Some Gronwall–Bellman Inequalities on Time Scales and Their Continuous Forms: A Survey

Abstract

Some generalizations of the Gronwall–Bellman (G–B) inequality are presented in this paper in continuous form and on time scales. After S. Hilger introduced the time scales theory in 1988, over the years many mathematicians have studied new versions of this inequality according to new results; the purpose of this paper is to present some of them. Therefore, in the Introduction, some generalizations of G–B inequality in continuous forms, linear and nonlinear are presented. In the second section, some important and interesting results on time scales theory are given. In the third and main part of our paper, G–B inequalities on time scales and their possible connection with G–B inequalities presented in the introduction are investigated. In particular, in the third section of this work, more attention is given to G–B type inequalities on time scales discussed in the last four years.