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

- Shuorui Zhang
^{1}and - Jitao Sun
^{1}Email author

**2014**:194

https://doi.org/10.1186/1687-1847-2014-194

© Zhang and Sun; licensee Springer. 2014

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

## 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 [1–8] and references therein. Since many real world phenomena exhibit the presence of sudden state changes, the study of impulsive differential equations is becoming more important nowadays. The theory of impulsive differential equations now not only is being recognized richer than the corresponding theory of differential 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, deterministic equations can hardly describe a real system precisely. In order to take random factors into account, many stochastic models were proposed and various achievements were obtained; see [5, 9–13] and references therein. Between them differential equations with random coefficients (see, *e.g.*, [13–15] ) offer a natural and rational approach (see [[16], Chapter 1]), 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.

(with some notations to be given later), where $(\mathrm{\Omega},\mathcal{F},P)$ is a complete probability space, $\omega \in \mathrm{\Omega}$, *A* is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators (*i.e.*, ${C}_{0}$-semigroup) $T(t)$ in a Banach space *X*, $f:I\times X\times \mathrm{\Omega}\to X$, $g:PC(I,X)\times \mathrm{\Omega}\to X$, ${I}_{k}:X\times \mathrm{\Omega}\to X$, $k=1,2,\dots ,m$, are given random functions which represent random nonlinear effects of the system.

The investigation of nonlocal initial value problems was initiated by Byszewski in [17]. Later on, Byszewski’s work was followed by many others and a lot of work was presented in [7, 8, 18–23], since in some physics models the nonlocal problems play a better role than the classical ones. The corresponding deterministic case of problem (1), with or without impulses, has been studied by many authors. For instance, Ntouyas and Tsamatos in [18] investigated the case of compactness conditions of *f* and $T(t)$. Xue in [19] studied the problem when *f* is compact. For impulsive semilinear nonlocal problems, in [20], Liang *et al.* obtained existence and uniqueness results by various assumptions of compactness or Lipschitz on $T(t)$, *f*, *g* or ${I}_{k}$. Very recently, in [7, 8] the authors put forward sufficient conditions on the existence of nonlocal impulsive differential equations by supposing the semigroup $T(t)$ 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.*, [8, 15], 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.*, [24] and references therein, in this paper we study impulsive semilinear nonlocal problem with random effects of another type. We prove existence results without compactness on $T(t)$ or *f* and the Lipschitz condition on *f*, and we weaken assumptions on impulsive functions ${I}_{k}$.

The rest of this paper is organized as follows. In Section 2, we recall some notations and important lemmas. In Section 3, we give sufficient conditions on the existence of mild solutions of problem (1). Finally, the conclusion is draw in Section 4.

## 2 Notations and preliminaries

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

Let $(X,\parallel \cdot \parallel )$ be a real separable Banach space, let $(\mathrm{\Omega},\mathcal{F},P)$ be a complete probability space, and let $I=[0,T]$ be a bounded closed interval in ℝ, $0={t}_{0}<{t}_{1}<\cdots <{t}_{m}<{t}_{m+1}=T$. By $C(I,X)$ we denote the Banach space of all continuous mappings from *I* into *X* with the norm ${\parallel f\parallel}_{C}=sup\{\parallel f(t)\parallel :t\in I\}$. Let $PC(I,X)$ = {$y:I\to X:y$ is continuous for $t\ne {t}_{k}$, $y({t}_{k}^{-})$ and $y({t}_{k}^{+})$ exist with $y({t}_{k}^{-})=y({t}_{k})$, $k=1,\dots ,m$}. It is easy to verify that $PC(I,X)$ is a separable Banach space with the norm ${\parallel f\parallel}_{PC}=sup\{\parallel f(t)\parallel :t\in I\}$. We denote by ${L}^{1}(I,X)$ the space of *X*-valued Bochner functions on *I* with the norm $\parallel x\parallel ={\int}_{0}^{T}\parallel x(s)\parallel \phantom{\rule{0.2em}{0ex}}ds$. Thus $C(I,X)\subseteq PC(I,X)\subseteq {L}^{1}(I,X)$.

