Existence and exponential stability of pseudo almost periodic solutions for impulsive nonautonomous partial stochastic evolution equations

R E S E A R C H

Open Access

Existence and exponential stability of

pseudo almost periodic solutions for

impulsive nonautonomous partial stochastic

evolution equations

Zuomao Yan

_{and Fangxia Lu}

*_{Correspondence:} [email protected]

Department of Mathematics, Hexi University, Zhangye, Gansu 734000, P.R. China

Abstract

In this paper, the concept ofp-mean piecewise pseudo almost periodic for stochastic processes is first introduced. Using the exponential dichotomy techniques and a fixed point strategy with stochastic analysis theory, we establish the existence ofp-mean piecewise pseudo almost periodic mild solutions for a class of impulsive

nonautonomous partial stochastic evolution equations in Hilbert spaces. Moreover, the exponential stability ofp-mean piecewise pseudo almost periodic mild solutions is investigated. Finally, an example is provided to illustrate the obtained theory.

MSC: 34A37; 60H10; 35B15; 34F05

Keywords: impulsive nonautonomous partial stochastic evolution equations; p-mean piecewise pseudo almost periodic functions; exponential dichotomy; exponential stability; fixed point theorems

1 Introduction

The almost periodic functions plays an important role in describing the phenomena that are similar to the periodic oscillations which can be observed frequently in many fields, such as celestial mechanics, nonlinear vibration, electromagnetic theory, and so on (see []). The concept of pseudo almost periodic functions is a natural generalization of almost periodic functions. The study of the existence of pseudo almost periodic solutions is one of the most interesting topics in the qualitative theory of differential equations both due to its mathematical interest as well as due to their applications in physics, mathematical biology, and other areas [–]. In the real world, stochastic perturbation is unavoidable. Therefore, we must move from deterministic problems to stochastic ones. The stochas-tic differential equations with delays and without delays have been extensively studied in the last decades (see [–]). Particularly, some authors focused on the existence of al-most periodic or pseudo alal-most periodic solutions to stochastic differential equations in Hilbert spaces [–]. Among them, Bezandry and Diagana [, ] studied the exis-tence of square-mean almost periodic solutions nonautonomous stochastic differential equations. In [, ], the authors introduced the concepts ofp-mean pseudo almost pe-riodicity, and studied the existence ofp-mean pseudo almost periodic mild solutions to

a nonautonomous semilinear stochastic differential equations. Diopet al.[] obtained the existence, uniqueness and global attractiveness of anp-mean pseudo almost periodic solution for stochastic evolution equation driven by a fractional Brownian motion.

The theory of impulsive partial differential equations has become an active area of in-vestigation due to their applications in fields such as mechanics, electrical engineering, medicine biology (see [–]). The existence, uniqueness and stability of almost periodic solutions for impulsive differential equations have been considered in abstract spaces by many authors. For example, Henríquezet al.[], Stamovet al.[, ] discussed the exis-tence and uniqueness of piecewise almost periodic solutions for a class of abstract impul-sive semilinear differential equations. Stamov [] established the existence and asymp-totic stability of piecewise almost periodic solutions of impulsive differential equations with time-varying delay. Liu and Zhang [] studied the existence and exponential stabil-ity of piecewise almost periodic solutions to abstract impulsive differential equation. The authors in [, ] introduced the concept of piecewise pseudo almost periodic functions on a Banach space and established the existence, uniqueness and exponential stability of piecewise pseudo almost periodic solutions to impulsive differential equations. Bainov and Simeonov [] concerned with the asymptotic equivalence of impulsive differential equations. However, besides impulse effects and delays, stochastic effects likewise exist in real systems. In recent years, several interesting results on impulsive partial stochas-tic systems have been reported in many publications [–] and the references therein. Further, Zhanget al.[] obtained the existence and uniqueness of almost periodic solu-tions for a class of impulsive stochastic differential equasolu-tions with delay by mean of the Banach contraction principle. In [], the authors investigated the existence and stabil-ity of square-mean piecewise almost periodic solutions for nonlinear impulsive stochastic differential equations by using Schauder fixed point theorem. In this paper, we consider the existence and exponential stability ofp-mean piecewise pseudo almost periodic mild solutions to the following impulsive nonautonomous partial stochastic evolution equa-tions:

dx(t) =A(t)x(t) +gt,x(t)dt+ft,x(t)dW(t), t∈R,t=ti,i∈Z, (.)

x(ti) =x

t+_i–xt_i–=Ii

x(ti)

, i∈Z, (.)

whereA(t) :D(A(t))⊆Lp_(P,H)→Lp_{(P,H) is a family of densely defined closed linear}

op-erators satisfying the so-called ‘Acquistapace-Terrani’ conditions, andW(t) is a two-sided standard one-dimensional Brownian motion defined on the filtered probability space (,F,P,Ft), whereFt =σ{W(u) –W(v);u,v≤t}.g, f, Ii,ti satisfy suitable conditions

which will be established later.x(t+

i),x(ti–) represent the right-hand side and the left-hand

side limits ofx(·) atti, respectively.

accurate description of the actual phenomenon in the real world. So it is natural to extend the concept of pseudo almost periodicity of mild solutions to dynamical systems repre-sented by these impulsive systems. To the authors knowledge, no results are available for the existence and exponential stability ofp-mean piecewise pseudo almost periodic mild solutions for nonlinear impulsive stochastic system (.)-(.). The systems have variable structures subject to stochastic abrupt changes, which may result from abrupt phenomena such as stochastic failures and repairs of the components, changes in the interconnections of subsystems, sudden environment changes,etc.Therefore, the system (.)-(.) involves a wide area of applications in physics and mathematics. Motivated by the above consid-eration, we will study these interesting problems, which are natural generalizations of the concept of pseudo almost periodicity for stochastic differential equations well known in the theory of infinite dimensional systems.

