Existence of mild solutions for the impulsive semilinear nonlocal problem with random effects

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R E S E A R C H

Open Access

Existence of mild solutions for the impulsive

semilinear nonlocal problem with random

effects

Shuorui Zhang and Jitao Sun

*

*_{Correspondence: [email protected]} Department of Mathematics, Tongji University, Shanghai, 200092, China

Abstract

In this paper, we study the existence of impulsive semilinear nonlocal problems with random eﬀects. Random perturbations are taken into consideration in abstract spaces. By applying the measure of noncompactness and a random ﬁxed point theorem with stochastic domain, we get some existence results which improve and generalize many known results. Some weaker assumptions are established. Besides, we do not claim the semigroup to be compact.

Keywords: random ﬁxed point; existence; nonlocal conditions; measure of noncompactness; impulsive diﬀerential equations

1 Introduction

During the past few decades impulsive differential equations have been widely studied and significant progress has been made in [–] and references therein. Since many real world phenomena exhibit the presence of sudden state changes, the study of impulsive differen-tial equations is becoming more important nowadays. The theory of impulsive differendifferen-tial equations now not only is being recognized richer than the corresponding theory of differ-ential equations but also represents a more natural framework for mathematical modeling of many real world phenomena.

On the other hand, due to a combination of uncertainties and complexities, determin-istic equations can hardly describe a real system precisely. In order to take random fac-tors into account, many stochastic models were proposed and various achievements were obtained; see [, –] and references therein. Between them differential equations with random coefficients (see,e.g., [–] ) offer a natural and rational approach (see [, Chap-ter ]), since sometimes we can get the random distributions of some main disturbances by historical experiences and data rather than take all random disturbances into account and assume the noise to be white noises.

The main purpose of this paper is to discuss the existence of mild solutions for the non-local initial value problem with impulses and random eﬀects of the form

⎧ ⎪ ⎨ ⎪ ⎩

x(t,ω) =Ax(t,ω) +f(t,x(t,ω),ω), t=tk,k= , , . . . ,m,

x(t+_k,ω) –x(t–

k,ω) =Ik(x(tk,ω),ω), k= , , . . . ,m,

x(,ω) =g(x,ω)

()

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(with some notations to be given later), where (,F,P) is a complete probability space,

ω∈,Ais the inﬁnitesimal generator of a strongly continuous semigroup of bounded linear operators (i.e.,C-semigroup) T(t) in a Banach space X, f :I×X×→X,g :

PC(I,X)×→X, Ik :X×→X,k= , , . . . ,m, are given random functions which represent random nonlinear eﬀects of the system.

The investigation of nonlocal initial value problems was initiated by Byszewski in []. Later on, Byszewski’s work was followed by many others and a lot of work was presented in [, , –], since in some physics models the nonlocal problems play a better role than the classical ones. The corresponding deterministic case of problem (), with or without impulses, has been studied by many authors. For instance, Ntouyas and Tsamatos in [] investigated the case of compactness conditions of f andT(t). Xue in [] studied the problem whenf is compact. For impulsive semilinear nonlocal problems, in [], Liang

et al.obtained existence and uniqueness results by various assumptions of compactness or Lipschitz onT(t),f,g orIk. Very recently, in [, ] the authors put forward sufficient conditions on the existence of nonlocal impulsive differential equations by supposing the semigroupT(t) to be equicontinuous, but in their papers impulsive functions are always assumed to be compact or Lipschitzian. The goal of this paper is to continue the study of these authors by making full use of the Hausdorff measure of noncompactness and take random effects into consideration. The nonlocal problem with impulses and differential equations with random effects were studied in,e.g., [, ], respectively. In this paper, we study the nonlocal problem with impulses combined with random effects. Though the nonlocal problem with impulses and random effects was studied in,e.g., [] and refer-ences therein, in this paper we study impulsive semilinear nonlocal problem with ran-dom effects of another type. We prove existence results without compactness on T(t) or f and the Lipschitz condition onf, and we weaken assumptions on impulsive func-tionsIk.

The rest of this paper is organized as follows. In Section , we recall some notations and important lemmas. In Section , we give suﬃcient conditions on the existence of mild solutions of problem (). Finally, the conclusion is draw in Section .

2 Notations and preliminaries

In this section, we introduce some basic deﬁnitions and results which will be used in the paper.

Let (X, · ) be a real separable Banach space, let (,F,P) be a complete probability space, and letI= [,T] be a bounded closed interval inR,  =t<t<· · ·<tm<tm+=T.

