Journal of Applied Mathematics
Volume 2013 (2013), Article ID 853476, 10 pages
Research Article

A Generalized Nonlinear Gronwall-Bellman Inequality with Maxima in Two Variables

Department of Mathematics, Sichuan University for Nationalities, Kangding, Sichuan 626001, China

Received 15 November 2012; Accepted 20 January 2013

Academic Editor: Jitao Sun

Copyright © 2013 Yong Yan. 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.


This paper deals with a generalized form of nonlinear retarded Gronwall-Bellman type integral inequality in which the maximum of the unknown function of two variables is involved. This form includes both a nonconstant term outside the integrals and more than one distinct nonlinear integrals. Requiring neither monotonicity nor separability of given functions, we apply a technique of monotonization to estimate the unknown function. Our result can be used to weaken conditions for some known results. We apply our result to a boundary value problem of a partial differential equation with maxima for uniqueness.