Integrodifferential Inequality for Stability of Singularly Perturbed Impulsive Delay Integrodifferential Equations

Abstract

The exponential stability of singularly perturbed impulsive delay integrodifferential equations (SPIDIDEs) is concerned. By establishing an impulsive delay integrodifferential inequality (IDIDI), some sufficient conditions ensuring the exponentially stable of any solution of SPIDIDEs for sufficiently small are obtained. A numerical example shows the effectiveness of our theoretical results.

1. Introduction

Integrodifferential equations (IDEs) arise from many areas of science (from physics, biology, medicine, etc.), which have extensive scientific backgrounds and realistic mathematical models, and hence have been emerging as an important area of investigation in recent years, see [1–6]. Correspondingly, the stability of impulsive delay integrodifferential equations has been studied quite well, for example, [7–9]. However, besides delay and impulsive effects, singular perturbation likewise exists in a wide models for physiological processes or diseases [10]. And many good results on the stability of singularly perturbed delay differential equations have been reported, see, for example, [11–14]. Therefore, it is necessary to consider delay, impulse and singular perturbation on the stability of integrodifferential equations. However, to the best of our knowledge, there are no results on the problems of the exponential stability of solutions for SPIDIDEs due to some theoretical and technical difficulties. Based on this, this article is devoted to the discussion of this problem.

Applying differential inequalities, in [14–17], authors investigated the stability of impulsive differential equations. In [14], Zhu et al. established a delay differential inequality with impulsive initial conditions and derived some sufficient conditions ensuring the exponential stability of solutions for the singular perturbed impulsive delay differential equations (SPIDDEs). In this paper, we will improve the inequality established in [14] such that it is effective for SPIDIDEs. By establishing an IDIDI, some sufficient conditions ensuring the exponential stability of any solution of SPIDIDEs for sufficiently small are obtained. The results extend and improve the earlier publications, and which will be shown by the Remarks 3.2 and 3.5 provided later. An example is given to illustrate the theory.

2. Preliminaries

Throughout this letter, unless otherwise specified, let be the space of -dimensional real column vectors and be the set of real matrices. . For or , means that each pair of corresponding elements of and satisfies the inequality " ()". Especially, is called a nonnegative matrix if , and is called a positive vector if .

denotes the space of continuous mappings from the topological space to the topological space . In particular, let denote the family of all bounded continuous -valued functions defined on with the norm , where is Euclidean norm of .

for exist for for all but points , where is an interval, and denote the left limit and right limit of scalar function , respectively. Especially, let .

For and or , we define

(21)

In this paper, we consider a class of SPIDIDEs described by

(22)

with the initial conditions

(23)

where , , , , , is a small parameter, and is a strictly increasing sequence such that .

Definition 2.1.

The solution of (2.2) is said to be exponentially stable for sufficiently small if there exist finite constant vectors and , which are independent of for some , and a constant such that for and for any initial perturbation satisfying . Here is the solution of (2.2) corresponding to the initial condition .

3. Main Results

In order to prove the main result in this paper, we first need the following technique lemma.

Lemma 3.1.

Assume that , satisfy

(31)

where for and , for , .

If there exist a positive constant and a positive vector and two positive diagonal matrices , with such that

(32)

Then one has

(33)

where the positive constant is defined as

(34)

for the given .

Proof.

Note that the result is trivial if . In the following, we assume that . Denote

(35)

then for any given , we have

(36)

the first inequality and the second inequality are from (3.2), the last inequality is because , , , .

We also have

(37)

So by (3.6) and (3.7), for any , there is a unique positive such that

(38)

Therefore, from the definition of , one can know that .

Next, we will show that .

If this is not true, fix satisfying and , , there exist a and some integer such that , where , such that

(39)

Then, we have

(310)

this contradiction shows that , so there at least exists a positive constant such that , that is, the definition of for (3.3) is reasonable.

