Open Access Research

Exponential stability of mild solutions to impulsive stochastic neutral partial differential equations with memory

Hua Yang1 and Feng Jiang2

Author Affiliations

1 School of Mathematics & Computer Science, Wuhan Polytechnic University, Wuhan, Hubei, China

2 School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan, Hubei, China

Advances in Difference Equations 2013, 2013:148  doi:10.1186/1687-1847-2013-148


The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2013/1/148


Received:20 March 2013
Accepted:7 May 2013
Published:23 May 2013

© 2013 Yang and Jiang; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we study the exponential stability in the pth moment of mild solutions to impulsive stochastic neutral partial differential equations with memory. Sufficient conditions ensuring the stability of the impulsive stochastic system are obtained by establishing a new integral inequality. The results obtained here generalize and improve some well-known results.

1 Introduction

At present, the study of stochastic partial different equations in a separable Hilbert space has become an important area of investigation in the past two decades because of their applications to various problems arising in physics, biology, engineering etc.[1,2]. The existence, uniqueness and stability of solutions of stochastic partial differential equations have been considered by many authors [2-12]. The stability of strong solutions of stochastic differential equations also have been discussed extensively [13-15]. However, there are a number of difficulties encountered in the study of stability by the Lyapunov second method. By the Banach fixed point theory, [16] studied a linear scalar neutral stochastic differential equation with variable delays and gave conditions to ensure that the zero solution is asymptotically mean square stable. Further [17] considered the stability of stochastic partial differential equations with delays by using the Banach fixed point theory.

On the other hand, the impulsive effects exist in many evolution processes, in which states are changed abruptly at certain moments of time, involved in such fields as medicine and biology, economics, mechanics, electronics [18,19]. In recent years, the investigation of impulsive stochastic differential equations attracts great attention, especially as regards stability. For example, [20] discussed the stability of impulsive stochastic systems. [21,22] discussed the exponential stability in mean square of impulsive stochastic difference equations by establishing difference inequalities. Jiang and Shen [23] discussed the asymptotic stability of impulsive stochastic neutral partial differential equations with infinite delays.

As known, although the Lyapunov second method is a powerful technique in proving the stability theorems, it is not so suitable in the non-delay case. A difficulty is that mild solutions do not have stochastic differentials, so that one cannot apply the Itô formula to them. Meanwhile, the difficulty of the method of the fixed point theory comes from finding an appropriate fixed point theorem. Therefore, the techniques and the methods for the stability of mild solutions should be developed and explored. In this present work, motivated by [21-23], we study the exponential stability of impulsive stochastic neutral partial differential equations with memory by establishing a new integral inequality. The results obtained here generalize the main results from [3,6,17] to cover a class of more general impulsive stochastic neutral systems.

The rest of this paper is organized as follows. In Section 2, we introduce some preliminaries. In Section 3 sufficient conditions ensuring the stability of the impulsive stochastic system are obtained by establishing a new integral inequality.

2 Preliminaries

Throughout this paper, let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M1">View MathML</a> be a complete probability space with a normal filtration <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M2">View MathML</a> satisfying the usual conditions (i.e., it is increasing and right-continuous while <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M3">View MathML</a> contains all P-null sets). Moreover, let X, Y be two real separable Hilbert spaces and let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M4">View MathML</a> denote the space of all bounded linear operators from Y into X.

For simplicity, we use the notation <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M5">View MathML</a> to denote the norm in X, Y and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M6">View MathML</a> to denote the operator norm in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M7">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M4">View MathML</a>. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M9">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M10">View MathML</a> denote the inner products of X, Y, respectively. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M11">View MathML</a> denote a Y-valued Wiener process defined on the probability space <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M12">View MathML</a> with a covariance operator Q, that is, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M13">View MathML</a>, for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M14">View MathML</a>, where Q is a positive, self-adjoint, trace class operator on Y. In particular, we denote by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M15">View MathML</a> a Y-valued Q-Wiener process with respect to <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M2">View MathML</a>. We assume that there exists a complete orthonormal system <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M17">View MathML</a> in Y, a bounded sequence of nonnegative real numbers <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M18">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M19">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M20">View MathML</a> , and a sequence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M21">View MathML</a> of independent Brownian motions such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M22">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M23">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M24">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M25">View MathML</a> is the σ-algebra generated by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M26">View MathML</a>. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M27">View MathML</a> be the space of all Hilbert-Schmidt operators from <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M28">View MathML</a> to X with the inner product <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M29">View MathML</a>; see, for example, [2].

