Volterra Theory on Time Scales

Abstract

This paper is concerned with the existence and uniqueness of solutions to generalized Volterra integral equations on time scales. Unlike previous papers published on this subject, we can weaken the continuity property of the kernel function since the method we introduce here to guarantee existence and uniqueness does not make use of the Banach fixed point theorem. This allows us to construct a bridge between the solutions of Volterra integral equations and of dynamic equations. The paper also covers results concerning the following concepts: the notion of the resolvent kernel, and its role in the formulation of the solution, the reciprocity property of kernels, Picard iterates, the relation between linear dynamic equations and Volterra integral equations, and some special types of kernels together with several illustrative examples.