In this paper, we introduce and develop the notion ofp-mean piecewise pseudo almost periodic for stochastic processes, which generalizes in a natural fashion the concept of piecewise almost periodic andp-mean almost periodic stochastic processes. As an ap-plication, we study the existence and exponential stability ofp-mean piecewise pseudo almost periodic mild solution for the impulsive stochastic evolution equation (.)-(.) with pseudo almost periodic coefficients. In order to obtain the existence of pseudo almost periodic mild solutions for differential equations, most of the previous research on com-position theorems for pseudo almost periodic functions was based upon a Lipschitz con-dition. It is obvious that the conditions for contraction mapping principle are too strong. In this paper we establish a new composition theorem forp-mean pseudo almost periodic functions under conditions which are different from Lipschitz conditions. Then, using this new composition theorem together with the Leray-Schauder nonlinear alternative and the exponential dichotomy techniques with stochastic analysis theory, we get new existence and exponential stability results. The well-known results that appeared in [–, , , , ] are generalized to the impulsive stochastic systems settings and the case of piecewise pseudo almost periodicity. Moreover, the results are also new for deterministic systems with impulse.

The paper is organized as follows. In Section , we introduce some notations and neces-sary preliminaries. In Section , we give the existence ofp-mean piecewise pseudo almost periodic mild solutions for (.)-(.). In Section , we establish the exponential stability ofp-mean piecewise pseudo almost periodic mild solutions for (.)-(.). In Section , an interesting example is given to illustrate our results. Finally, concluding remarks are given in Section .

2 Preliminaries

Throughout the paper,N,Z,RandR+_{stand for the set of natural numbers, integers, real}

numbers, positive real numbers, respectively. We assume that (H, · ), (K, · K) are real separable Hilbert spaces and (,F,P) is supposed to be a filtered complete probability space. DefineLp(P,H), forp≥, to be the space of allH-valued random variablesxsuch thatExp₌

xpdP<∞. ThenLp(P,H) is a Banach space when it is equipped with its natural norm · p defined byxp= (

ExpdP)/p<∞for eachx∈Lp(P,H). We let L(K,H) be the space of all linear bounded operators fromK intoH, equipped with the usual operator norm · L(K,H); in particular, this is simply denoted byL(H) when

KtoHwith the normψ

L_ =Tr(ψQψ

∗_{) <∞for anyψ ∈L(K,H). LetC(R,L}p_(P,H)),

BC(R,Lp_{(P,H)) stand for the collection of all continuous functions fromRintoL}p_(P,H),

the Banach space of all bounded continuous functions from Rinto Lp_{(P,H), equipped}

with the sup norm, respectively.

Definition .([]) A stochastic processx:R→Lp_{(P,H) is said to be continuous}

pro-vided that, for anys∈R,

lim

t→sEx(t) –x(s) p

= .

Definition . ([]) A stochastic processx:R→Lp_{(P,H) is said to be stochastically}

bounded provided that

lim

N→∞lim sup_t∈R

Px(t)>N = .

LetTbe the set consisting of all real sequences{ti}i∈Zsuch thatα=infi∈Z(ti+–ti) >

,limi→∞ti=∞, andlimi→–∞ti= –∞. For{ti}i∈Z∈T, letPC(R,Lp(P,H)) be the space

consisting of all stochastically bounded piecewise continuous functionsf :R→Lp_(P,H)

such that f(·) is stochastically continuous att for anyt∈ {/ ti}i∈Z andf(ti) =f(ti–) for all

i∈Z; letPC(R×Lp_(P,K),Lp_{(P,H)) be the space formed by all stochastically piecewise}

continuous functionsf :R×Lp_(P,K)→Lp_{(P,H) such that, for anyx∈L}p_{(P,K),f(·,x)∈}

PC(R,Lp_{(P,H)) and for anyt∈R,f(t,·) is stochastically continuous atx∈L}p_(P,K).

Definition .([]) A functionf ∈C(R,Lp_{(P,H)) is said to bep-mean almost periodic}

if, for each ε> , there exists anl(ε) > , such that every intervalJ of lengthl(ε) con-tains a numberτ with the property thatEf(t+τ) –f(t)p _{<εfor allt∈R. Denote by}

AP(R,Lp(P,H)) the set of such functions.

Definition .(Compare with []) A sequence{xn}is calledp-mean almost periodic if,

for anyε> , there exists a relatively dense set of itsε-periods,i.e., there exists a natural numberl=l(ε), such that, fork∈Z, there is at least one numberqin [k,k+l], for which inequalityExn+q–xnp<εholds for alln∈N. Denote byAP(Z,Lp(P,H)) the set of such

sequences.

Definel∞(Z,Lp_{(P,H)) ={x:Z→L}p_{(P,H) :x=sup}

n∈Z(Ex(n)p)/p<∞}, and

PAP

Z,Lp(P,H)=

x∈l∞Z,Lp(P,H): lim

n→∞

 n

n

j=–n

Ex(n)pdt= 

.

Definition . A sequence {xn}n∈Z ∈l∞(Z,X) is called p-mean pseudo almost peri-odic if xn =xn +xn, where xn ∈AP(Z,Lp(P,H)), xn ∈ PAP(Z,Lp(P,H)). Denote by

PAP(Z,Lp(P,H)) the set of such sequences.

