A New Delay Vector Integral Inequality and Its Application

Abstract

A new delay integral inequality is established. Using this inequality and the properties of nonnegative matrix, the attracting sets for the nonlinear functional differential equations with distributed delays are obtained. Our results can extend and improve earlier publications.

1. Introduction

The attracting set of dynamical systems have been extensively studied over the past few decades and various results are reported. It is well known that one of the most researching tools is inequality technique. With the establishment of various differential and integral inequalities [1–12], the sufficient conditions on the attracting sets for different differential systems are obtained [12–16]. However, the inequalities mentioned above are ineffective for studying the attracting sets of a class of nonlinear functional differential equations with distributed delays.

Motivated by the above discussions, in this paper, a new delay vector integral inequality is established. Applying this inequality and the properties of nonnegative matrix, some sufficient conditions ensuring the global attracting set for a class of nonlinear functional differential equations with distributed delays are obtained. The result in [16] is extended.

2. Preliminaries

In this section, we introduce some notations and recall some basic definitions.

means unit matrix; is the set of real numbers. means that each pair of corresponding elements of and satisfies the inequality "". Especially, is called a nonnegative matrix if .

For a nonnegative matrix , let denote the spectral radius of . Then is an eigenvalue of and its eigenspace is denoted by

(2.1)

which includes all positive eigenvectors of provided that the nonnegative matrix has at least one positive eigenvector (see [17]).

denotes the space of continuous mappings from the topological space to the topological space . Especially, let , where .

Let . For ,,,, we define , , , , ,,. For , we define .

Definition 2.1 (Xu [4]).

means that and for any given and any there exist positive numbers , , and satisfying

(2.2)

Especially, if and .

Lemma 2.2 (Lasalle [18]).

If and , then .

3. Delay Integral Inequality

Theorem 3.1.

Let be a solution of the delay integral inequality

(3.1)

(3.2)

where , ,, ,. Assume that the following conditions are satisfied:

as ,,, ,,,, and there exists a nonnegative matrix such that

(3.3)

.

If the initial condition satisfies

(3.4)

where , , and , then there exists a constant such that

(3.5)

Proof.

By the condition and Lemma 2.2, together with , this implies that exists and .

From , there is a such that

(3.6)

By the continuity of , together with (3.2) and (3.4), there exists a constant such that

(3.7)

In the following, we will prove that

(3.8)

If this is not true, from (3.7) and the continuity of , then there must be a constant and some integer such that

(3.9)

(3.10)

where .

Using (3.1), (3.3), (3.6), (3.10), and , we obtain that

(3.11)

This contradicts the equality in (3.9), and so (3.8) holds. The proof is complete.

4. Applications

The delay integral inequality obtained in Section 3 can be widely applied to study the attracting set of the nonlinear functional differential equations. To illustrate the theory, we consider the following differential equation with distributed delays

(4.1)

where , , , , ,, ,, . We always assume that for any , the system (4.1) has at least one solution through denoted by or simply if no confusion should occur.

In order to study the attracting set, we rewrite (4.1) as

(4.2)

where be a fundamental matrix of the linear equation .

Throughout this section, we suppose the following:

, where ,,, , ,

,,. , .. . There are two matrices and such that

(4.3)

.

Definition 4.1.

The set is called a global attracting set of (4.1), if for any initial value , the solution converges to as . That is,

(4.4)

where , is the distance of to in .

Theorem 4.2.

Assume that hold. Then is a global attracting set of (4.1).

Proof.

It follows from and (4.2) that

(4.5)

For the initial conditions , where , we have

(4.6)

where ,,, and so

(4.7)

By (4.5)-(4.7), and Theorem 3.1, then there exists a constant such that

(4.8)

From (4.8), there exists a constant vector , such that

(4.9)

Next we will show that . From , , and , for any and , there exist a positive number and a positive constant matrix such that for all

(4.10)

(4.11)

(4.12)

(4.13)

According to the definition of superior limit and , there exists sufficient large , such that for any ,

(4.14)

So, from , (4.5), and (4.10)-(4.14), when , we obtain

(4.15)

Due to (4.9) and the definition of superior limit, there exists , such that . So,

(4.16)

Letting , we have , that is , and the proof is completed.

Corollary 4.3.

If is an equilibrium point of system (4.1), suppose that the conditions of Theorem 4.2. hold and , then the equilibrium is globally asymptotically stable.

Remark 4.4.

In Corollary 4.3., if , , , ,,, ,,, then we can get Theorems 3 and 4 in [16]. In fact , Theorems 3 and 4 in [16] satisfy all conditions of Corollary 4.3.