΨ(t)∈C(R_,X_{). ApplyingΨk∈P AP0(}X_{, ρ)and the inequality}

1 m(r, ρ)

r

Z

−r

kΨ(t)kρ(t)dt≤

≤ 1

m(r, ρ)

r

Z

−r

Ψ(t)− n

X

k=1 Ψk(t)

ρ(t)dt+

n

X

k=1 1 m(r, ρ)

r

Z

−r

kΨk(t)kρ(t)dt,

we deduce that the uniformly limit Ψ(·) =P∞_k=1Ψk(t)∈P AP0(X, ρ). Thus we have Λx∈W P AP(X_).

Now to complete the rest of the proof, we need to show that Λ is continuous onW P AP(X_{). Let}{xn} ⊂ W P AP(X)be a sequence which converges to some x∈

W P AP(X_{), that is,kxn−xk →0asn→ ∞. We may find a bounded subsetK⊂}X

such that xn(t), x(t) ∈ K for t ∈R, n = 1,2,· · ·. By (H4)(i), for any ε > 0, there

existsδ >0 such thatx, y∈K andkx−yk< δ imply that

kf(t, x)−f(t, y)k< δε

M for each t∈R,

where δ, M are given in (H1). For the above δ > 0, there exists N > 0 such that

kBxn(t)−Bx(t)k< δ for alln > N and allt∈R. Therefore,

kf(t, Bxn(t))−f(t, Bx(t))k<

δε M

for alln > Nand allt∈R_{. Then by the Lebesgue’s Dominated Convergence theorem,}

we have

k(Λxn)(t)−(Λx)(t)k=

t

Z

−∞

T(t−s)[f(s, Bxn(s))−f(s, Bx(s))]ds

≤

≤M

t

Z

−∞

e−δ(t−s)_{kf(s, Bx}

n(s))−f(s, Bx(s))kds <

< M

t

Z

−∞

e−δ(t−s)δε Mds≤ε

for alln > Nand allt∈R_{. This implies thatΛis continuous. The proof is completed.}

Now, we are ready to state our first main result.

Theorem 3.3. Assume the conditions(H1)–(H5)are satisfied, then the problem(1.1)

Proof. We define the nonlinear operatorΓ :W P AP(X_)→C(R_,X_)by

(Γx)(t) =

t

Z

−∞

T(t−s)f(s, Bx(s))ds, t∈R_.

From the statement of the above lemma it is clear that the nonlinear operatorΓis well defined and continuous. Moreover, from Lemma 3.2 we infer thatΓx∈ W P AP(X_),

that is,ΓmapsW P AP(X_{)into itself. For the sake of convenience, we break the proof}

into several steps.

Step 1.Let B₌{x∈W P AP(X_{) :}kxk∞ ≤L}. ThenBis a closed convex subset of

W P AP(X_{). We claim thatΓ}B⊂B_{. In fact, for} x∈B_andt∈R_{, we get}

k(Γx)(t)k ≤

∞ X

k=1

t−k+1

Z

t−k

T(t−s)f(s, Bx(s))ds

≤

≤

∞ X

k=1 M

t−k+1

Z

t−k

e−δ(t−s)kf(s, Bx(s))kds≤

≤

∞ X

k=1 M

t−Zk+1

t−k

e−qδ(t−s)ds

1q t−Zk+1

t−k kf

s, Bx(s)

kp_ds

p1 ≤

≤

∞ X

k=1 M q

s

eqδ₋₁

qδ e

−δk_M f ≤L,

which implies thatkΓxk∞≤L. ThusΓB⊂B.

Step 2. Next we prove that the operatorΓ is completely continuous on B_{. It surfies}

to prove that the following statements are true.

(i)V(t) ={(Γx)(t) :x∈B_{is relatively compact in} X_{for eacht∈}R_.

First we show that (i) holds. Let 0< ε <1 be given. For each t∈R_andx∈B_,

we define

(Γεx)(t) = t−ε

Z

−∞

T(t−s)f(s, Bx(s))ds=

=T(ε)

t−ε

Z

−∞

T(t−ε−s)f(s, Bx(s))ds=

=T(ε)[(Γx)(t−ε)].

Since T(t)(t >0) is compact, then the set Vε(t) := {(Γεx)(t) :x ∈B} is relatively

compact inX_{for each} t∈R_{. Moreover, for each}x∈B_{, we get}

k(Γx)(t)−(Γεx)(t)k=

t

Z

t−ε

T(t−s)f(s, Bx(s))ds

≤

≤M

t

Z

t−ε

e−δ(t−s)kf(s, Bx(s))kds≤

≤M

Zt

t−ε

e−qδ(t−s)ds

1qZt

t−ε

kf(s, Bx(s))kp_ds

1p ≤

≤M Mf

Zt

t−ε

e−qδ(t−s)ds

1q

.

Therefore, lettingε→0, it follows that there are relatively compact setsVε(t)

arbi-trarily close toV(t)and henceV(t)is also relatively compact inX_{for each} t∈R_.

Next we prove that (ii) holds. Letε >0be small enough and−∞< t1< t2<∞.

