Copyright © 2010 Aneta Sikorska-Nowak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We prove existence theorems for integro-differential equations xΔ(t)=f(t,x(t),∫0tk(t,s,x(s))Δs), x(0)=x0, t∈Ia=[0,a]∩T, a∈R+, where T denotes a time scale (nonempty closed subset of real numbers R), and Ia is a time scale interval. The functions f,  k are weakly-weakly sequentially continuous with values in a Banach space E, and the integral is taken in the sense of Henstock-Kurzweil-Pettis delta integral. This integral generalizes the Henstock-Kurzweil delta integral and the Pettis integral. Additionally, the functions f and k satisfy some boundary conditions and conditions expressed in terms of measures of weak noncompactness. Moreover, we prove Ambrosetti's lemma.