Estimation of unknown function of a class of retarded iterated integral inequalities

Abstract

It is well known that differential equations and integral equations are important tools to discuss the rule of natural phenomena. Various generalizations of the Gronwall-Bellman inequality are important tools in the study of existence, uniqueness, boundedness, stability, continuous dependence on the initial value and parameters, and other qualitative properties of solutions of differential equations and integral equation. In this paper, we discuss a class of retarded iterated integral inequalities, which includes a nonconstant term outside the integrals. By adopting novel analysis techniques, the upper bound of the embedded unknown function is estimated explicitly. The derived result can be applied in the study of solutions of ordinary differential equations and integral equations. Introduction It is well known that differential equations and integral equations are important tools to discuss the rule of natural phenomena. In the study of the existence, uniqueness, boundedness, stability, oscillation and other qualitative properties of solutions of differential equations and integral equations, one often deals with certain integral inequalities. One of the best known and widely used inequalities in the study of nonlinear differential equations is GronwallBellman inequality [1,2], which can be stated as follows: If u and f are non-negative continuous functions on an interval [a, b] satisfying ∫ ∈ + ≤ t a b a t ds s u s f c t u ] , [ , ) ( ) ( ) ( , (1) for some constant 0 ≥ c , then ] , [ , ) ( ) ( exp ) ( b a t ds s u s f c t u t a ∈       ≤ ∫ . Pachpatte in [3] investigated the retarded inequality ∫ ∫ + + ≤ ) ( ) ( ) ( ) ( ) ( ) ( t