Gronwall-Bellman-Type Integral Inequalities and Applications to BVPs
© Chur-Jen Chen et al. 2009
Received: 9 May 2008
Accepted: 29 January 2009
Published: 5 February 2009
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.
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 ).
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 ).
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 ).
Theorem 1.4 (see ).
Very recently, the above results have been further generalized by Cheung to the following.
Theorem 1.5 (see ).
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  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.
It is easily seen that Theorem 2.1 generalizes Theorems 1.3 and 1.4.
Theorem 2.7 is a generalization of Theorem 1.5.
3. Application to Boundary Value Problems
Our first result deals with the boundedness of solutions.
Consider BVP (3.1). If
then all positive solutions to BVP (3.1) satisfy
The next result is about the quantitative property of solutions.
Consider BVP (3.1) and BVP (3.13). If
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).
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