A More Generalized Gronwall-like Integral Inequality with Applications

This paper deals with a new Gronwall-like integral inequality which is a generalization of integral inequalities proved by Engler (1989) and Pachpatte (1992). The new Gronwall-like integral inequality can be used in various problems in the theory of certain class of ordinary and integral equations. 1. Introduction. It is well known that integral inequalities play a very crucial role in the study of differential equations, integral equations, functional-differential equations, and integro-differential equations. Besides the famous Gronwall-Bellman inequality and its first nonlinear generalization by Bihari (see Bellman and Cooke [1]), there are several other very useful Gronwall-like inequalities. Haraux [3, Corollary 16, page 139] derived one Gronwall-like inequality and used it to prove the existence of solutions of wave equations with logarithmic nonlinearities. On the other hand, Engler [2] utilized the following slight variant of inequality due to Haraux [3, page 139] in the study of global regular solutions for the dynamic antiplane shear problem in nonlinear viscoelasticity.