Suppose that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M30">View MathML</a> is an analytic semigroup with its infinitesimal generator A; for literature relating to semigroup theory, we suggest Pazy [24]. We suppose <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M31">View MathML</a>, the resolvent set of −A. For any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M32">View MathML</a>, it is possible to define the fractional power <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M33">View MathML</a> which is a closed linear operator with its domain <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M34">View MathML</a>.

In this paper, we consider the following impulsive stochastic neutral partial differential equations with memory:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M35">View MathML</a>

(1)

in a real separable Hilbert space X, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M36">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M37">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M38">View MathML</a> are all Borel measurable; <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M39">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M40">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M41">View MathML</a> are continuous; A is the infinitesimal generator of a semigroup of bounded linear operators <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M42">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M43">View MathML</a>, in X; <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M44">View MathML</a>. Furthermore, the fixed moments of time <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M45">View MathML</a> satisfy <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M46">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M47">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M48">View MathML</a> represent the right and left limits of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M49">View MathML</a> at <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M50">View MathML</a>, respectively. Also, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M51">View MathML</a>, represents the jump in the state x at time <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M45">View MathML</a> with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M53">View MathML</a> determining the size of the jump. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M54">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M55">View MathML</a> denote the family of all right continuous functions with left-hand limits η from <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M56">View MathML</a> to X. The space C is assumed to be equipped with the norm <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M57">View MathML</a>. Here <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M58">View MathML</a> is the family of all almost surely bounded, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M3">View MathML</a>-measurable, continuous random variables from <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M56">View MathML</a> to X.

Definition 2.1 A process <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M61">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M62">View MathML</a>, is called a mild solution of Eq. (1) if

(i) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M49">View MathML</a> is adapted to <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M64">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M43">View MathML</a> with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M66">View MathML</a> a.s.;

(ii) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M67">View MathML</a> has càdlàg paths on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M68">View MathML</a> a.s. and for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M69">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M49">View MathML</a> satisfies the integral equation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M71">View MathML</a>

(2)

and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M72">View MathML</a>.

Definition 2.2 Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M73">View MathML</a> be an integer. Equation (1) is said to be exponentially stable in the pth mean if for any initial value φ, there exists a pair of positive constants λ and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M74">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M75">View MathML</a>

(3)

In particular, if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M76">View MathML</a>, then Eq. (1) is said to be mean-square exponentially stable.

To establish the exponential stability of the mild solution of Eq. (1), we employ the following assumptions.

(H1) A is the infinitesimal generator of a semigroup of bounded linear operators <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M42">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M43">View MathML</a>, in X satisfying <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M79">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M43">View MathML</a>, for some constants <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M81">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M82">View MathML</a>.

(H2) The mappings f and g satisfy the following Lipschitz condition: there exists a constant K for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M83">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M43">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M85">View MathML</a>

(H3) The mapping <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M86">View MathML</a> satisfies that there exists a number <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M87">View MathML</a> and a positive constant <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M88">View MathML</a> such that for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M89">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M43">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M91">View MathML</a> and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M92">View MathML</a>

(H4) There exists a constant <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M93">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M94">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M95">View MathML</a>, for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M89">View MathML</a>.

Moreover, for the purposes of stability, we always assume that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M97">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M98">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M99">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M100">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M101">View MathML</a>). Hence Eq. (1) has a trivial solution when <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M102">View MathML</a>.

Lemma 2.1[24]

If (H1) holds, then for any<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M103">View MathML</a>:

(i) For each<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M104">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M105">View MathML</a>;

(ii) There exist positive constants<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M106">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M107">View MathML</a>such that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M108">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M109">View MathML</a>.

3 Stability of mild solutions

In this section, to establish sufficient conditions ensuring the exponential stability in p-moment (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M73">View MathML</a>) for a mild solution to Eq. (1), we firstly establish a new integral inequality to overcome the difficulty when the neutral term and impulsive effects are present.

Lemma 3.1For any<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M111">View MathML</a>, assume that there exist some positive constants<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M112">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M113">View MathML</a>), <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M114">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M101">View MathML</a>) and a function<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M116">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M117">View MathML</a>

(4)

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M118">View MathML</a>

(5)

for each<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M43">View MathML</a>. If

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M120">View MathML</a>

(6)

then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M121">View MathML</a>

(7)

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M122">View MathML</a>is the unique solution to the equation: <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M123">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M124">View MathML</a>.

Proof Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M125">View MathML</a>, then by (6) and the existence theorem of the root, there exists a positive constant <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M126">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M127">View MathML</a>.