Definition .(Compare with []) For{ti}i∈Z∈T, the functionf ∈PC(R,Lp(P,H)) is

(i) {t_ij=ti+j–ti},j∈Z, is equipotentially almost periodic, that is, for anyε> , there exists a relatively dense setQεofRsuch that for eachτ∈Qεthere is an integer

q∈Zsuch that|ti+q–ti–τ|<εfor alli∈Z.

(ii) For anyε> , there exists a positive numberδ˜=δ˜(ε)such that if the pointstandt

belong to a same interval of continuity ofϕand|t–t|<δ, then˜

Ef(t) –f(t)p_<ε.

(iii) For everyε> , there exists a relatively dense set˜(ε)inRsuch that ifτ∈ ˜(ε), then

Ef(t+τ) –f(t)p<ε

for allt∈Rsatisfying the condition|t–ti|>ε,i∈Z. The numberτ is called ε-translation number off.

We denote byAPT(R,Lp(P,H)) the collection of all thep-mean piecewise almost

pe-riodic functions. Obviously, the spaceAPT(R,Lp(P,H)) endowed with the sup norm

de-fined by f∞=supt∈R(Ef(t)p)/p for anyf ∈APT(R,Lp(P,H)) is a Banach space. Let

UPC(R,Lp_{(P,H)) be the space of all stochastic functionsf ∈PC(R,L}p_{(P,H)) such thatf}

satisfies the condition (ii) in Definition ..

Definition .(Compare with []) The functionf ∈PC(R×Lp_(P,K),Lp_{(P,H)) is said to}

bep-mean piecewise almost periodic int∈Runiform inx∈Lp(P,K) if, for every compact subsetK⊆Lp(P,K),{f(·,x) :x∈K}is uniformly bounded, and givenε> , there exists a relatively dense subsetεsuch that

Ef(t+τ,x) –f(t,x)p<ε

for allx∈K,τ∈ε, andt∈Rsatisfying|t–ti|>ε. Denote byAPT(R×Lp(P,K),Lp(P,H))

the set of all such functions.

Denote

PC_TR,Lp(P,H)=f ∈PCR,Lp(P,H): lim

t→∞Ef(t) p

= ,

PAP_TR,Lp(P,H)=

f∈PCR,Lp(P,H): lim

r→∞

 r

r

–r

Ef(t)pdt= 

,

PAP_TR×Lp(P,K),Lp(P,H) =

f∈PCR×Lp(P,K),Lp(P,H):

lim

r→∞

 r

r

–r

Ef(t,x)pdt=  uniformly with respect tox∈K,

whereKis an arbitrary compact subset ofLp(P,K)

.

Definition . A function f ∈PC(R,Lp(P,H)) is said to bep-mean piecewise pseudo almost periodic if it can be decomposed asf =h+ϕ, whereh∈APT(R,Lp(P,H)) and ϕ∈PAP_T(R,Lp_{(P,H)). Denoted byPAP}

PAPT(R,Lp(P,H)) is a Banach space with the sup norm · ∞.

Similar to [, ], one has the following.

Remark .

(i) PAP_T(R,Lp(P,H))is a translation invariant set ofPC(R,Lp(P,H)). (ii) PC_T(R,Lp(P,H))⊂PAP_T(R,Lp(P,H)).

Lemma . Let{fn}n∈N⊂PAPT(R,Lp(P,H))be a sequence of functions.If fnconverges

uni-formly to f,then f ∈PAP

T(R,Lp(P,H)).

One can refer to Lemma . in [] for the proof of Lemma ..

Definition . A functionf ∈PC(R×Lp_(P,K),Lp_{(P,H)) is said to be p-mean}

piece-wise pseudo almost periodic if it can be decomposed asf =h+ϕ, whereh∈APT(R×

Lp_(P,K),Lp_{(P,H)) andϕ∈PAP}

T(R×Lp(P,K),Lp(P,H)). Denoted byPAPT(R×Lp(P,K),

Lp_{(P,H)) the set of all such functions.}

We need the following composition ofp-mean pseudo almost periodic processes.

Lemma . Assume f ∈PAPT(R×Lp(P,K),Lp(P,H)).Assume that the following

condi-tions hold:

(i) {f(t,x) :t∈R,x∈K}is bounded for every bounded subsetK⊂Lp_(P,K).

(ii) f(t,·)is uniformly continuous in each bounded subset ofLp(P,K)uniformly int∈R.

Ifφ(·)∈PAPT(R,Lp(P,K))such thatR(φ)⊂Lp(P,K),then f(·,φ(·))∈PAPT(R,Lp(P,H)).

Lemma . Assume the sequence of vector-valued functions{Ii}i∈Zis pseudo almost

peri-odic,i.e.,for any x∈Lp_(P,H),{I

i(x),i∈Z}is a pseudo almost periodic sequence.Suppose

{Ii(x) :i∈Z,x∈K}is bounded for every bounded subset K ⊂Lp(P,H),Ii(x)is uniformly

continuous in x∈Lp_{(P,H)uniformly in∈Z.Ifφ∈PAP}

T(R,Lp(P,H))∩UPC(R,Lp(P,H))

such thatR(φ)⊂Lp_{(P,K),then I}

i(φ(ti))is pseudo almost periodic.

One can refer to Lemmas ., . in [] for the proof of Lemmas . and ..

Next, we introduce a useful compactness criterion onPC(R,Lp_{(P,H)). Leth:R→R}+

be a continuous function such thath(t)≥ for allt∈Randh(t)→ ∞as|t| → ∞. Define

PC_hR,Lp(P,H)=

f ∈PCR,Lp(P,H): lim

|t|→∞

Ef(t)p h(t) = 

endowed with the normfh=supt∈REf(t) p

h(t) , it is a Banach space.

Similarly as the proof of Lemma . in [], one has the following.

