# A New Nonlinear Retarded Integral Inequality and Its Application

- Wu-Sheng Wang
^{1}Email author, - Ri-Cai Luo
^{1}and - Zizun Li
^{2}

**2010**:462163

https://doi.org/10.1155/2010/462163

© Wu-Sheng Wang et al. 2010

**Received: **28 April 2010

**Accepted: **15 August 2010

**Published: **18 August 2010

## Abstract

The main objective of this paper is to establish a new retarded nonlinear integral inequality with two variables, which provide explicit bound on unknown function. This inequality given here can be used as tool in the study of integral equations.

## 1. Introduction

(1.7) was studied by Gripenberg in [10].

However, the bound given on such inequality in [8] is not directly applicable in the study of certain retarded differential and integral equations. It is desirable to establish new inequalities of the above type, which can be used more effectively in the study of certain classes of retarded differential and integral equations.

We will prove importance of (1.8) in achieving a desired goal.

## 2. Main Result

Throughout this paper, are given numbers, and . For functions , denotes the derivative of , and denotes the partial derivative on . Consider (1.8), and suppose that

( ) is a strictly increasing function with and as ;

( ) are nondecreasing in each variable;

( ) are nondecreasing with for ;

( ) and are nondecreasing such that and ;

Theorem 2.1.

Proof.

Since is arbitrary, from (2.14), we get the required estimation (2.2).

## 3. Applications

where is a strictly increasing function with and as , is a given positive constant, are bounded functions and nondecreasing in each variable, functions and satisfy hypothesis , i=1,2, and is nondecreasing on such that for .

The integral equation (3.1) is obviously more general than (1.7) considered in [10]. When keeping fixed, let , then integral equation (3.1) reduces to integral equation (1.7) in [10].

Corollary 3.1.

Proof.

Clearly, inequality (3.5) is in the form of (1.8). Thus, the estimate (3.2) of the solution in this corollary is obtained immediately by our Theorem 2.1.

## Declarations

### Acknowledgments

The authors are very grateful to the editor and the referees for their helpful comments and valuable suggestions. This paper is supported by the Natural Science Foundation of Guangxi Autonomous Region (0991265), the Scientific Research Foundation of the Education Department of Guangxi Autonomous Region (200707MS112), and the Key Discipline of Applied Mathematics of Hechi University of China (200725).

## Authors’ Affiliations

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## Copyright

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