On some new discrete generalizations of Gronwall's inequality

AbstractThis paper derives new discrete generalizations of the Gronwall-Bellman integral inequality. These generalizations should have wide application in the study of finite difference equations and numerical analysis. The main result (Theorem 3) concerns a very general form of linear Bellman-type discrete inequalities in one independent variable. It is a discrete analogue of an integral inequality obtained by the present author in [J Math. Anal. Appl. 103 (1984), 184–197, Theorem 4] and it extends many discrete inequalities of Agarwal and Thandapani, Pachpatte, and Sugiyama. Two nonlinear extensions of Theorem 3 are also established here