Lemma . A set B⊆PC

h(R,Lp(P,H))is relatively compact if and only if it verifies the

following conditions:

(i) lim_|t|→∞Ef(t)

p

h(t) = uniformly forf∈B.

(ii) B(t) ={f(t) :f ∈B}is relatively compact inLp_{(P,H)for everyt∈R.} (iii) The setBis equicontinuous on each interval(ti,ti+)(i∈Z).

Definition .([]) A family of bounded linear operators{U(t,s) :t≥s,t,s∈R}on Lp_{(P,H) associated withA(t) is said to be an evolution family of operators if the following}

conditions hold:

(a) U(s,s) =I,U(t,s) =U(t,τ)U(τ,s)fort≥τ≥sandt,τ,s∈R. (b) (t,s)→U(t,s)∈L(Lp_{(P,H))is strongly continuous fort>s.}

Definition .([]) An evolution familyUis called hyperbolic (or has exponential di-chotomy) if there are projectionsP(t),t∈R, uniformly bounded and strongly continuous int, and constantsM,δ>  such that

(a) U(t,s)P(s) =P(t)U(t,s)for allt≥s;

(b) the restrictionUQ(t,s) :Q(s)Lp(P,H)→Q(t)Lp(P,H)is invertible for allt≥s(and we setUQ(s,t) =UQ(t,s)–);

(c) U(t,s)P(s) ≤Me–δ(t–s)_andU

Q(s,t)Q(t) ≤Me–δ(t–s)for allt≥s.

Here and belowQ:=I–P. IfP(t) =Ifort∈R, then (U(t,s))t≥sis exponentially stable (see

[, ]).

Definition .([]) IfUis a hyperbolic evolution family, then

(t,s) :=

U(t,s)P(s) ift≥s,t,s∈R, –UQ(t,s)Q(s) ift<s,t,s∈R,

is called Green’s function corresponding toUandP(·).

Lemma .(Leray-Schauder nonlinear alternative []) Let X be a Banach space with D⊂X closed and convex.Assume V is a relatively open subset of D with∈V and: V→D is a compact map,then either

(i) has a fixed point inV,or

(ii) there is a pointx∈∂V andλ∈(, )withx∈λ(x).

3 Existence

In this section, we investigate the existence ofp-mean piecewise pseudo almost periodic mild solution for system (.)-(.). We first introduce the notion of mild solution to system (.)-(.).

Definition . AnFt-progressively measurable process{x(t)}t∈Ris called a mild solution

of system (.)-(.) if, for anyt∈R,t>s,s=ti,i∈Z,

x(t) =U(t,s)x(s) + t

s

U(t,τ)gτ,x(τ)dτ

+

t

s

U(t,τ)fτ,x(τ)dW(τ) +

s<ti<t

U(t,ti)Ii

x(ti)

. (.)

In order to obtain our main results, we make the following hypotheses:

(H) There exist constantsλ> ,θ∈(π_,π),K,K≥, andα,α∈(, ]with

α+α> such that

θ∪ {} ⊂ρ

A(t) –λ

, Rλ,A(t) –λ≤

K

and

A(t) –λ

Rλ,A(t) –λ

Rλ,A(t)

–Rλ,A(s)≤K|t–s|α|λ|–α

fort,s∈R,λ∈θ:={λ∈C\ {}:|argλ| ≤θ}.

(H) R(λ,A(·))∈AP(L(R,Lp(P,H))).

(H) The evolution familyU(t,s)generated byA(t)has an exponential dichotomy with constantsM,δ> , dichotomy projectionsP(t),t∈R, and Green’s function(t,s). Moreover,U(t,s)is compact fort>s.

(H) For eachx∈H,U(t+h,t)x→xash→+uniformly fort∈R. (H) The functionsg∈PAPT(R×Lp(P,K),Lp(P,H)),

f ∈PAPT(R×Lp(P,K),Lp(P,L)), andg(t,·),f(t,·)are uniformly continuous in

each bounded subset ofLp_{(P,K)uniformly int∈R;I}

iis a pseudo almost periodic sequence,Ii(x)is uniformly continuous inx∈Lp(P,K)uniformly ini∈Z.

(H) There exists a continuous nondecreasing function: [,∞)→(,∞)such that

sup

t∈R

Eg(t,x)p+Ef(t,x)p_L 

≤xp, x∈Lp(P,K),

and there exist continuous nondecreasing functions˜i: [,∞)→(,∞),i∈Z, such that

EIi(x) p

≤ ˜i

Exp, x∈Lp(P,K).

(H) There exists a constantM∗> such that

M∗

N(M∗) +Nsupi∈Z˜i(M∗)

> ,

where forp> ,N= p–Mp[δp+Cp(

p–

pδ )

p– p 

pδ],N= 

p–_Mp 

(–e–δα₎p, and for p= ,N= M[_δ +__δ],N= M_(–e–δα₎.

Remark . Assumption (H) is usually called the ‘Acquistapace-Terreni’ condition, which was first introduced in [] and widely used to investigate nonautonomous evo-lution equations in [–, ]. If (H) holds, then there exists a unique evoevo-lution family {U(t,s),t≥s> –∞}onLp_(P,H).

Lemma .([]) Assume that(H)-(H)hold. If U(·,·)has an exponential dichotomy with constants M,δ> ,then for eachε> and h> there is a relatively dense setε,h

such that

(t+τ,s+τ) –(t,s)≤εe–δ|t–s|_, |_t–s|_>h,t,s∈R_,τ∈_ε,h.

We abbreviate this property by writing∈AP(L(H)).