Since is bounded, we always can choose a sufficiently large such that

(311)

In order to prove (3.3), we first prove for any given ,

(312)

If (3.12) is not true, then by continuity of , there must exist some integer and such that

(313)

(314)

So, by (3.1), the equality of (3.13), (3.14) and and , , for , and the definition of , we derive that

(315)

which contradicts the inequality in (3.13), and so (3.12) holds for all . Letting , then (3.3) holds, and the proof is completed.

Remark 3.2.

If in Lemma 3.1, then we get [14, Lemma  1].

Theorem 3.3.

Assume that for and , further suppose the following

For any , there exist nonnegative matrices and , , such that

(316)

For any ,there exist nonnegative constant matrices such that

(317)

There exist a positive constant and a positive vector and two positive diagonal matrices , , with , such that

(318)

where , .

There exists a positive constant satisfying

(319)

where satisfy

(320)

and is defined as

(321)

for the given .

Then there exists a small such that the solution of (2.2) is exponentially stable for sufficiently small .

Proof.

By a similar argument with (3.4), one can know that the defined by (3.21) is reasonable. For any , let , be two solutions of (2.2) through , , respectively. Since are bounded, we can always choose a positive vector such that

(322)

Calculating the upper right derivative along the solution of (2.2), by condition , we have

(323)

From condition , we have

(324)

Therefore, (3.23) and (3.24) imply that all the assumptions of Lemma 3.1 are true. So we have

(325)

where is determined by (3.21) and the positive constant vector is determined by (3.18).

Using the discrete part of (2.2), condition , (3.20) and (3.25), we can obtain that

(326)

and so, we have

(327)

By a similar argument with (3.25), we can use (3.27) derive that

(328)

Therefore, by simple induction, we have

(329)

From (3.19) and (3.29), we obtain

(330)

For any , let be defined as the unique positive zero of

(331)

Differentiate both sides of (3.31) with respect to the variable , we have

(332)

so is monotonically decreasing with respect to the variable , which implies that is also monotonically decreasing with respect to the variable . So we can choose the in (3.21) satisfying the same monotonicity with , for example, , where . Hence we can deduce that there exists a small such that the solution of (2.2) is exponentially stable for sufficiently small . The proof is completed.

Remark 3.4.

Suppose that in Theorem 3.3, then we can easily get [14, Theorem  1]. In fact, "" of condition in [14, Theorem  1] ensure that the above (3.20) holds.

Remark 3.5.

If , , that is there have no impulses in (2.2), then by Theorem 3.3, we can obtain the following result.

Corollary 3.6.

Assume that for and , , further suppose that and hold. Then there exists a small such that the solution of (2.2) is exponentially stable for sufficiently small .

Remark 3.7.

From Lemma 3.1 and the proof of Theorem 3.3, it is obvious that the results obtained in this paper still hold for . So this type of exponential stability can obviously be applied to general impulsive delay integrodifferential equations.

Remark 3.8.

When and , the global exponential stability criteria for (2.2) have been established in [18] by utilizing the Lyapunov functional method. However, the additional assumption that is bounded is required in [18].

4. An Illustrative Example

In this section, we will give an example to illustrate the exponential stability of (2.2).

Example 4.1.

Consider the following SPIDIDEs:

(41)

where , are constants, , , , .

We can easily find that conditions and are satisfied with

(42)

So there exist , , and such that

(43)

Let , we can obtain satisfy .

Case 1.

Let , , , , then we obtain that there exists an such that

(44)

and for , the positive constant is determined by the following equations:

(45)

So for a given , we can obtain the corresponding by (4.5). By the proof of Theorem 3.3, we know that is monotonically decreasing with respect to the variable , then there exists an such that for any , we have . Therefore, all the conditions of Theorem 3.3 are satisfied, we conclude that the solution of (4.1) is exponentially stable for sufficiently small .

Case 2.

Let and , then (4.1) becomes the singularly perturbed delay integrodifferential equations without impulses. So by Corollary 3.6, the solution of (4.1) is exponentially stable for sufficiently small .

Remark 4.2.

Obviously, the delay differential inequality which established in [14] is ineffective for studing the stability of SPIDIDEs (4.1).