Existence Theory for Integrodifferential Equations and Henstock-Kurzweil Integral in Banach Spaces

We prove existence theorems for the integrodifferential equation x'(t)=f(t,x(t),∫0tk(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 measureof noncompactness