- Research Article
- Open Access

# Gronwall-Bellman-Type Integral Inequalities and Applications to BVPs

- Chur-Jen Chen
^{1}, - Wing-Sum Cheung
^{1}Email author and - Dandan Zhao
^{1}

**2009**:258569

https://doi.org/10.1155/2009/258569

© Chur-Jen Chen et al. 2009

**Received: **9 May 2008

**Accepted: **29 January 2009

**Published: **5 February 2009

## Abstract

We establish some new nonlinear Gronwall-Bellman-Ou-Iang type integral inequalities with two variables. These inequalities generalize former results and can be used as handy tools to study the qualitative as well as the quantitative properties of solutions of differential equations. Example of applying these inequalities to derive the properties of BVPs is also given.

## Keywords

## 1. Introduction

Inequality (1.2) provides an explicit bound to the unknown function and hence furnishes a handy tool in the study of quantitative and qualitative properties of solutions of differential and integral equations. Because of its fundamental importance over the years many generalizations and analogous results of (1.2) have been established (see, e.g., [1–20]). Among various of Gronwall-Bellman-type inequalities, a very useful one is the following.

Theorem 1.1 (see [21]).

Inequality (1.4) is called Ou-Iang's inequality, which was established by Ou-Iang during his study of the boundedness of certain kinds of second-order differential equations.

Recently, Pachpatte established the following generalization of Ou-Iang-type inequality.

Theorem 1.2 (see [20]).

for all , where is the inverse function of and is chosen such that for all .

Bainov-Simeonov and Lipovan gave the following Gronwall-Bellman-type inequalities, which are useful in the study of global existence of solutions of certain integral equations and functional differential equations.

Theorem 1.3 (see [1]).

for all , where is defined as in Theorem 1.2 and is chosen such that for all .

Theorem 1.4 (see [13]).

for all , where is defined as in Theorem 1.2, and is chosen such that for all .

Very recently, the above results have been further generalized by Cheung to the following.

Theorem 1.5 (see [7]).

Let and be constants. Let , , , and be functions satisfying

(ii) is nondecreasing with for .

In this paper, we intend to establish some new nonlinear Gronwall-Bellman-Ou-Iang type integral inequalities with two variables. The setup is basically along the line of [7] but this is by all means a nontrivial improvement of the results there. Examples are also given to illustrate the usefulness of these inequalities in the study of qualitative as well as the quantitative properties of solutions of BVPs.

## 2. Main Results

Throughout this paper, are two fixed numbers. Let , , and , here we allow or to be . We denote by the set of all -times continuously differentiable functions of into , and . Partial derivatives of a function are denoted by , and so forth. The identity function will be denoted as and so, in particular, is the identity function of onto itself.

Note that we allow and to be here.

Theorem 2.1.

Suppose . Let be a constant. If , , , and are functions satisfying

(ii) is nondecreasing with for ;

(iii) is strictly increasing with and as ;

Proof.

Corollary 2.2.

Suppose . Let be a constant. If , , , and are functions satisfying

(ii)h is strictly increasing with and as ;

Proof.

Let , then . The corollary now follows immediately from Theorem 2.1.

Remark 2.3.

It is easily seen that Theorem 2.1 generalizes Theorems 1.3 and 1.4.

Theorem 2.4.

Suppose . Let be a constant. If , , , and are functions satisfying

(ii) is nondecreasing with for ;

(iii) and are strictly increasing with as ;

Proof.

Since is arbitrary, this concludes the proof of the theorem.

Corollary 2.5.

Suppose . Let be a constant. If , , and are functions satisfying

(ii) is nondecreasing with for ;

(iii) and are strictly increasing with as ;

Remark 2.6.

If we choose then Corollary 2.5 reduces to Theorem 1.2.

Theorem 2.7.

Suppose . Let be a constant. If , , , and are functions satisfying

(ii) is nondecreasing with for ;

(iii) , , and , are strictly increasing with when and as ;

for all , where are the same as in Theorem 2.4, , and is chosen such that for all

Proof.

that is,

Remark 2.8.

Theorem 2.7 is a generalization of Theorem 1.5.

## 3. Application to Boundary Value Problems

where is defined as in Theorem 2.1, , , and are given.

Our first result deals with the boundedness of solutions.

Theorem 3.1.

Consider BVP (3.1). If

then all positive solutions to BVP (3.1) satisfy

In particular, if is bounded on , then every solution to BVP (3.1) is bounded on .

Proof.

The next result is about the quantitative property of solutions.

Theorem 3.2.

for some , then (BVP) (3.1) has at most one solution on .

Proof.