ByC(I,X) we denote the Banach space of all continuous mappings fromI intoX with the norm fC=sup{f(t):t∈I}. Let PC(I,X) ={y:I→X:yis continuous fort=

tk,y(t–k) andy(tk+) exist withy(t–k) =y(tk),k= , . . . ,m}. It is easy to verify thatPC(I,X) is a separable Banach space with the normfPC=sup{f(t):t∈I}. We denote byL(I,X) the space of X-valued Bochner functions on I with the normx=_Tx(s)ds. Thus

C(I,X)⊆PC(I,X)⊆L₍_I_,_X_).

AC-semigroupT(t) is said to be compact ifT(t) is compact for anyt> . Ift→T(t)x

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LetY be a separable Banach space with the Borelσ-algebraBY. A mapping y:→Y is said to be a random variable with values inYif for eachB∈BY,y–(B)∈F. A mapping

T :×Y→Y is called a random operator ifT(·,y) is measurable for eachy∈Yand is generally expressed asT(ω,y) :=T(ω)y; we will use these two expressions alternatively.

Next, we will give a very useful random ﬁxed point theorem with stochastic domain.

Deﬁnition (see []) LetCbe a mapping frominto Y_{. A mapping}_T _:_{₍_ω_,_y₎_|_ω_∈

∧y∈C(ω)} →Yis called ‘random operator with stochastic domainC’ iﬀCis measur-able (i.e., for all closedA⊆Y,{ω∈|C(ω)∩A=∅} ∈F) and for all openD⊆Y and all

y∈Y,{ω∈|y∈C(ω)∧T(ω,y)∈D} ∈F.T will be called ‘continuous’ if everyT(ω) is continuous. For a random operatorT, a mappingy:→Yis called ‘random (stochastic) ﬁxed point ofT’ iﬀ forP-almost allω∈,y(ω)∈C(ω) andT(ω)y(ω) =y(ω) and for all openD⊆Y,{ω∈|y(ω)∈D} ∈F(‘yis measurable’).

Remark  IfC(ω)≡Y, then the deﬁnition of random operator with stochastic domain coincides with the deﬁnition of random operator.

Lemma (see []) Let C:→Y _{be measurable with C}₍_ω₎_closed_,_{convex and solid} (i.e.,int C(ω)=∅)for allω∈.We assume that there exists measurable y:→Y with y∈intC(ω)for allω∈.Let T be a continuous random operator with stochastic domain C such that for everyω∈,{y∈C(ω)|T(ω)y=y} =∅.Then T has a stochastic ﬁxed point.

Letξ be a mapping ofI×intoX.ξ is said to be a stochastic process if for eacht∈I,

ξ(t,·) is measurable and if, in addition, for eachω∈,ξ(·,ω)∈PC(I,X),ξis said to satisfy condition (PC,). Ifξsatisﬁes condition (PC,), thenξis considered to be a mapping of

intoPC(I,X).

By some similar techniques in [], without proof we state the following proposition.

Proposition  ξ satisﬁes condition(PC,)if and only ifξis measurable as a mapping of

into PC(I,X).

Deﬁnition  A stochastic processxis said to be a mild random solution of problem () if

it satisﬁes condition (PC,) and forω∈,

x(t,ω) =T(t)gx(·,ω)+ t



T(t–s)fs,x(s,ω),ωds+

<tk<t

T(t–tk)Ik

x(tk,ω),ω

for allt∈I.

Letαdenote the Hausdorﬀ measure of noncompactness both onXandPC(I,X), then we have the following ﬁxed point theorem.

Lemma ([]) Let F be a closed and convex subset of a real Banach space,let A:F→F be a continuous operator and A(F)be bounded.For each bounded subset B⊆F,set

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If there exist a constant≤k< and a positive integer nsuch that for each bounded subset B⊆F,

αAn₍_B₎≤_k_α₍_B_),

then A has a ﬁxed point in F.

For speciﬁc properties of measures of noncompactness, we refer the readers to []. To prove the existence results in this paper, we need the following lemmas concerning the relationship in measures of noncompactness.

Lemma ([]) If Y is a bounded subset of Banach space X,then for each> ,there is a sequence{yk}∞k=⊆Y such that

α(Y)≤α{yk}∞k=

+.

Lemma ([]) If{uk}k∞=⊆L(I,X)is uniformly integrable,thenα({uk(t)}∞k=)is measur-able and

α

t



uk(s)ds

∞

k=

≤ t



αuk(s) ∞

k=

ds for t∈[,T].

Lemma ([]) If the semigroup T(t)is equicontinuous andη∈L₍_I_,_R+_),_{then the set} {t→t

T(t–s)x(s)ds;x(s) ≤η(s),for a.e. s∈[,T]}is equicontinuous on[,T].

