# 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.

## Keywords

## 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

## References

- Agarwal RP, Deng S, Zhang W: Generalization of a retarded Gronwall-like inequality and its applications.
*Applied Mathematics and Computation*2005, 165(3):599–612. 10.1016/j.amc.2004.04.067MathSciNetView ArticleMATHGoogle Scholar - Agarwal RP, Kim Y-H, Sen SK: New retarded integral inequalities with applications.
*Journal of Inequalities and Applications*2008, 2008:-15.Google Scholar - Cheung W-S: Some new nonlinear inequalities and applications to boundary value problems.
*Nonlinear Analysis*2006, 64(9):2112–2128. 10.1016/j.na.2005.08.009MathSciNetView ArticleMATHGoogle Scholar - Chen C-J, Cheung W-S, Zhao D: Gronwall-Bellman-type integral inequalities and applications to BVPs.
*Journal of Inequalities and Applications*2009, 2009:-15.Google Scholar - Ma Q-H, Yang E-H: Some new Gronwall-Bellman-Bihari type integral inequalities with delay.
*Periodica Mathematica Hungarica*2002, 44(2):225–238. 10.1023/A:1019600715281MathSciNetView ArticleMATHGoogle Scholar - Pachpatte BG: On some new nonlinear retarded integral inequalities. JIPAM. Journal of Inequalities in Pure and Applied Mathematics 2004, 5(3, Article 80):-8.Google Scholar
- Wang W-S: A generalized retarded Gronwall-like inequality in two variables and applications to BVP.
*Applied Mathematics and Computation*2007, 191(1):144–154. 10.1016/j.amc.2007.02.099MathSciNetView ArticleMATHGoogle Scholar - Pachpatte BG: On some new inequalities related to certain inequalities in the theory of differential equations.
*Journal of Mathematical Analysis and Applications*1995, 189(1):128–144. 10.1006/jmaa.1995.1008MathSciNetView ArticleMATHGoogle Scholar - Pachpatte BG: On a new inequality suggested by the study of certain epidemic models.
*Journal of Mathematical Analysis and Applications*1995, 195(3):638–644. 10.1006/jmaa.1995.1380MathSciNetView ArticleMATHGoogle Scholar - Gripenberg G: On some epidemic models.
*Quarterly of Applied Mathematics*1981, 39(3):317–327.MathSciNetMATHGoogle Scholar

## Copyright

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