A ${C}_{0}$-semigroup $T(t)$ is said to be compact if $T(t)$ is compact for any $t>0$. If $t\to T(t)x$ is equicontinuous at all $t>0$ with respect to *x* in all bounded subsets of *X*, then the semigroup $T(t)$ is called equicontinuous. Note that if the semigroup $T(t)$ is compact, then it must be equicontinuous (*cf.* [25]).

Let *Y* be a separable Banach space with the Borel *σ*-algebra ${\mathcal{B}}_{Y}$. A mapping y: $\mathrm{\Omega}\to Y$ is said to be a random variable with values in *Y* if for each $B\in {\mathcal{B}}_{Y}$, ${y}^{-1}(B)\in \mathcal{F}$. A mapping $T:\mathrm{\Omega}\times Y\to Y$ is called a random operator if $T(\cdot ,y)$ is measurable for each $y\in Y$ and is generally expressed as $T(\omega ,y):=T(\omega )y$; we will use these two expressions alternatively.

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

**Definition 1** (see [26])

Let *C* be a mapping from Ω into ${2}^{Y}$. A mapping $T:\{(\omega ,y)|\omega \in \mathrm{\Omega}\wedge y\in C(\omega )\}\to Y$ is called ‘random operator with stochastic domain *C*’ iff *C* is measurable (*i.e.*, for all closed $A\subseteq Y$, $\{\omega \in \mathrm{\Omega}|C(\omega )\cap A\ne \mathrm{\varnothing}\}\in \mathcal{F}$) and for all open $D\subseteq Y$ and all $y\in Y$, $\{\omega \in \mathrm{\Omega}|y\in C(\omega )\wedge T(\omega ,y)\in D\}\in \mathcal{F}$. *T* will be called ‘continuous’ if every $T(\omega )$ is continuous. For a random operator *T*, a mapping $y:\mathrm{\Omega}\to Y$ is called ‘random (stochastic) fixed point of *T*’ iff for *P*-almost all $\omega \in \mathrm{\Omega}$, $y(\omega )\in C(\omega )$ and $T(\omega )y(\omega )=y(\omega )$ and for all open $D\subseteq Y$, $\{\omega \in \mathrm{\Omega}|y(\omega )\in D\}\in \mathcal{F}$ (‘*y* is measurable’).

**Remark 1** If $C(\omega )\equiv Y$, then the definition of random operator with stochastic domain coincides with the definition of random operator.

**Lemma 1** (see [26])

*Let* $C:\mathrm{\Omega}\to {2}^{Y}$ *be measurable with* $C(\omega )$ *closed*, *convex and solid* (*i*.*e*., $int\phantom{\rule{0.25em}{0ex}}C(\omega )\ne \mathrm{\varnothing}$) *for all* $\omega \in \mathrm{\Omega}$. *We assume that there exists measurable* ${y}_{0}:\mathrm{\Omega}\to Y$ *with* ${y}_{0}\in int\phantom{\rule{0.25em}{0ex}}C(\omega )$ *for all* $\omega \in \mathrm{\Omega}$. *Let* *T* *be a continuous random operator with stochastic domain* *C* *such that for every* $\omega \in \mathrm{\Omega}$, $\{y\in C(\omega )|T(\omega )y=y\}\ne \mathrm{\varnothing}$. *Then* *T* *has a stochastic fixed point*.

Let *ξ* be a mapping of $I\times \mathrm{\Omega}$ into *X*. *ξ* is said to be a stochastic process if for each $t\in I$, $\xi (t,\cdot )$ is measurable and if, in addition, for each $\omega \in \mathrm{\Omega}$, $\xi (\cdot ,\omega )\in PC(I,X)$, *ξ* is said to satisfy condition $(PC,\mathrm{\Omega})$. If *ξ* satisfies condition $(PC,\mathrm{\Omega})$, then *ξ* is considered to be a mapping of Ω into $PC(I,X)$.