Since {T(t)}t≥0 is a C0-semigroup and T(t)is compact for t > 0, there exists ω = ω(ε)< τεsuch thatt2−t1< ω implies that

T(

t 2)−T

t

2+t2−t1

<

ε

L for each t >0,

whereL_{= 3M Mf} q

r

2(eqδ2 −1)

qδ

P∞

k=1e−

Indeed, forx∈B_{andt2−t1< ω, we have}

k(Γx)(t2)−(Γx)(t1)k ≤

t1−τε Z

−∞

[T(t2−s)−T(t1−s)]f(s, Bx(s))ds

+ + t1 Z

t1−τε

[T(t2−s)−T(t1−s)]f(s, Bx(s))ds

+ + t2 Z t1

T(t2−s)f(s, Bx(s))ds

≤ ≤ ∞ Z τε

[T(t2−t1+s)−T(s)]f(t1−s, Bx(t1−s))ds

+ +M t1 Z

t1−τε

[e−δ(t2−s)+e−δ(t1−s)]kf(s, Bx(s))kds+

+M

t2

Z

t1

e−δ(t2−s)kf s, Bx(s)

kds≤

≤ ∞ Z τε

[T(t2−t1+ s 2)−T(

s

2)]f(t1−s, Bx(t1−s))ds

+ +M Zt1

t1−τε

[e−δ(t2−s)+e−δ(t1−s)]qds

1q Zt1

t1−τε

kf(s, Bx(s))kp_ds

1p

+

+M

Zt2

t1

e−qδ(t2−s)ds

1qZt2

t1

kf(s, Bx(s))kp_ds

1p ≤ ≤ ε LM ∞ Z τε

e−δs2kf(t₁−s, Bx(t₁−s))kds+ 2M τ 1 q

εMf+M ω 1 qM f ≤ ≤ ε LM ∞ X k=1

τε+k Z

τε+k−1

e−δs2k_f(t

1−s, Bx(t1−s))kds+

+ 2M

ε 6M Mf

q1q

Mf+M τ 1 q εMf ≤

≤ ε

LM

∞ X

k=1

τZε+k

τε+k−1

e−qδs2 _ds

1q τZε+k

τε+k−1 kf

t1−s, Bx(t1−s)

kp_ds

1p + +ε 3 + ε 6 ≤ ≤ ε

LM Mf

q s

2(eqδ2 −1)

qδ

∞ X

k=1

e−δ(τε2+k) +ε

3 + ε 6 ≤ ε 3+ ε 3 + ε 6 < ε.

Now we denote the closed convex hull of ΓB_byconvΓB_{. SinceΓ}B_⊂B _andB _is

closed convex,convΓB_⊂B_{. Thus,Γ(convΓ}B_)⊂ΓB_⊂convΓB_{. This implies thatΓis}

a continuous mapping fromconvΓB_{to convΓ}B_{. It is easy to verify thatconvΓ}B_has

the properties (i) and (ii). More explicitly,{x(t) :x∈convΓB_{}is relatively compact in} X_{for eacht∈}R_{, andconvΓ}B_⊂BC(R_,X_{)is uniformly bounded and equicontinuous.}

By the Ascoli-Arzelà theorem, the restriction ofconvΓB_{to every compact subset}K′ of R_{, namely{x(t) :} x∈convΓB_}x∈K′ is relatively compact in C(K′,X). Thus, by the conditions (H4)(iii) and Lemma 3.2 we deduce that Γ :convΓB →convΓB _{is a}

compact operator. So by the Schauder fixed point theorem, we conclude that there is a fixed pointx(·)forΓinconvΓB_{. That is Eq. (1.1) has at least one weighted pseudo}

almost periodic mild solutionx∈B_{. This completes the proof.}

Proposition 3.4. Assume that {T(t)}t≥0 is an exponentially stable C0-semigroup.

Let f ∈W P AP Sp_{(X)and there exists a positive number Lsuch that}

kf(t, x)−f(t, y)k ≤Lkx−yk for all t∈R, x, y∈X.

If M LkBkL(X)

δ < 1, then (1.1) has a unique weighted pseudo almost periodic mild

solution onR.

Proof. We can easily obtain this result by using Lemma 3.2 combined with the Banach fixed point theorem.

Acknowledgments

The second author’s work was supported by NNSF of China (10901075), the Key Project of Chinese Ministry of Education (210226), China Postdoctoral Science Foundation funded project (20100471645), the Scientific Research Fund of Gansu Provincial Education Department (0804–08), and Qing Lan Talent Engineering Funds

(QL–05–16A)by Lanzhou Jiaotong University.

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Zhi-Han Zhao

zhaozhihan841110@126.com

Lanzhou Jiaotong University Department of Mathematics

Lanzhou, Gansu 730070, P.R. China

Yong-Kui Chang lzchangyk@163.com

Lanzhou Jiaotong University Department of Mathematics Lanzhou 730070, P.R. China

G.M. N’Guérékata nguerekata@aol.com

Gaston.N’Guerekata@morgan.edu

Morgan State University Department of Mathematics

1700 E. Cold Spring Lane, Baltimore, M.D. 21251, USA