Lemma .(Compare with []) Assume that f ∈APT(R,Lp(P,H)),the sequence{xi}i∈Z∈

(i) Ef(t+τ) –f(t)p_{<εfor allt∈R,|t–t}

i|>ε,τ∈εandi∈Z.

(ii) (t+τ,s+τ) –(t,s)p_{<εfor allt,s∈R,|t–s|> ,|s–t}

i|>ε,|t–ti|>ε,τ∈ε

andi∈Z.

(iii) Exi+q–xip<εfor allq∈εandi∈Z.

(iv) Exq_i –τp_{<εfor allq,τ∈}

εandi∈Z.

Also, we need to introduce a few preliminary and important results.

Lemma . Assume that(H)-(H)hold.If g∈PAPT(R,Lp(P,H))and if G is the function

defined by

G(t) := t

–∞U(t,

τ)P(τ)g(τ)dτ–

+∞

t

UQ(t,τ)Q(τ)g(τ)dτ (.)

for each t∈R,then G∈PAPT(R,Lp(P,H)).

Proof Sinceg ∈PAPT(R,Lp(P,H)), there existg ∈APT(R,Lp(P,H)) and g∈PAPT(R,

Lp_{(P,H)), such thatg=g}

+g, thenG(t) can be decomposed as

G(t) =

t

–∞U(t,

τ)P(τ)g(τ)dτ–

+∞

t

UQ(t,τ)Q(τ)g(τ)dτ

+

t

–∞U(t,τ)P(τ)g(τ)dτ–

+∞

t

UQ(t,τ)Q(τ)g(τ)dτ

=:G(t) +G(t).

Next we show thatG(t)∈APT(R,Lp(P,H)) andG(t)∈PAPT(R,Lp(P,H)). Thus, the

fol-lowing verification procedure is divided into three steps. Step.G∈UPC(R,Lp(P,H)).

Lett,t∈(ti,ti+),i∈Z,t<t. By (H), for anyε> , there exists  <ξ< (_ε_g˜)/p such

that  <t–t<ξ, we have

Ut,t–Ip≤δ

p_ε

g˜

, UQ

t,t–Ip≤δ

p_ε

g˜

,

whereg˜= p–Mpgp∞. Using Hölder’s inequality, we have

EG

t–G

tp

≤p–E

t

–∞

Ut,t–IUt,τP(τ)g(τ)dτ

p

+ p–E t

t

Ut,τP(τ)g(τ)dτ

p

+ p–E

+∞

t

UQ

t,t–IUQ

t,τQ(τ)g(τ)dτ

p

+ p–E t

t

UQ

t,τQ(τ)g(τ)dτ

p

≤p–MpUt,t–Ip

t

–∞

e–δ(t–τ)_dτ

p– t

–∞

e–δ(t–τ)_Eg (τ)

p

dτ

+ p–Mp

Using Hölder’s inequality, it follows that

+

i–

j=–∞

tj+η

tj

e–δ(t–τ)Eg

τ+τ–g(τ)

p

dτ

+

i–

j=–∞

tj+

tj+–η

e–δ(t–τ)Eg

τ+τ–g(τ)pdτ

+

t

ti

e–δ(t–τ)Eg

τ+τ–g(τ)

p

dτ

.

Sinceg∈APT(R,Lp(P,H)), one has

Eg(t+τ) –g(t)

p

<ε

for allt∈[tj+η,tj+–η],j∈Z,j≤i, andt–τ≥t–ti+ti– (tj+–η)≥t–ti+α(i–  –j) +η.

Then

i–

j=–∞

tj+–η

tj+η

e–δ(t–τ)Eg

τ+τ–g(τ)

p

dτ

≤ε i–

j=–∞

tj+–η

tj+η

e–δ(t–τ)dτ

≤ε

δ i–

j=–∞

e–δ(t–tj++η)

≤ε

δ i–

j=–∞

e–δα(i–j–)

≤ ε

δ( –e–δα₎,

i–

j=–∞

tj+η

tj

e–δ(t–τ)Eg

τ+τ–g(τ)

p

dτ

≤p–sup

s∈R

Eg(τ)p

i–

j=–∞

tj+η

tj

e–δ(t–τ)dτ

≤p–gp_∞εeδη

i–

j=–∞

e–δ(t–tj)

≤p–gp_∞εeδηe–δ(t–ti)

i–

j=–∞

e–δα(i–j)

≤p–gp∞eδα/ε

 –e–δα . Similarly, one has

i–

j=–∞

tj+

tj+–η

e–δ(t–τ)Eg

τ+τ–g(τ)

p

dτ≤ ˜Mε,

t

ti

e–δ(t–τ)Eg

whereM˜,M˜are some positive constants. Therefore, we getJ≤ ¯NεforN¯> . Using

Forr> , by Hölder’s inequality, we have

Lemma . Assume that(H)-(H)hold.If f ∈PAPT(R,Lp(P,L))and if F is the function

+ M

Using Hölder’s inequality and the Ito integral, we have, forp> ,

for alls∈R. Using Lebesgue’s dominated convergence theorem, we haveF∈PAPT(R,

Lp_{(P,H)). This completes the proof.}

Lemma . Assume that(H)-(H)hold.Ifγi∈PAP(Z,Lp(P,H)),i∈Z,and ifγ˜iis the

function defined by

˜

γi(t) :=

ti<t

U(t,ti)P(ti)γi–

t<ti

UQ(t,ti)Q(ti)γi (.)

for each t∈R,thenγ˜i∈PAPT(R,Lp(P,H)).