By some similar techniques in [27], without proof we state the following proposition.

**Proposition 1** *ξ* *satisfies condition* $(PC,\mathrm{\Omega})$ *if and only if* *ξ* *is measurable as a mapping of* Ω *into* $PC(I,X)$.

**Definition 2**A stochastic process

*x*is said to be a mild random solution of problem (1) if it satisfies condition $(PC,\mathrm{\Omega})$ and for $\omega \in \mathrm{\Omega}$,

for all $t\in I$.

Let *α* denote the Hausdorff measure of noncompactness both on *X* and $PC(I,X)$, then we have the following fixed point theorem.

**Lemma 2** ([28])

*Let*

*F*

*be a closed and convex subset of a real Banach space*,

*let*$A:F\to F$

*be a continuous operator and*$A(F)$

*be bounded*.

*For each bounded subset*$B\subseteq F$,

*set*

*If there exist a constant*$0\le k<1$

*and a positive integer*${n}_{0}$

*such that for each bounded subset*$B\subseteq F$,

*then* *A* *has a fixed point in* *F*.

For specific properties of measures of noncompactness, we refer the readers to [29].

To prove the existence results in this paper, we need the following lemmas concerning the relationship in measures of noncompactness.

**Lemma 3** ([30])

*If*

*Y*

*is a bounded subset of Banach space*

*X*,

*then for each*$\u03f5>0$,

*there is a sequence*${\{{y}_{k}\}}_{k=1}^{\mathrm{\infty}}\subseteq Y$

*such that*

**Lemma 4** ([31])

*If*${\{{u}_{k}\}}_{k=1}^{\mathrm{\infty}}\subseteq {L}^{1}(I,X)$

*is uniformly integrable*,

*then*$\alpha ({\{{u}_{k}(t)\}}_{k=1}^{\mathrm{\infty}})$

*is measurable and*

**Lemma 5** ([32])

*If the semigroup* $T(t)$ *is equicontinuous and* $\eta \in {L}^{1}(I,{\mathbb{R}}^{+})$, *then the set* $\{t\to {\int}_{0}^{t}T(t-s)x(s)\phantom{\rule{0.2em}{0ex}}ds;\parallel x(s)\parallel \le \eta (s),\mathit{\text{for a.e.}}s\in [0,T]\}\mathit{\text{is equicontinuous on}}[0,T]$.

**Lemma 6** ([7])

*If*$W\subseteq PC(I,X)$

*is bounded*,

*then*$\alpha (W(t))\le \alpha (W)$

*for all*$t\in I$,

*where*$W(t)=\{u(t);u\in W\}\subseteq X$.

*Furthermore*,

*suppose that the following conditions are satisfied*:

- (1)
*W**is equicontinuous on*${J}_{0}=[0,{t}_{1}]$*and each*${J}_{i}=({t}_{i},{t}_{i+1}]$, $i=1,\dots ,m$; - (2)
*W**is equicontinuous at*$t={t}_{i}^{+}$, $i=1,\dots ,m$.

*Then* ${sup}_{t\in I}\alpha (W(t))=\alpha (W)$.

## 3 Main results

In this section, we give existence results for the nonlocal problem (1). First we list the following conditions.

(HA) The ${C}_{0}$-semigroup $T(t)$ generated by *A* is equicontinuous and we denote $N=sup\{\parallel T(t)\parallel :t\in I\}$.

(Hg) $g:PC(I,X)\times \mathrm{\Omega}\to X$. For each $\omega \in \mathrm{\Omega}$, $g(\cdot ,\omega )$ is continuous and compact.

