- Research Article
- Open Access

# The Existence and Exponential Stability for Random Impulsive Integrodifferential Equations of Neutral Type

- Huabin Chen
^{1}Email author, - Xiaozhi Zhang
^{1}and - Yang Zhao
^{1}

**2010**:540365

https://doi.org/10.1155/2010/540365

© Huabin Chen et al. 2010

**Received:**24 March 2010**Accepted:**28 July 2010**Published:**19 August 2010

## Abstract

By applying the Banach fixed point theorem and using an inequality technique, we investigate a kind of random impulsive integrodifferential equations of neutral type. Some sufficient conditions, which can guarantee the existence, uniqueness, and exponential stability in mean square for such systems, are obtained. Compared with the previous works, our method is new and our results can generalize and improve some existing ones. Finally, an illustrative example is given to show the effectiveness of the proposed results.

## Keywords

- Exponential Stability
- Impulsive Differential Equation
- Neutral Type
- Neutral Differential Equation
- Neutral Functional Differential Equation

## 1. Introduction

Since impulsive differential systems have been highly recognized and applied in a wide spectrum of fields such as mathematical modeling of physical systems, technology, population and biology, etc., some qualitative properties of the impulsive differential equations have been investigated by many researchers in recent years, and a lot of valuable results have been obtained (see, e.g., [1–10] and references therein). For the general theory of impulsive differential systems, the readers can refer to [11, 12]. For an impulsive differential equations, if its impulsive effects are random variable, their solutions are stochastic processes. It is different from the deterministic impulsive differential equations and stochastic differential equations. Thus, the random impulsive differential equations are more realistic than deterministic impulsive systems. The investigation for the random impulsive differential equations is a new area of research. Recently, the -moment boundedness, exponential stability and almost sure stability of random impulsive differential systems were studied by using the Lyapunov functional method in [13–15], respectively. In [16] Wu and Duan have investigated the oscillation, stability and boundedness in mean square of second-order random impulsive differential systems; Wu et al. in [17] studied the existence and uniqueness of the solutions to random impulsive differential equations, and in [18] Zhao and Zhang discussed the exponential stability of random impulsive integro-differential equations by employing the comparison theorem. Very recently, the existence, uniqueness and stability results of random impulsive semilinear differential equations, the existence and uniqueness for neutral functional differential equations with random impulses are discussed by using the Banach fixed point theorem in [19, 20], respectively.

It is well known that the nonlinear impulsive delay differential equations of neutral type arises widely in scientific fields, such as control theory, bioscience, physics, etc. This class of equations play an important role in modeling phenomena of the real world. So it is valuable to discuss the properties of the solutions of these equations. For example, Xu et al. in [21], have considered the exponential stability of nonlinear impulsive neutral differential equations with delays by establishing singular impulsive delay differential inequality and transforming the -dimensional impulsive neutral delay differential equation into a -dimensional singular impulsive delay differential equations; and the results about the global exponential stability for neutral-type impulsive neural networks are obtained by using the linear matrix inequality (LMI) in [9, 10], respectively.

However, most of these studies are in connection with deterministic impulses and finite delay. And, to the best of author's knowledge, there is no paper which investigates the existence, uniqueness and exponential stability in mean square of random impulsive integrodifferential equation of neutral type. One of the main reason is that the methods to discuss the exponential stability of deterministic impulsive differential equations of neutral type and the exponential stability for random differential equations can not be directly adapted to the case of random impulsive differential equations of neutral type, especially, random impulsive integrodifferential equations of neutral type. That is, the methods proposed in [15, 16] are ineffective for the exponential stability in mean square for such systems. Although the exponential stability of nonlinear impulsive neutral integrodifferential equations can be derived in [22], the method used in [22] is only suitable for the deterministic impulses. Besides, the methods introduced to deal with the exponential stability of random impulsive integrodifferential equations in [18] and study the exponential stability in mean square of random impulsive differential equations in [19], can not be applied to deal with our problem since the neutral item arises. So, the technique and the method dealt with the exponential stability in mean square of random impulsive integrodifferential equations of neutral type are in need of being developed and explored. Thus, with these aims, we will make the first attempt to study such problems to close this gap in this paper.

The format of this work is organized as follows. In Section 2, some necessary definitions, notations and lemmas used in this paper will be introduced. In Section 3, The existence and uniqueness of random impulsive integrodifferential equations of neutral type are obtained by using the Banach fixed point theorem. Some sufficient conditions about the exponential stability in mean square for the solution of such systems are given in Section 4. Finally, an illustrative example is provided to show the obtained results.

## 2. Preliminaries

Let denote the Euclidean norm in . If is a vector or a matrix, its transpose is denoted by ; and if is a matrix, its Frobenius norm is also represented by . Assumed that is a nonempty set and is a random variable defined from to for all , where . Moreover, assumed that and are independent with each other as for .

