It is known that a large number of research works dealing with discrete inequalities and their applications have appeared during the last three decades. Furthermore recurrent inequalities involving sequences of real numbers considered as discrete analogues of the Gronwall-Bellman inequality or its variants have been used in the analysis of the finite difference equations such as those of infinity type given by B. G. Pachpatte in {On certain new finite difference inequalities} (Indian J. Pure Appl. Math. 24(1993), 373-384). The aim of the present paper is to study the structure of three analogues of the Gronwall-Bellman inequalities of infinity type given by B. G. Pachpatte by introducing a general form for such inequalities. According to author's approach it is given two generalized inequalities in the case of one dimensional functions defined on Z to R_+. (The original results given in section 2, Theorem 1 and 2 generalize the corresponding Theorem 1 in the abovementioned B. G. Pachpatte's paper) and a generalized inequality for functions of two-variables defined on Z|times Z into P_+. (The original result given in section 3, Theorem 1 generalize the analogous Theorem 4 in the B. G. Pachpatte's paper). The previous generalizations are supported by using the mean value theorem instead of the elementary calculations that are used in B. G. Pachpatte's paper.

Στοιχεία Άρθρου

Περιοδικό	Δελτίο της Ελληνικής Μαθηματικής Εταιρίας
Αρ. Τεύχους	Τεύχος 43
Περίοδος	2000
Συγγραφέας	Stevo Stevic
Αρ. Αρθρου	10
Σελίδες	137-146
Γλώσσα	-
Λέξεις Κλειδιά	Gronwall-Bellman inequality

A Generalization of Pachpatte Difference Inequalities