- (i)
$f:I\times X\times \mathrm{\Omega}\to X$. For each $\omega \in \mathrm{\Omega}$, $f(\cdot ,\cdot ,\omega ):I\times X\to X$ satisfies the Caratheodory condition,

*i.e.*, $f(\cdot ,x,\omega )$ is measurable for all $x\in X$, and $f(t,\cdot ,\omega )$ is continuous for a.e. $t\in I$. - (ii)There exist functions $\varphi :{\mathbb{R}}^{+}\times \mathrm{\Omega}\to {\mathbb{R}}^{+}$ and $m:I\times \mathrm{\Omega}\to {\mathbb{R}}^{+}$ such that for each $\omega \in \mathrm{\Omega}$, $\varphi (\cdot ,\omega )$ is nondecreasing and continuous and $m(\cdot ,\omega )$ integrable with$\parallel f(t,x,\omega )\parallel \le m(t,\omega )\varphi (\parallel x\parallel ,\omega )$

- (iii)There exists a function $L:I\times \mathrm{\Omega}\to {\mathbb{R}}^{+}$ with $L(\cdot ,\omega )\in {L}^{1}(I,{\mathbb{R}}^{+})$ for each $\omega \in \mathrm{\Omega}$ such that for any bounded $B\subseteq X$,$\alpha (f(t,B,\omega ))\le L(t,\omega )\alpha (B)$(2)

for a.e. $t\in I$.

for $k=1,2,\dots ,m$.

(Hm) For each $t\in I$, $x\in X$, $f(t,x,\cdot )$ and ${I}_{k}(x,\cdot )$, $k=1,2,\dots ,m$, are measurable; for each $x\in PC(I,X)$, $g(x,\cdot )$ is measurable.

for each $\omega \in \mathrm{\Omega}$, where ${B}_{r}$ denotes the closed ball in $PC(I,X)$ centered at zero and with radius *r*.

^{∗})

**Remark 2** Conditions (2) or (3) can be derived by the Lipschitz conditions on *f* or ${I}_{k}$, $k=1,2,\dots ,m$, see [23].

**Theorem 1** *Suppose that hypotheses* (HA), (Hf), (Hg), (HI), (Hm), (HR) *and* (H^{∗}) *are valid*, *then the nonlocal random impulsive problem* (1) *has at least one mild random solution*.

*Proof*Consider the random operator $\mathcal{P}:\mathrm{\Omega}\times PC(I,X)\to PC(I,X)$ defined by

For each $\omega \in \mathrm{\Omega}$ and each $x\in PC(I,X)$, by the definition of $\mathcal{P}$, clearly $\mathcal{P}(\omega )(x)\in PC(I,X)$. Thus the mapping is well defined.

*X*, satisfies condition $(PC,\mathrm{\Omega})$,

*i.e.*,

- (a)
for any $\omega \in \mathrm{\Omega}$, $\mathcal{P}(\omega ,x)(t)\in PC(I,X)$;

- (b)
for any $t\in I$, $\mathcal{P}(\cdot ,x)(t):\mathrm{\Omega}\to X$ is measurable;

(a) is satisfied since the mapping is well defined, (b) can be fulfilled by assumption (Hm).

By some usual techniques (see, *e.g.*, [7]), it is easy to show that the random operator $\mathcal{P}$ is continuous.

Let $W:\mathrm{\Omega}\to {2}^{PC(I,X)}$ be defined by $W(\omega )=\{x\in PC(I,X):{\parallel x\parallel}_{PC}\le R(\omega )\}$ with $W(\omega )$ bounded, closed, convex and solid for all $\omega \in \mathrm{\Omega}$. Then *W* is measurable by Lemma 17 in [26].

This implies that $\mathcal{P}$ is a random operator with stochastic domain *W* and $\mathcal{P}(\omega ):W(\omega )\to W(\omega )$ for each $\omega \in \mathrm{\Omega}$.

Next, let $\omega \in \mathrm{\Omega}$ be fixed (therefore we do not write ‘*ω*’ in the sequel) but arbitrary.

where $a=N{\sum}_{k=1}^{m}{\mathcal{L}}_{k}+4N\gamma $, $b=4NM$.