Lemma  ([]) If W ⊆PC(I,X)is bounded,then α(W(t))≤α(W)for all t∈I,where W(t) ={u(t);u∈W} ⊆X.Furthermore,suppose that the following conditions are satis-ﬁed:

() W is equicontinuous onJ= [,t]and eachJi= (ti,ti+],i= , . . . ,m;

() Wis equicontinuous att=t+

i,i= , . . . ,m.

Thensup_t_∈_Iα(W(t)) =α(W). 3 Main results

In this section, we give existence results for the nonlocal problem (). First we list the following conditions.

(HA) The C-semigroup T(t) generated byAis equicontinuous and we denote N=

sup{T(t):t∈I}.

(Hg)g:PC(I,X)×→X. For eachω∈,g(·,ω) is continuous and compact. (Hf )

(i) f :I×X×→X. For eachω∈,f(·,·,ω) :I×X→Xsatisﬁes the Caratheodory condition,i.e.,f(·,x,ω)is measurable for allx∈X, andf(t,·,ω)is continuous for a.e.t∈I.

(ii) There exist functionsφ:R+_×_→_R+_and_m_:_I_×_→_R+_{such that for each} ω∈,φ(·,ω)is nondecreasing and continuous andm(·,ω)integrable with

f(t,x,ω)≤m(t,ω)φx,ω

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(iii) There exists a functionL:I×→R+_with_L₍_·_,_ω₎_∈_L₍_I_,_R+₎_{for each}_ω_∈_such

that for any boundedB⊆X,

αf(t,B,ω)≤L(t,ω)α(B) ()

for a.e.t∈I.

(HI)Ik:X×→Xis sample path continuous and there existsLk:→R+such that forω∈and all bounded subsetB⊆X,

αIk(B,ω)

≤Lk(ω)α(B) ()

fork= , , . . . ,m.

(Hm) For eacht∈I,x∈X,f(t,x,·) andIk(x,·),k= , , . . . ,m, are measurable; for each

x∈PC(I,X),g(x,·) is measurable.

(HR) There exists a random functionR:→R+\ {}such that

sup

x∈BR(ω)

g(x,ω)+φR(ω),ω

T



m(s,ω)ds+ m

k= sup

x∈BR(ω)

Ik

x(tk),ω≤

R(ω)

N

for eachω∈, whereBrdenotes the closed ball in PC(I,X) centered at zero and with radiusr.

(H∗)

N

m

k=

Lk(ω) < , ∀ω∈.

Remark  Conditions () or () can be derived by the Lipschitz conditions onf or Ik,

k= , , . . . ,m, see [].

Theorem  Suppose that hypotheses(HA), (Hf), (Hg), (HI), (Hm), (HR)and(H∗)are valid,

then the nonlocal random impulsive problem()has at least one mild random solution.

Proof Consider the random operatorP:×PC(I,X)→PC(I,X) deﬁned by

P(ω)x(t) =T(t)g(x,ω) + t



T(t–s)fs,x(s),ωds

+

<tk<t

T(t–tk)Ik

x(tk),ω

, t∈I, ()

and we can dividePinto three parts,

F(ω)x(t) =T(t)g(x,ω),

G(ω)x(t) = t



T(t–s)fs,x(s),ωds,

H(ω)x(t) =

<tk<t

T(t–tk)Ik

x(tk),ω

.

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Then we show that the mapping deﬁned by () is a random operator. To do this, we need

(a) is satisfied since the mapping is well defined, (b) can be fulfilled by assumption (Hm). By some usual techniques (see,e.g., []), it is easy to show that the random operatorP is continuous.

LetW :→PC(I,X) _{be deﬁned by}_W₍_ω_{) =}_{x_∈_PC₍_I_,_X_{) :}_x

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αHB(t)=α

Thus, since>  is arbitrary,

αPB(t)≤

From Lemma  and Lemma , it is easy to verify the equicontinuity ofPB, thus

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≤N

Hence, by mathematical induction, for any positive integernandt∈[,T], we obtain

αPnB(t)≤

SincePn_B_{is equicontinuous on}_I_{, consequently we have}

αPnB≤

Sincea< , from Lemma . in [], there exists a positive integernsuch that

an₊_C Lemma ,P has a stochastic ﬁxed point,i.e., the nonlocal initial value problem has at

least one mild solution which completes the proof.

Remark  IfIkis sample path continuous and compact, thenLk=  fork= , , . . . ,m. Thus hypotheses (HI) and (H∗) hold naturally.

