- Research Article
- Open Access

# A New Delay Vector Integral Inequality and Its Application

- Li Xiang
^{1}Email author, - Lingying Teng
^{2}and - Hua Wu
^{3}

**2010**:927059

https://doi.org/10.1155/2010/927059

© Li Xiang et al. 2010

**Received:**22 September 2010**Accepted:**14 December 2010**Published:**21 December 2010

## 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.

## Keywords

- Differential Equation
- Dynamical System
- Spectral Radius
- Differential System
- Early Publication

## 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.

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

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

Definition 2.1 (Xu [4]).

Especially, if and .

Lemma 2.2 (Lasalle [18]).

If and , then .

## 3. Delay Integral Inequality

Theorem 3.1.

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

.

Proof.

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

where .

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

## 4. Applications

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.

where be a fundamental matrix of the linear equation .

Throughout this section, we suppose the following:

, where , , , , ,

.

Definition 4.1.

where , is the distance of to in .

Theorem 4.2.

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

Proof.

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.

## Declarations

### Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant no. 10971147. Thanks are due to the instruction of Professor Xu.

## Authors’ Affiliations

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## Copyright

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