*ϵ*is arbitrary,

*n*and $t\in [0,T]$, we obtain

*I*, consequently we have

It follows from Lemma 2 that for each $\omega \in \mathrm{\Omega}$, $\mathcal{P}$ has at least one fixed point in *W*. Since ${\bigcap}_{\omega \in \mathrm{\Omega}}intW(\omega )\ne \mathrm{\varnothing}$, the hypothesis that a measurable selector of int*W* exists holds. By Lemma 1, $\mathcal{P}$ has a stochastic fixed point, *i.e.*, the nonlocal initial value problem has at least one mild solution which completes the proof. □

**Remark 3** If ${I}_{k}$ is sample path continuous and compact, then ${\mathcal{L}}_{k}=0$ for $k=1,2,\dots ,m$. Thus hypotheses (HI) and (H^{∗}) hold naturally.

Actually, (HR) is a very rough condition. If we specify some features on ${I}_{k}$ and *g*, $k=1,2,\dots ,m$, we have the following corollary.

(Hg′) $g:PC(I,X)\times \mathrm{\Omega}\to X$. For each $\omega \in \mathrm{\Omega}$, $g(\cdot ,\omega )$ is continuous and compact, there exists ${c}_{0}:{\mathbb{R}}^{+}\times \mathrm{\Omega}\to {\mathbb{R}}^{+}$ such that $\parallel g(x,\omega )\parallel \le {c}_{0}({\parallel x\parallel}_{PC},\omega )$, $\mathrm{\forall}x\in PC(I,X)$.

for $k=1,2,\dots ,m$.

for each $\omega \in \mathrm{\Omega}$.

**Corollary 1** *Suppose that hypotheses* (HA), (Hf), (Hg′), (HI′), (Hm), (HR′) *and* (H^{∗}) *are valid*, *then the nonlocal random impulsive problem* (1) *has at least one mild random solution*.

**Theorem 2**

*Suppose that hypotheses*(HA), (Hf), (Hg′), (HI′), (Hm)

*and*(H

^{∗})

*are valid*,

*then the nonlocal random impulsive problem*(1)

*has at least one mild random solution provided that*

*Furthermore*, *if* ${I}_{k}$ *is compact for* $k=1,2,\dots ,m$, *then* (H^{∗}) *is not essential*.

*Proof* Since the proof is similar to Theorem 1, we omit it. □

**Remark 4** If we do not consider the random effects, *i.e.*, taking $\mathrm{\Omega}=\{{\omega}_{0}\}$ as a single point set, since hypotheses on *f* and ${I}_{k}$ are weaker, we generalize the results in [7]. Furthermore, if let the impulsive functions ${I}_{k}$ be 0, for $k=1,2,\dots ,m$, then since we relax restrictions on *g*, our results generalize the results in [23]. Besides, since there are no compactness assumptions on the semigroup $T(t)$ or *f*, our work extends and improves many main results such as those in [19–21].

## 4 An example

Suppose that $X={L}^{2}[0,\pi ]$, $(\mathrm{\Omega},\mathcal{F},P)$ is a complete probability space. Let $A:X\to X$ be defined by $Ay={y}^{\prime}$ with the domain $D(A)=\{y\in X:y\text{is absolutely continuous}{y}^{\prime}\in X,y(0)=y(\pi )=0\}$. It is well known that *A* is an infinitesimal generator of a semigroup $T(t)$ defined by $T(t)y(s)=y(t+s)$ for each $y\in X$. $T(t)$ is not a compact semigroup on *X* and $\alpha (T(t)D)\le \alpha (D)$, where *α* is the Hausdorff MNC. We assume the following.

We take $F(t,x(\xi ),\omega )={c}_{0}(\omega )sin(x(\xi ))$, ${c}_{0}$ is a real-valued random variable. *F* is Lipschitz continuous for the second variable. Then *f* satisfies hypothesis (Hf).

We take ${I}_{k}(x(\xi ),\omega )={\int}_{0}^{\pi}{d}_{0}(\omega ){\rho}_{k}(\xi ,y){cos}^{2}(x(y))\phantom{\rule{0.2em}{0ex}}dy$, $x\in X$, ${\rho}_{k}\in C([0,\pi ]\times [0,\pi ],R)$ for $k=1,2,\dots ,m$, ${d}_{0}$ is a real-valued random variable. Then ${I}_{k}$ is compact thus satisfies hypothesis (HI).

with $\phi (s)(\xi )=x(s,\xi )$. Then *g* is a compact operator and satisfies hypothesis (Hg). Under these assumptions, the partial differential system (7) can be reformulated as the abstract problem (1). By Theorem 1 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.

