Sharp Gronwall-Bellman type integral inequalities with delay

Abstract

Various attempts have been made to give an upper bound for the solutions of the delayed version of the Gronwall-Bellman integral inequality, but the obtained estimations are not sharp. In this paper a new approach is presented to get sharp estimations for the nonnegative solutions of the considered delayed inequalities. The results are based on the idea of the generalized characteristic inequality. Our method gives sharp estimation, and therefore the results are more exact than the earlier ones.

Keywords

• delayed gronwall-bellman integral inequality
• sharp solution
• generalized characteristic inequality