Existence theory for integrodifferential equations and Henstock-Kurzweil integral in Banach spaces

Abstract

We prove existence theorems for the integrodifferential equation x′ (t) = f(t, x(t), ∫0t k(t, s, x(s)) ds), x(0) = x0, t ∈ Ia = [0, a], a > 0, where f, k, x are functions with values in a Banach space E and the integral is taken in the sense of HL. Additionally, the functions f and k satisfy certain boundary conditions expressed in terms of the measure of noncompactness.