Proof Sinceγi∈PAP(Z,Lp_{(P,H)), there exist γ}

,i∈AP(Z,Lp(P,H)) and γ,i ∈PAP(Z,

Lp_{(P,H)), such thatγ}

i=γ,i+γ,i, thenγ˜i(t) can be decomposed as

˜

γi(t) =

ti<t

U(t,ti)P(ti)γ,i–

t<ti

UQ(t,ti)Q(ti)γ,i

+

ti<t

U(t,ti)P(ti)γ,i–

t<ti

UQ(t,ti)Q(ti)γ,i

=:,i(t) +,i(t).

Next we show that,i(t)∈APT(R,Lp(P,H)) and,i(t)∈PAPT(R,Lp(P,H)). Thus, the

verification procedure is divided into the following three steps. Step.,i∈UPC(R,Lp(P,H)).

Lett,t∈(ti,ti+),i∈Z,t<t. By (H), for anyε> , we have

Ut,t–Ip≤( –e

–δα₎p_ε

γ˜ , UQ

t,t–Ip≤( –e

–δα₎p_ε

γ˜ ,

whereγ˜_{= }p–_Mp_γ

,ip∞. Using Hölder’s inequality, we have

E,i

t–,i

tp

≤p–E

ti<t

Ut,t–IUt,ti

P(ti)γ,i

p

+ p–E

t<ti

UQ

t,t–IUQ

t,ti

Q(ti)γ,i

p

≤p–_Mp_Ut,t–Ip ti<t

e–δ(t–ti)

p–

ti<t

e–δ(t–ti)_Eγ

,ip

+ p–MpUQ

t,t–Ip

t<ti eδ(t–ti)

p–

t<ti

eδ(t–ti)_Eγ

,ip

≤p–MpUt,t–Ip

ti<t e–δ(t–ti)

p

sup

i∈ZEγ,i

p

+ p–MpUQ

t,t–Ip

t<ti eδ(t–ti)

p

sup

i∈ZEγ,i

≤p–Mp( –e

For anyε> , by Lemma ., there exist relative dense sets of real numbersεand integers

and

ˆ

J≤p–E

ti<t

Ut+τ,ti+q

P(ti+q) –U(t,ti)P(ti)γ,i

p

≤p–εpE

ti<t e–δ(t–ti)

p–

ti<t

e–δ(t–ti)_γ

,ip

≤p–εp ti<t

e–δ(t–ti) p

sup

i∈Z

Eγ,ip

≤ p–εp

( –e–δ_γ₎psup_i∈ZEγ,i p_.

Similar to the proof ofJˆ, ˆJ we have ˆJ ≤ ˆNε, Jˆ≤ ˆNε forNˆ,Nˆ> . Hence, ,i∈

APT(R,Lp(P,H)).

Step.,i∈PAPT(R,Lp(P,H)).

Forr> , we have

 r

r

–r

E,i(t) p

dt≤p– 

r

r

–r

E

ti<t

U(t,ti)P(ti)γ,i

pdt

+ p–  r

r

–r

E

t<ti

UQ(t,ti)Q(ti)γ,i

pdt.

For a giveni∈Z, define the functionv(t) byv(t) =U(t,ti)P(ti)γ,i,ti<t≤ti+, then

lim

t→∞Ev(t) p

=lim

t→∞EU(t,ti)P(ti)γ,i p

≤ lim

t→∞M

p_e–pδ(t–ti)_sup

i∈Z

Eγ,ip= .

Thusv∈PC

T(R,Lp(P,H))⊂PAPT(R,Lp(P,H)). Definevj:R→Lp(P,H) by

vj(t) =U(t,ti–j)P(ti–j)γ,i–j, ti<t≤ti+,j∈N.

Sovj∈PAPT(R,Lp(P,H)). Moreover,

Evj(t) p

=EU(t,ti–j)P(ti–j)γ,i–j p

≤Mpe–pδ(t–ti–j)_sup

i∈ZEγ,i

p

≤Mpe–pδ(t–ti)_e–pδαj_sup

i∈ZEγ,i

p_.

Therefore, the series∞_j=vjis uniformly convergent onR. By Lemma ., one has

ti<t

U(t,ti)P(ti)γ,i= ∞

j=

vj(t)∈PAPT

R,Lp(P,H),

that is,

p–  r

r

–r

E

ti<t

U(t,ti)P(ti)γ,i

Similarly, we have

p– 

r

r

–r

E

ti<t

UQ(t,ti)Q(ti)γ,i

pdt→ asr→ ∞.

Hence,,i∈PAPT(R,Lp(P,H)) and, we getγ˜i∈PAPT(R,Lp(P,H)). This completes the

proof.

Theorem . Assume that(H)-(H)are satisfied.Then system(.)-(.)has at least one p-mean piecewise pseudo almost periodic mild solution onR.

Proof Consider the operator:PAPT(R,Lp(P,H))∩UPC(R,Lp(P,H))→PC(R,Lp(P,H))

defined by

(x)(t) =

t

–∞

U(t,τ)P(τ)gτ,x(τ)dτ+ t

–∞

U(t,τ)P(τ)fτ,x(τ)dW(τ)

+

ti<t

U(t,ti)P(ti)Ii(xi)

+

–

+∞

t

UQ(t,τ)Q(τ)g

τ,x(τ)dτ

–

+∞

t

UQ(t,τ)Q(τ)f

τ,x(τ)dW(τ) –

t<ti

UQ(t,ti)Q(ti)Ii(xi)

=: (x)(t) + (x)(t), t∈R.

We next show thathas a fixed point inPAPT(R,Lp(P,H))∩UPC(R,Lp(P,H)) and divide

the proof into several steps.

Step. For everyx∈PAPT(R,Lp(P,H))∩UPC(R,Lp(P,H)),x∈PAPT(R,Lp(P,H))∩

UPC(R,Lp_(P,H)).

