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

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.

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 .

For , we introduce the following norm:

(2.1)

In this paper, we consider the following random impulsive integrodifferential equations of neutral type:

(2.2)

(2.3)

(2.4)

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.

Firstly, define the space consisting of -valued stochastic process with the norm

(2.5)

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.

The solution of system (2.2) with conditions (2.3) and (2.4) is said to be exponentially stable in mean square, if there exist two positive constants and such that

(2.6)

Lemma 2.4 (see [23]).

For any two real positive numbers , then

(2.7)

where .

Lemma 2.5 (see [23]).

Let and be three real continuous functions defined on and , for , and assumed that on , one has the inequality

(2.8)

If is differentiable, then

(2.9)

for all .

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

(H) The function satisfies the Lipschitz condition: there exists a positive constant such that

(2.10)

for , , and .

(H) The function satisfies the following condition: there also exist a positive constant and a function with two important properties, and (), such that

(2.11)

for ,  , and .

(H) is uniformly bounded. That is, there exists a positive constant such that

(2.12)

for all and  .

(H).

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.

Let and be two continuous functions. Then, is the unique solution of the random impulsive integrodifferential equations of neutral type:

(3.1)

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

(i),  ,

• (ii)
  

  (3.2)

for all , where as , , and denotes the index function, that is,

(3.3)

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.

Define the iterative sequence as follows:

(3.4)

Thus, due to Lemma 2.4, it follows that

(3.5)

From condition (H), we have

(3.6)

In view of Lemma 2.5, it yields that

(3.7)

where . Furthermore,

(3.8)

By (3.4), we can obtain that

(3.9)

(3.10)

From the Gronwall inequality, (3.9) implies that

(3.11)

where and .

From (3.8) and (3.11), we have

(3.12)

for all , where

(3.13)

From (3.4), it follows that

(3.14)

By virtue of condition (H) and Lemma 2.5,

(3.15)

Now, for all and , we claim that

(3.16)

We will show (3.16) by mathematical induction. From (3.12), it is easily seen that (3.16) holds as . Under the inductive assumption that (3.16) holds for some . We will prove that (3.16) still holds when . Notice that

(3.17)

From condition (H), we have

(3.18)

In view of Lemma 2.5 and (3.16), it yields that

(3.19)

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

For any , it follows that

(3.20)

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).

Uniqueness. Let be two solutions of system (2.2) with conditions (2.3) and (2.4). By the same ways as above, we can yield that

(3.21)

Applying the Gronwall inequality into (3.21), it follows that

(3.22)

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.

Supposed that the conditions of Theorem 3.2 holds, then the solution of the system (2.2) with conditions (2.3) and (2.4) is exponential stable in mean square if the inequality

(4.1)

holds.

Proof.

From (3.2) and Lemma 2.4, we derive that

(4.2)

Thus, it follows that

(4.3)

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

For the convenience, setting , , and , it implies from (4.3) that

(4.4)

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

For any and letting

(4.5)

Now, in order to show our main result, we only claim that (4.4) implies

(4.6)

It is easily seen that (4.6) holds for any . Assume, for the sake of contradiction, that there exists a and

(4.7)

Then, it, from (4.4), implies that

(4.8)

From the definitions of and , we have

(4.9)

Thus, (4.8) yields that

(4.10)

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

As is arbitrarily small, in view of (4.6), it follows that

(4.11)

where .

That is,

(4.12)

where and

(4.13)

The proof is completed.

In particular, when , and , system (2.2) is turned into the following form:

(4.14)

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.

Under conditions (H) and (H), the existence, uniqueness, and exponential stability in mean square for the solution of system (4.14) can be obtained only if the inequality

(4.15)

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 .

Consider the random impulsive integrodifferential equations of neutral type:

(5.1)

as ,   and for all ,

(5.2)

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

The corresponding linear homogeneous equations:

(5.3)

And, the fundamental solution matrix of system (5.3) can be given by

(5.4)

Then, it is easily obtain that

(5.5)

where and , and it is easily seen that we can derive that the functions and satisfy conditions (H) and (H) with the Lipschitz coefficients and , respectively. On the other hand, hypothesis (H) and (H) are easily verified. In view of Theorems 3.2 and 4.1, the existence, uniqueness, and exponential stability in mean square of system (5.1) with (5.2) are obtained if the constants and satisfy the following inequality:

(5.6)