Finally, we investigate the continuous dependence of the solutions of BVP (3.1) on the functional and the boundary data and . For this, we consider the following variation of BVP (3.1):

where is defined as in Theorem 2.1, , , and are given.

Theorem 3.3.

Consider BVP (3.1) and BVP (3.13). If

where is as defined in Theorem 3.1. Hence depends continuously on , , and .

Proof.

Let and be solutions to BVP (3.1) and BVP (3.13), respectively. Then

for some for all . Therefore, depends continuously on , , and .

Remark 3.4.

thus , and so Theorem 3.2 can be viewed as a corollary of Theorem 3.3.

## Declarations

### Acknowledgments

The authors express their gratitude to the referees for their careful reading and many useful comments and suggestions. Research is supported in part by the Research Grants Council of the Hong Kong SAR, China (Project no. HKU7016/07P).

## Authors’ Affiliations

## References

- Bainov D, Simeonov P:
*Integral Inequalities and Applications, Mathematics and Its Applications*.*Volume 57*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1992:xii+245.View ArticleMATHGoogle Scholar - Beckenbach EF, Bellman R:
*Inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge: Heft 30*. Springer, Berlin, Germany; 1961:xii+198.Google Scholar - Bellman R:
**The stability of solutions of linear differential equations.***Duke Mathematical Journal*1943,**10**(4):643–647. 10.1215/S0012-7094-43-01059-2MathSciNetView ArticleMATHGoogle Scholar - Bihari I:
**A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations.***Acta Mathematica Academiae Scientiarum Hungaricae*1956,**7**(1):81–94. 10.1007/BF02022967MathSciNetView ArticleMATHGoogle Scholar - Cheung W-S:
**On some new integrodifferential inequalities of the Gronwall and Wendroff type.***Journal of Mathematical Analysis and Applications*1993,**178**(2):438–449. 10.1006/jmaa.1993.1317MathSciNetView ArticleMATHGoogle Scholar - Cheung W-S:
**Some discrete nonlinear inequalities and applications to boundary value problems for difference equations.***Journal of Difference Equations and Applications*2004,**10**(2):213–223. 10.1080/10236190310001604238MathSciNetView ArticleMATHGoogle 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.2005.08.009MathSciNetView ArticleMATHGoogle Scholar - Cheung W-S, Ma Q-H:
**Nonlinear retarded integral inequalities for functions in two independent variables.***Journal of Concrete and Applicable Mathematics*2004,**2**(2):119–134.MathSciNetMATHGoogle Scholar - Cheung W-S, Ma Q-H, Pecaric J:
**Some discrete nonlinear inequalities and applications to difference equations.***Acta Mathematica Scientia*2008,**28**(2):417–430. 10.1016/S0252-9602(08)60044-2MathSciNetView ArticleGoogle Scholar - Cheung W-S, Ren J:
**Discrete non-linear inequalities and applications to boundary value problems.***Journal of Mathematical Analysis and Applications*2006,**319**(2):708–724. 10.1016/j.jmaa.2005.06.064MathSciNetView ArticleMATHGoogle Scholar - Dafermos CM:
**The second law of thermodynamics and stability.***Archive for Rational Mechanics and Analysis*1979,**70**(2):167–179.MathSciNetView ArticleMATHGoogle Scholar - Haraux A:
*Nonlinear Evolution Equations: Global Behavior of Solutions, Lecture Notes in Mathematics*.*Volume 841*. Springer, Berlin, Germany; 1981:xii+313.Google Scholar - Lipovan O:
**A retarded Gronwall-like inequality and its applications.***Journal of Mathematical Analysis and Applications*2000,**252**(1):389–401. 10.1006/jmaa.2000.7085MathSciNetView ArticleMATHGoogle Scholar - Ma Q-H, Yang E-H:
**On some new nonlinear delay integral inequalities.***Journal of Mathematical Analysis and Applications*2000,**252**(2):864–878. 10.1006/jmaa.2000.7134MathSciNetView ArticleMATHGoogle 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 - Mitrinovic DS:
*Analytic Inequalities, Die Grundlehren der Mathematischen Wissenschaften, Band 16*. Springer, New York, NY, USA; 1970:xii+400.Google Scholar - Mitrinovic DS, Pecaric J, 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:
**Explicit bounds on certain integral inequalities.***Journal of Mathematical Analysis and Applications*2002,**267**(1):48–61. 10.1006/jmaa.2001.7743MathSciNetView 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 - 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 - Ou-Iang L:
**The boundedness of solutions of linear differential equations .***Shuxue Jinzhan*1957,**3:**409–415.MathSciNetGoogle 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.