For any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M128">View MathML</a>, let

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M129">View MathML</a>

(8)

To now prove the result, we only claim that (4) and (5) imply

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M130">View MathML</a>

(9)

Clearly, for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M131">View MathML</a>, (9) holds. By the contradiction, assume that there is a positive constant <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M132">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M133">View MathML</a>

(10)

This, together with (5), yields (note that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M134">View MathML</a>)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M135">View MathML</a>

(11)

By (8), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M136">View MathML</a>

(12)

Hence, by (11), we obtain <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M137">View MathML</a>, which contradicts (10). Therefore (9) holds.

Since ε is arbitrarily small, so (7) holds. This completes the proof. □

We can now state our main result of this paper.

Theorem 3.1If (H1)-(H4) hold for some<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M138">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M139">View MathML</a>, then the mild solution of Eq. (1) is exponentially stable in thepth moment, provided

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M140">View MathML</a>

(13)

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M141">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M142">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M143">View MathML</a>is defined in Lemma 3.1.

Proof From the condition (13), we can always find a number <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M144">View MathML</a> small enough such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M145">View MathML</a>

On the other hand, recall the inequalities <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M146">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M147">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M148">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M144">View MathML</a>. Then, for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M150">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M151">View MathML</a>

(14)

From (2) and (14),

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M152">View MathML</a>

(15)

Now we compute the right-hand terms of (15). Firstly, by (H1) and (H3), we can easily obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M153">View MathML</a>

(16)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M154">View MathML</a>

(17)

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M155">View MathML</a>

(18)

By (H4) and the Hölder inequality, for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M73">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M157">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M158">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M159">View MathML</a>

(19)

By (H3), Lemma 3.1 and the Hölder inequality,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M160">View MathML</a>

(20)

Similar to (20), by (H2) and the Hölder inequality, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M161">View MathML</a>

(21)

By Da Prato and Zabczyk [[2], Lemma 7.7, p.194], similar to (20), (H2) and the Hölder inequality, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M162">View MathML</a>

(22)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M141">View MathML</a>.

Substituting (16)-(22) into (15) yields

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M164">View MathML</a>

(23)

This, together with Lemma 3.1 and (13), gives that there exist two positive constants <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M165">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M166">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M167">View MathML</a> for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M168">View MathML</a>. This completes the proof. □

If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M76">View MathML</a>, then we get the following corollary from Theorem 3.1.

Corollary 3.1If (H1)-(H4) hold for some<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M170">View MathML</a>, then the mild solution of Eq. (1) is mean-square exponentially stable, provided

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M171">View MathML</a>

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Remark 3.1 Unlike earlier studies, ours does not make use of general methods such as Lyapunov methods, fixed point theory and so forth. As we know, in general, it is impossible to construct a suitable Lyapunov function (functional) and to find an appropriate fixed point theorem for stochastic partial differential equations with memory, even for constant delays, to deal with stability. In this work, we use the new impulsive integral inequality to derive the sufficient conditions for stability.

Remark 3.2 Without delay and impulsive effect, Eq. (1) becomes stochastic neutral partial differential equations, which is investigated in [3]. Without the neutral term and impulsive effect, Eq. (1) reduces to stochastic partial differential delay equations, which is studied in [6,17]. Therefore, we generalize by the integral inequality the results to cover a class of more general impulsive stochastic neutral partial differential equations with memory. Moreover, unlike [6], we need not require the functions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M172">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M173">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M174">View MathML</a> to be differentiable.

Remark 3.3 In Eq. (1), provided <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/148/mathml/M175">View MathML</a>, Eq. (1) becomes stochastic neutral partial differential equations without impulsive effects, that is to say, our Theorem 3.1 is effective for it.

4 Conclusion

In this paper, we discuss the exponential stability in the pth moment of mild solutions to impulsive stochastic neutral partial differential equations with memory. By establishing a new integral inequality, we obtain sufficient conditions ensuring the stability of the impulsive stochastic system. The results generalize and improve earlier publications.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

HY gave the proof of the main result and drafted the manuscript. FJ established the new integral inequality and participated in the study of the main result of the paper. All authors read and approved the final manuscript.

Acknowledgements

The work is supported by the Research Funds of Wuhan Polytechnic University under Grant 2012Y16, the Fundamental Research Funds for the Central Universities under Grant 2722013JC080, China Postdoctoral Science Foundation funded project under Grant 2012M511615 and the Natural Science Foundation of Hubei Province of China.

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