## Declarations

### Acknowledgements

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

## Authors’ Affiliations

## References

- Lakshmikantham V, Bainov DD, Simeonov PS:
*Theory of Impulsive Differential Equations*. World Scientific, Singapore; 1989.View ArticleGoogle Scholar - Zhang S, Sun J: Stability analysis of second-order differential systems with Erlang distribution random impulses.
*Adv. Differ. Equ.*2013., 2013: Article ID 4Google Scholar - Chen L, Sun J: Nonlinear boundary value problem of first order impulsive functional differential equations.
*J. Math. Anal. Appl.*2006, 318: 726-741. 10.1016/j.jmaa.2005.08.012MathSciNetView ArticleGoogle Scholar - Lu J, Ho D, Cao J: A unified synchronization criterion for impulsive dynamical networks.
*Automatica*2010, 46: 1215-1221. 10.1016/j.automatica.2010.04.005MathSciNetView ArticleGoogle Scholar - Li C, Shi J, Sun J: Stability of impulsive stochastic differential delay systems and its application to impulsive stochastic neural networks.
*Nonlinear Anal., Theory Methods Appl.*2011, 74: 3099-3111. 10.1016/j.na.2011.01.026MathSciNetView ArticleGoogle Scholar - Pan L, Cao J: Exponential stability of impulsive stochastic functional differential equations.
*J. Math. Anal. Appl.*2011, 382: 672-685. 10.1016/j.jmaa.2011.04.084MathSciNetView ArticleGoogle Scholar - Ji S, Li G: A unified approach to nonlocal impulsive differential equations with the measure of noncompactness.
*Adv. Differ. Equ.*2012., 2012: Article ID 182Google Scholar - Zhu L, Dong Q, Li G: Impulsive differential equations with nonlocal conditions in general Banach spaces.
*Adv. Differ. Equ.*2012., 2012: Article ID 10Google Scholar - Li C, Sun R, Sun J: Stability analysis of a class of stochastic differential delay equations with nonlinear impulsive effects.
*J. Franklin Inst. Eng. Appl. Math.*2010, 347: 1186-1198. 10.1016/j.jfranklin.2010.04.017View ArticleGoogle Scholar - Bao H, Cao J: Stochastic global exponential stability for neutral-type impulsive neural networks with mixed time-delays and Markovian jumping parameters.
*Commun. Nonlinear Sci. Numer. Simul.*2011, 16: 3786-3791. 10.1016/j.cnsns.2010.12.027MathSciNetView ArticleGoogle Scholar - Yang X, Cao J, Lu J: Synchronization of delayed complex dynamical networks with impulsive and stochastic effects.
*Nonlinear Anal., Real World Appl.*2011, 12: 2252-2266. 10.1016/j.nonrwa.2011.01.007MathSciNetView ArticleGoogle Scholar - Liu B: Stability of solutions for stochastic impulsive systems via comparison approach.
*IEEE Trans. Autom. Control*2008, 53: 2128-2133.View ArticleGoogle Scholar - Stanescu D, Chen-Charpentier BM: Random coefficient differential equation models for bacterial growth.
*Math. Comput. Model.*2009, 50: 885-895. 10.1016/j.mcm.2009.05.017MathSciNetView ArticleGoogle Scholar - Csoergo S, Hatvani L: Stability properties of solutions of linear second order differential equations with random coefficients.
*J. Differ. Equ.*2010, 248: 21-49. 10.1016/j.jde.2009.08.001View ArticleGoogle Scholar - Charrier J: Strong and weak error estimates for elliptic partial differential equations with random coefficients.
*SIAM J. Numer. Anal.*2012, 50: 216-246. 10.1137/100800531MathSciNetView ArticleGoogle Scholar - Soong TT:
*Random Differential Equations in Science and Engineering*. Academic Press, New York; 1973.Google Scholar - Byszewski L: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem.
*J. Math. Anal. Appl.*1991, 162: 494-505. 10.1016/0022-247X(91)90164-UMathSciNetView ArticleGoogle Scholar - Ntouyas S, Tsamatos PC: Global existence for semilinear evolution equations with nonlocal conditions.
*J. Math. Anal. Appl.*1997, 210: 679-687. 10.1006/jmaa.1997.5425MathSciNetView ArticleGoogle Scholar - Xue X: Existence of solutions for semilinear nonlocal Cauchy problems in Banach spaces.
*Electron. J. Differ. Equ.*2005., 2005: Article ID 64Google Scholar - Liang J, Liu J, Xiao T: Nonlocal impulsive problems for nonlinear differential equations in Banach spaces.
*Math. Comput. Model.*2009, 49: 798-804. 10.1016/j.mcm.2008.05.046MathSciNetView ArticleGoogle Scholar - Fan Z: Impulsive problems for semilinear differential equations with nonlocal conditions.
*Nonlinear Anal., Theory Methods Appl.*2010, 72: 1104-1109. 10.1016/j.na.2009.07.049View ArticleGoogle Scholar - Olszowy L: Existence of mild solutions for the semilinear nonlocal problem in Banach spaces.
*Nonlinear Anal., Theory Methods Appl.*2013, 72: 211-223.MathSciNetView ArticleGoogle Scholar - Zhu T, Song C, Li G: Existence of mild solutions for abstract semilinear evolution equations in Banach spaces.
*Nonlinear Anal., Theory Methods Appl.*2012, 75: 177-181. 10.1016/j.na.2011.08.019MathSciNetView ArticleGoogle Scholar - Cui J, Yan L: Existence results for impulsive neutral second-order stochastic evolution equations with nonlocal conditions.
*Math. Comput. Model.*2013, 57: 2378-2387. 10.1016/j.mcm.2011.12.037MathSciNetView ArticleGoogle Scholar - Pazy A:
*Semigroups of Linear Operators and Applications to Partial Differential Equations*. Springer, New York; 1983.View ArticleGoogle Scholar - Engl HW: A general stochastic fixed-point theorem for continuous random operators on stochastic domains.
*J. Math. Anal. Appl.*1978, 66: 220-231. 10.1016/0022-247X(78)90279-2MathSciNetView ArticleGoogle Scholar - Itoh S: Random fixed point theorems with an application to random differential equations in Banach spaces.
*J. Math. Anal. Appl.*1979, 67: 261-273. 10.1016/0022-247X(79)90023-4MathSciNetView ArticleGoogle Scholar - Liu L, Guo F, Wu C, Wu Y: Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces.
*J. Math. Anal. Appl.*2005, 309: 638-649. 10.1016/j.jmaa.2004.10.069MathSciNetView ArticleGoogle Scholar - Deimling K:
*Nonlinear Functional Analysis*. Springer, Berlin; 1985.View ArticleGoogle Scholar - Bothe D: Multivalued perturbation of
*m*-accretive differential inclusions.*Isr. J. Math.*1998, 108: 109-138. 10.1007/BF02783044MathSciNetView ArticleGoogle Scholar - Mönch H: Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces.
*Nonlinear Anal., Theory Methods Appl.*1980, 4: 985-999. 10.1016/0362-546X(80)90010-3View ArticleGoogle Scholar - Xue X: Semilinear nonlocal differential equations with measure of noncompactness in Banach spaces.
*J. Nanjing Univ. Math. Biq.*2007, 24: 264-276.Google Scholar - Zhang X, Liu L, Wu Y: Global solutions of nonlinear second-order impulsive integro-differential equations of mixed type in Banach spaces.
*Nonlinear Anal., Theory Methods Appl.*2007, 67: 2335-2349. 10.1016/j.na.2006.08.033MathSciNetView ArticleGoogle Scholar

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