Letx(·)∈PAPT(R,Lp(P,H))∩UPC(R,Lp(P,H)), by (H) and Lemmas ., ., we

de-duce thatg(·,x(·)),f(·,x(·))∈PAPT(R,Lp(P,H)) andIi(x(ti))∈PAP(Z,Lp(P,H)) Similarly

as the proof of Lemmas .-., one hasx∈PAPT(R,Lp(P,H))∩UPC(R,Lp(P,H)).

Step.maps bounded sets into bounded sets inPAPT(R,Lp(P,H))∩UPC(R,Lp(P,H)).

Indeed, letr∗>  andx∈Br∗={x∈PAPT(R,Lp(P,H))∩UPC(R,Lp(P,H)) :Exp≤r∗}.

It is enough to show that there exists a positive constantL˜such that for eachx∈Br∗one

hasExp_{≤ ˜L. Letx∈B}

r∗, andt∈R. By (H), Hölder’s inequality, and the Ito integral,

we have, forp> ,

E(x)(t)

p

≤p–E t

–∞

U(t,τ)P(τ)gτ,x(τ)dτ p

+ p–E t

–∞U(t,

τ)P(τ)fτ,x(τ)dW(τ)

p

+ p–E

ti<t

U(t,ti)P(ti)Ii

x(ti)

p

≤p–Mp

t

–∞

e–δ(t–τ)_dτ

p– t

–∞

e–δ(t–τ)_Egτ,x(τ)p

dτ

+ p–CpMpE

t

–∞e

–δ(t–τ)_fτ,x(τ)

L_dτ

p/

+ p–MpE

ti<t e–δ(t–ti)

p–

ti<t

e–δ(t–ti)_I

i

x(ti)

p

≤p–Mp 

δp–

t

–∞

e–δ(t–τ)_Ex(τ)p

dτ

+ p–Mp

t

–∞

e– p

p–δ(t–τ)_dτ p–

p t

–∞

e–pδ(t–τ)_Ex(τ)p_dτ

+ p–_Mp 

( –e–δα₎p–

ti<t

e–δ(t–ti)˜

i

Ex(ti)

p

≤p–Mp 

δp

r∗+ p–Mp

p–  pδ

p– p _

pδ

r∗

+ p–Mp 

( –e–δα₎psup i∈Z

˜

i

r∗:=L.

Forp= , we have

E(x)(t)≤M



δ

r∗+ M

δ

r∗

+ M 

( –e–δα₎sup

i∈Z ˜

i

r∗:=L.

TakeL=max{L,L}. Then, for eachx∈Br∗, we haveExp≤L. Similarly, for each

x∈Br∗, we haveExp≤L. Hence, for eachx∈Br∗, we getExp≤L=L˜.

Step.:PAPT(R,Lp(P,H))∩UPC(R,Lp(P,H))→PAPT(R,Lp(P,H))∩UPC(R,Lp(P,

H)) is continuous.

Let{x(n)} ⊆Br∗ withx(n)→x(n→ ∞) inPAPT(R,Lp(P,H))∩UPC(R,Lp(P,H)), then

there exists a bounded subsetK⊆Lp_{(P,K) such thatR(x)⊆K,R(x}n_{)⊆K,n∈N. By the}

assumption (H), for anyε> , there exists ξ>  such thatx,y∈K andEx–yp<ξ

implies that

Egτ,x(τ)–gτ,y(τ)p<ε for allt∈R,

Efτ,x(τ)–fτ,y(τ)p_L

<ε for allt∈R,

and

EIi(x) –Ii(y) p

<ε for alli∈Z.

For the aboveξ there existsnsuch thatEx(n)(t) –x(t)p<εforn>nandt∈R, then,

forn>n, we have

Egτ,x(n)(τ)–gτ,x(τ)p<ε for allt∈R, Efτ,x(n)(τ)–fτ,x(τ)p_L

 <

and

Then, by Hölder’s inequality, we have, forp> ,

Ex(n)

Thusis continuous. Similarly, we can show thatis continuous and henceis

+ p–Mp

τ

τ

e–δ(τ–τ)_dτ p

r∗

+ p–MpCpU(τ,τ) –I

pp– 

pδ

p– p _

pδ

r∗

+ p–MpCp

τ

τ

e–pp–δ(τ–τ)_dτ p–

p τ

τ

e–pδ(τ–τ)_dτ

r∗

+ p–MpU(τ,τ) –Ip



( –e–δα₎psup i∈Z ˜

ir∗.

Forp= , we have

E(x)(τ) – (x)(τ)

≤MU(τ,τ) –I  

δ

r∗+ M

τ

τ

e–δ(τ–τ)_dτ 

r∗

+ MU(τ,τ) –I 

δ

r∗+ M

τ

τ

e–δ(τ–τ)_dτ

r∗

+ MU(τ,τ) –I  

( –e–δα₎sup

i∈Z

˜

i

r∗.

The right-hand side of the above inequality is independent ofx∈Br∗and tends to zero as τ→τ, since the compactness ofU(t,τ) fort–τ>  implies imply the continuity in the

uniform operator topology. Thus,mapsBr∗into an equicontinuous family of functions.

Similarly, we can show thatmapsBr∗into an equicontinuous family of functions and

hencemapsBr∗into an equicontinuous family of functions.

Step.(t) ={(x)(t) :x∈Br∗}is relatively compact inLp(P,H) for eacht∈R.

For eacht∈R, and letεbe a real number satisfying  <ε< . Forx∈Br∗, we define

_εx(t) =U(t,t–ε) t–ε

–∞

U(t–ε,τ)P(τ)gτ,x(τ)dτ

+ t–ε

–∞

U(t–ε,τ)P(τ)fτ,x(τ)dW(τ)

+

ti<t–ε

U(t–ε,ti)P(ti)Ii

x(ti)

.

