Generalized first order dynamic equations on time scales with Δ-Carathéodory functions

Abstract

In this paper we consider a first order dynamic equation on time scales in which the right hand side is a Δ -Carathéodory function, which is not necessarily continuous. We generalize this discontinuous dynamic equation using Henstock–Kurzweil Δ -integral and establish results concerning existence of solutions using simple analysis. Uniqueness of solutions is obtained using an Osgood type condition. Moreover we introduce the concept of Henstock–Kurzweil Δ -equi-integrability and study continuous dependence and convergence of solutions.