Let be the space of bounded and continuous mappings from the topological space into , and be the space of bounded and continuously differentiable mappings from the topological space into . In particular, Let and . is bounded and almost surely continuous for all but at most countable points and at these points , and exist, , where is an interval, and denote the right-hand and left-hand limits of the function , respectively. Especially, let . is bounded and almost surely continuously differentiable for all but at most countable points and at these points , and , , , where denote the derivative of . Especially, let .

where are two matrices of dimension ; and are two appropriate functions; is a matrix valued functions for each ; assume that is an arbitrary real number, and for ; obviously, ; ; is a bounded and continuous function and . for all . Let us denote by the simple counting process generated by , that is, , and present the -algebra generated by . Then, is a probability space.

It is easily shown that the space is a completed space.

Definition 2.1.

A function is said to be a solution of (2.2)–(2.4) if satisfies (2.2) and conditions (2.3) and (2.4).

Definition 2.2.

The fundamental solution matrix of the equation is said to be exponentially stable if there exist two positive numbers and such that , for all .

Definition 2.3.

Lemma 2.4 (see [23]).

Lemma 2.5 (see [23]).

In order to obtain our main results, we need the following hypotheses.

## 3. Existence and Uniqueness

In this section, to make this paper self-contained, we study the existence and uniqueness for the solution to system (2.2) with conditions (2.3) and (2.4) by using the Picard iterative method under conditions (H )–(H ). In order to prove our main results, we firstly need the following auxiliary result.

Lemma 3.1.

if and only if is a solution of impulsive integrodifferential equations:

Proof.

The approach of the proof is very similar to those in [17, 19, 20]. Here, we omit it.

Theorem 3.2.

Provided that conditions (H )–(H ) hold, then the system (2.2) with the conditions (2.3) and (2.4) has a unique solution on .

Proof.

That is, (3.16) holds for . Hence, by induction, (3.16) holds for all .

as . Thus, is a Cauchy sequence in Banach space . Denote the limit by . Now, letting in both sides of (3.4), we obtain the existence for the solution of system (2.2) with conditions (2.3) and (2.4).

That is, . So, the uniqueness is also proved. The proof of this theorem is completed.

## 4. Exponential Stability

In this section, the exponential stability in mean square for system (2.2) with initial conditions (2.3) and (2.4) is shown by using an integral inequality.

Theorem 4.1.

holds.

Proof.

and it is easily seen that there exists a positive number such that , for all .

From (4.4), letting , holds. That is, there exists a positive constant such that .

which contradicts (4.7), that is, (4.4) holds.

The proof is completed.

Remark 4.2.

Obviously, we can also give the existence, uniqueness, and exponential stability in mean square for the solution of system (4.14) by employing the Picard iterative method and a similar impulsive-integral inequality proposed in [24]. So, the following corollary can be given as follows.

Corollary 4.3.

holds.

Proof.

The proofs of this corollary are very similar to those of Theorems 3.2 and 4.1. So, we omit them.

Remark 4.4.

Recently, in [19], Anguraj and Vinodkumar have derived Corollary 4.3 by using the fixed point theorem. Obviously, our results are more general than those obtained in [19]. Thus, we can generalize and improve the results in [19].

## 5. An Illustrative Example

Let be a random variable defined in for all , where . Furthermore, assume that and are independent of each other as for .

where and for all , is a function of . Denote that and there is such that for all . and and . And , , .

## Declarations

### Acknowledgment

The authors would like to thank the referee and the editor for their careful comments and valuable suggestions on this work.