SinceU(t,τ) (t–τ > ) is compact, then the setε_(t) ={(_εx)(t) :x∈Br∗}is relatively

compact inLp(P,H) for eacht∈R. Moreover, for everyx∈Br∗, we have forp> ,

E(x)(t) –

_εx(t)p ≤p–E

t

t–ε

U(t,τ)P(τ)gτ,x(τ)dτ p

+ p–E t

t–ε

U(t,τ)P(τ)fτ,x(τ)dW(τ)

p

+ p–E

t–ε<ti<t

U(t,ti)P(ti)Ii

x(ti)

≤p–Mp

Therefore, lettingε→, it follows that there are relatively compact setsε

(t) arbitrarily

close to(t) ={(x)(t) :x∈Br∗}and hence(t) is also relatively compact inLp(P,H)

for each t∈R. Similarly, we can show that(t) ={(x)(t) :x∈Br∗}is also relatively

compact inLp_{(P,H) for eacht∈R. Hence(t) is relatively compact inL}p_{(P,H) for each}

t∈R. Since{x:x∈Br∗} ⊂PCh(R,Lp(P,H)), then{x:x∈Br∗}is a relatively compact

set by Lemma ., thenis a compact operator.

+λ

Then, by (H), Hölder’s inequality, and the Ito integral, we have, forp> ,

Ex(t)p≤p–Mp 

Therefore, we have, forp> ,

+

ti<t

U(t,ti)P(ti)Ii

x(ti)

–

s<ti<t

U(t,ti)P(ti)Ii

x(ti)

–

t<ti

UQ(t,ti)Q(ti)Ii

x(ti)

+

s<t<ti

UQ(t,ti)Q(ti)Ii

x(ti)

=x(t) – t

s

U(t,τ)gτ,x(τ)dτ– t

s

U(t,τ)fτ,x(τ)dW(τ)

–

s<ti<t

U(t,ti)Ii

x(ti)

.

Hencexis a piecewise pseudo almost periodic mild solution to the system (.)-(.). The

proof is complete.

4 Exponential stability

In this section, we present the exponential stability of a piecewise pseudo almost periodic solution of (.)-(.). To do this, we also need the following assumptions:

(B) There exist constants <β<δ,l> and a continuous functionm:R→(,∞)

withm(t)≤le–βt,l> , such that

Eg(t,x)p+Ef(t,x)p_L ≤lx

p_{+m(t), t∈R,x∈L}p_(P,K).

(B) There exists a constantci> ,i∈Z, such that

EIi(x) p

≤ciExp, x∈Lp(P,K).

Theorem . Assume that assumptions of Theorem.hold and,in addition,hypotheses (B), (B)are satisfied.Then the piecewise pseudo almost periodic mild solution of(.) -(.)is exponentially stable,provided that

p–Mp



δp–_(δ–β)+Cp

p– 

pδ

p– p _

pδ– β

l

+ 

( –e–δα₎p–_{( –e}–(δ–β)α₎sup

i∈Zci

< 

for p> ,and

Mp



δ(δ–β)+  δ–β

l+



( –e–δα_{)( –e}–(δ–β)α₎sup

i∈Z

ci

< 

for p= .

Proof Letx(·) be a fixed point of inPAPT(R,Lp(P,H))∩UPC(R,Lp(P,H)). By

Theo-rem ., any fixed point ofis a mild solution of the system (.)-(.). We now can choose a positive constantβsuch that  <β<δ, and

eβtEx(t)p≤p–eβtE

t

–∞U(t,

τ)P(τ)gτ,x(τ)dτ p

+ p–eβt_E

+∞

t

UQ(t,τ)Q(τ)g

+ p–eβtE

By (B) and Hölder’s inequality, we have

ν≤p–Mpeβt

By (B), Hölder’s inequality, and the Ito integral, we have

By (B) and Hölder’s inequality, we have

ν≤p–MpeβtE

ti<t e–δ(t–ti)

p–

ti<t

e–δ(t–ti)_I

i

x(ti)

p

≤p–Mp  ( –e–δα₎p–e

βt

ti<t

e–δ(t–ti)_c

iEx(ti)p

≤p–Mp  ( –e–δα₎p

ti<t

e–(δ–β)(t–ti)_c

ieβtiEx(ti) p

≤p–Mp 

( –e–δα₎p–_{( –e}–(δ–β)α₎

sup

i∈Zcie βti_Ex(t

i) p

.

Similarly, we have

ν= p–Mp



( –e–δα₎p–_{( –e}–(δ–β)α₎

sup

i∈Z

cieβtiEx(ti) p

.

Thus, from the above inequality, it follows that

eβt_Ex(t)p_≤L∗_sup

τ∈R

eβτ_Ex(τ)p_+L

.

SinceL∗< , we have

sup

t∈Re

βt_Ex(t)p

≤ L

 –L∗,

where

L∗= p–Mp



δp–_(δ–β)+Cp

p– 

pδ

p– p _

pδ– β

l

+ 

( –e–δα₎p–_{( –e}–(δ–β)α₎sup

i∈Z

ci

,

L= p–Mp



δp–_(δ–β)+Cp

p– 

pδ

p– p _

pδ– β

l

forp> , and

L∗= Mp



δ(δ–β)+  δ–β

l+



( –e–δα_{)( –e}–(δ–β)α₎sup

i∈Z

ci

,

L= M



δ(δ–β)+  δ–β

l

forp= . Then we getEx(t)p_≤ L

–L∗e

–βt_{, which implies that the piecewise pseudo almost}