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(HR) There exists a random functionR:→R+_{\ {}__}_{such that}

cR(ω),ω+φR(ω),ω

T



m(s,ω)ds+ m

k= ck

R(ω),ω≤R

₍_ω₎

N

for eachω∈.

Corollary  Suppose that hypotheses(HA), (Hf), (Hg), (HI), (Hm), (HR)and(H∗)are valid,then the nonlocal random impulsive problem()has at least one mild random solu-tion.

Theorem  Suppose that hypotheses(HA), (Hf), (Hg), (HI), (Hm)and(H∗)are valid,

then the nonlocal random impulsive problem()has at least one mild random solution provided that

T



m(s,ω)ds≤lim inf

r→+∞

r–Nm_k_=ck(r,ω)

φ(r,ω) . ()

Furthermore,if Ikis compact for k= , , . . . ,m,then(H∗)is not essential.

Proof Since the proof is similar to Theorem , we omit it.

Remark  If we do not consider the random eﬀects,i.e., taking={ω}as a single point

set, since hypotheses onfandIkare weaker, we generalize the results in []. Furthermore, if let the impulsive functionsIk be , fork= , , . . . ,m, then since we relax restrictions ong, our results generalize the results in []. Besides, since there are no compactness as-sumptions on the semigroupT(t) orf, our work extends and improves many main results such as those in [–].

4 An example

Consider the impulsive partial diﬀerential system with nonlocal conditions and random eﬀects of the form

∂

∂tx(t,ξ,ω) = ∂

∂ξx(t,ξ,ω) +m(ξ)u(t,ξ,ω) +F

t,x(t,ξ,ω),ω

forξ∈[,π],t∈[,T],t=tk,k= , . . . ,m,

x(t, ,ω) =x(t,π,ω) = , t∈[,T],

xt_k+,ξ,ω–xt_k–,ξ,ω=Ik

xt_k–,ξ,ω,ω, k= , , . . . ,m,

x(,ξ,ω) = T



h(s,ω)log +x(s,ξ,ω)ds.

()

Suppose thatX=L_[,_π_{], (}_,_F_,_P_{) is a complete probability space. Let}_A_:_X_→_X_be

deﬁned byAy=ywith the domainD(A) ={y∈X:yis absolutely continuousy∈X,y() =

y(π) = }. It is well known thatAis an inﬁnitesimal generator of a semigroupT(t) deﬁned byT(t)y(s) =y(t+s) for eachy∈X.T(t) is not a compact semigroup onXandα(T(t)D)≤

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(H)f : [,T]×X×→Xis a continuous random function deﬁned by

f(t,x,ω)(ξ) =Ft,x(ξ),ω, t∈[,T],ξ∈[,π].

We takeF(t,x(ξ),ω) =c(ω)sin(x(ξ)),cis a real-valued random variable.F is Lipschitz continuous for the second variable. Thenf satisﬁes hypothesis (Hf ).

(H)Ik:X×→Xis a continuous random function fork= , , . . . ,m, deﬁned by

Ik(x,ω)(ξ) =Ik

x(ξ),ω.

We takeIk(x(ξ),ω) = π

 d(ω)ρk(ξ,y)cos(x(y))dy,x∈X,ρk∈C([,π]×[,π],R) fork=

, , . . . ,m,dis a real-valued random variable. ThenIkis compact thus satisﬁes hypothesis (HI).

(H)g:PC([,T];X)×→Xis a continuous random function deﬁned by

g(ϕ,ω)(ξ) = T



h(s,ω)log +ϕ(s)(ξ)ds, ϕ∈PC[,T];X

withϕ(s)(ξ) =x(s,ξ). Theng is a compact operator and satisﬁes hypothesis (Hg). Under these assumptions, the partial diﬀerential system () can be reformulated as the abstract problem (). By Theorem  we conclude that there exists at least one random solution for this system.

5 Conclusion

In this paper, we have studied existence results for impulsive semilinear nonlocal problems with random effects. Random perturbations have been taken into consideration in abstract spaces. By applying the measure of noncompactness and a random fixed point theorem with stochastic domain, we have obtained some existence results for impulsive semilinear nonlocal problems with random effects. Our results have improved and generalized many known results. Some weaker assumptions have been established.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

SZ completed the proof and wrote the initial draft. JS provided the problem and gave some suggestions on the amendment. SZ then ﬁnalized the manuscript. Correspondence was mainly done by JS. All authors read and approved the ﬁnal manuscript.

Acknowledgements

This work is supported by the NSF of China under grant No. 61174039.

Received: 29 October 2013 Accepted: 9 July 2014 Published:23 Jul 2014

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10.1186/1687-1847-2014-194