- Research Article
- Open Access

# On a Generalized Retarded Integral Inequality with Two Variables

- Wu-Sheng Wang
^{1, 2}Email author and - Cai-Xia Shen
^{1}

**2008**:518646

https://doi.org/10.1155/2008/518646

© W.-S. Wang and C.-X. Shen. 2008

**Received: **16 November 2007

**Accepted: **22 April 2008

**Published: **5 May 2008

## Abstract

This paper improves Pachpatte's results on linear integral inequalities with two variables, and gives an estimation for a general form of nonlinear integral inequality with two variables. This paper does not require monotonicity of known functions. The result of this paper can be applied to discuss on boundedness and uniqueness for a integrodifferential equation.

## Keywords

- Differential Equation
- Continuous Function
- Integral Equation
- Positive Constant
- Planar Region

## 1. Introduction

for boundedness and uniqueness of solutions.

for all . Obviously, appears linearly in (1.1), but in our (1.3) it is generalized to nonlinear terms: and . Our strategy is to monotonize functions s with other two nondecreasing ones such that one has stronger monotonicity than the other. We apply our estimation to an integrodifferential equation, which looks similar to (1.2) but includes delays, and give boundedness and uniqueness of solutions.

## 2. Main Result

Throughout this paper, are given numbers. Let and . Consider inequality (1.3), where we suppose that is strictly increasing such that , and are nondecreasing, such that and , , and are given, and are functions satisfying and for all .

Furthermore, let which is also nondecreasing in for each fixed , and and satisfies .

Theorem 2.1.

Here denotes the domain of a function.

Proof.

for all , where is defined by (2.8).

which proves the claimed (2.12).

for all . Letting , we obtain (2.6) because of continuity of in and continuity of , and . This completes the proof.

## 3. Applications

for all , where , and are supposed to be as in Theorem 2.1; , , , and are all continuous functions such that . Obviously, the estimation obtained in [11] cannot be applied to (3.2).

Corollary 3.1.

and , and are defined as in Theorem 2.1.

on , then every solution of (3.2) is bounded on .

Next, we give the condition of the uniqueness of solutions for (3.2).

Corollary 3.2.

on . Then, (3.2) has at most one solution on , where are defined as in Theorem 2.1.

## Declarations

### Acknowledgments

This work is supported by the Scientific Research Fund of Guangxi Provincial Education Department (no. 200707MS112), the Natural Science Foundation (no. 2006N001), and the Applied Mathematics Key Discipline Foundation of Hechi College of China.

## Authors’ Affiliations

## References

- Bellman R:
**The stability of solutions of linear differential equations.***Duke Mathematical Journal*1943,**10**(4):643–647. 10.1215/S0012-7094-43-01059-2MATHMathSciNetView ArticleGoogle Scholar - Gronwall TH:
**Note on the derivatives with respect to a parameter of the solutions of a system of differential equations.***The Annals of Mathematics*1919,**20**(4):292–296. 10.2307/1967124MATHMathSciNetView ArticleGoogle Scholar - 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.2004.04.067MATHMathSciNetView ArticleGoogle Scholar - Bihari I:
**A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations.***Acta Mathematica Hungarica*1956,**7:**81–94. 10.1007/BF02022967MATHMathSciNetView ArticleGoogle Scholar - Cheung W-S:
**Some new nonlinear inequalities and applications to boundary value problems.***Nonlinear Analysis: Theory, Methods & Applications*2006,**64**(9):2112–2128. 10.1016/j.na.2005.08.009MATHMathSciNetView ArticleGoogle Scholar - Dafermos CM:
**The second law of thermodynamics and stability.***Archive for Rational Mechanics and Analysis*1979,**70**(2):167–179.MATHMathSciNetView ArticleGoogle Scholar - Dannan FM:
**Integral inequalities of Gronwall-Bellman-Bihari type and asymptotic behavior of certain second order nonlinear differential equations.***Journal of Mathematical Analysis and Applications*1985,**108**(1):151–164. 10.1016/0022-247X(85)90014-9MATHMathSciNetView ArticleGoogle Scholar - Medina R, Pinto M:
**On the asymptotic behavior of solutions of certain second order nonlinear differential equations.***Journal of Mathematical Analysis and Applications*1988,**135**(2):399–405. 10.1016/0022-247X(88)90163-1MATHMathSciNetView ArticleGoogle Scholar - Mitrinović DS, Pečarić JE, Fink AM:
*Inequalities Involving Functions and Their Integrals and Derivatives, Mathematics and Its Applications*.*Volume 53*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1991:xvi+587.View ArticleMATHGoogle Scholar - Pachpatte BG:
*Inequalities for Differential and Integral Equations, Mathematics in Science and Engineering*.*Volume 197*. Academic Press, San Diego, Calif, USA; 1998:x+611.MATHGoogle Scholar - B. G. Pachpatte,
**Bounds on certain integral inequalities**,*Journal of Inequalities in Pure and Applied Mathematics*, vol. 3, no. 3, article 47, 10 pages, 2002.Google Scholar - Wang W-S:
**A generalized sum-difference inequality and applications to partial difference equations.***Advances in Difference Equations*2008,**2008:**-12.Google Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.