## Authors’ Affiliations

## References

- Anokhin A, Berezansky L, Braverman E:
**Exponential stability of linear delay impulsive differential equations.***Journal of Mathematical Analysis and Applications*1995,**193**(3):923-941. 10.1006/jmaa.1995.1275MathSciNetView ArticleMATHGoogle Scholar - Hino Y, Murakani S, Naito T:
*Functional Differential Equations with Infinite Delay, Lecture Notes in Mathematics*.*Volume 1473*. Springer, Berlin, Germany; 1991.Google Scholar - Hale J, Verguyn Lunel SM:
*Introduction to Functional Differential Equations, Applied Mathematical Science*.*Volume 99*. Springer, New York, NY, USA; 1993.View ArticleGoogle Scholar - Henderson J, Ouadhab A:
**Local and global existence and uniqueness results for second and higher order impulsive functional differential equations with infinite delay.***The Australian Journal of Mathematical Analysis and Applications*2007,**4:**1-26.MathSciNetGoogle Scholar - Korzeniowski A, Ladde GS:
**Modeling hybrid network dynamics under random perturbations.***Nonlinear Analysis: Hybrid Systems*2009,**3**(2):143-149. 10.1016/j.nahs.2008.12.001MathSciNetMATHGoogle Scholar - Liu X, Shen X, Zhang Y:
**A comparison principle and stability for large-scale impulsive delay differential systems.***The ANZIAM Journal*2005,**47**(2):203-235. 10.1017/S1446181100009998MathSciNetView ArticleMATHGoogle Scholar - Liu X:
**Stability results for impulsive differential systems with applications to population growth models.***Dynamics & Stability of Systems. Series A*1994,**9**(2):163-174.View ArticleMathSciNetMATHGoogle Scholar - Liu X, Liao X:
**Comparison method and robust stability of large-scale dynamic systems.***Dynamics of Continuous, Discrete & Impulsive Systems. Series A*2004,**11**(2-3):413-430.MathSciNetView ArticleMATHGoogle Scholar - Rakkiyappan R, Balasubramaniam P, Cao J:
**Global exponential stability results for neutral-type impulsive neural networks.***Nonlinear Analysis: Real World Applications*2010,**11**(1):122-130. 10.1016/j.nonrwa.2008.10.050MathSciNetView ArticleMATHGoogle Scholar - Samidurai R, Anthoni SM, Balachandran K:
**Global exponential stability of neutral-type impulsive neural networks with discrete and distributed delays.***Nonlinear Analysis: Hybrid Systems*2010,**4**(1):103-112. 10.1016/j.nahs.2009.08.004MathSciNetMATHGoogle Scholar - Lakshmikantham V, Baĭnov DD, Simeonov PS:
*Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics*.*Volume 6*. World Scientific, Teaneck, NJ, USA; 1989:xii+273.View ArticleGoogle Scholar - Samoĭlenko AM, Perestyuk NA:
*Impulsive Differential Equations, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises*.*Volume 14*. World Scientific, River Edge, NJ, USA; 1995:x+462.Google Scholar - Wu S, Meng X:
**Boundedness of nonlinear differential systems with impulsive effect on random moments.***Acta Mathematicae Applicatae Sinica*2004,**20**(1):147-154. 10.1007/s10255-004-0157-zMathSciNetView ArticleMATHGoogle Scholar - Wu S, Guo X, Zhou Y:
**-moment stability of functional differential equations with random impulses.***Computers & Mathematics with Applications*2006,**52**(12):1683-1694. 10.1016/j.camwa.2006.04.026MathSciNetView ArticleMATHGoogle Scholar - Wu S, Guo X, Zhai R:
**Almost sure stability of functional differential equations with random impulses.***Dynamics of Continuous, Discrete & Impulsive Systems. Series A*2008,**15**(3):403-415.MathSciNetMATHGoogle Scholar - Wu S, Duan Y:
**Oscillation, stability, and boundedness of second-order differential systems with random impulses.***Computers & Mathematics with Applications*2005,**49**(9-10):1375-1386. 10.1016/j.camwa.2004.12.009MathSciNetView ArticleMATHGoogle Scholar - Wu S, Guo X, Lin S:
**Existence and uniqueness of solutions to random impulses.***Acta Mathematicae Applicatae Sinica*2004,**22:**595-600.Google Scholar - Zhao D, Zhang L:
**Exponential asymptotic stability of nonlinear Volterra equations with random impulses.***Applied Mathematics and Computation*2007,**193**(1):18-25. 10.1016/j.amc.2007.03.029MathSciNetView ArticleMATHGoogle Scholar - Anguraj A, Vinodkumar A:
**Existence, uniqueness and stability results of random impulsive semilinear differential systems.***Nonlinear Analysis: Hybrid Systems*2010,**4:**475-483. 10.1016/j.nahs.2009.11.004MathSciNetMATHGoogle Scholar - Anguraj A, Vinodkumar A:
**Existence and uniqueness of neutral functional differential equations with random impulses.***International Journal of Nonlinear Science*2009,**8**(4):412-418.MathSciNetMATHGoogle Scholar - Xu D, Yang Z, Yang Z:
**Exponential stability of nonlinear impulsive neutral differential equations with delays.***Nonlinear Analysis: Theory, Methods & Applications*2007,**67**(5):1426-1439. 10.1016/j.na.2006.07.043MathSciNetView ArticleMATHGoogle Scholar - Xu L, Xu D:
**Exponential stability of nonlinear impulsive neutral integro-differential equations.***Nonlinear Analysis: Theory, Methods & Applications*2008,**69**(9):2910-2923. 10.1016/j.na.2007.08.062MathSciNetView ArticleMATHGoogle Scholar - Walter W:
*Differential and Integral Inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 55*. Springer, New York, NY, USA; 1970:x+352.Google Scholar - Chen H:
**Impulsive-integral inequality and exponential stability for stochastic partial differential equations with delays.***Statistics & Probability Letters*2010,**80**(1):50-56. 10.1016/j.spl.2009.09.011MathSciNetView ArticleMATHGoogle Scholar

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