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

Abstract

We prove existence theorems for the integrodifferential equation $x\text{'}(t)=f\left(t,x\left(t\right),{\int }_0^{t}k\left(t,s,x\left(s\right)\right)ds\right)$, $x\left(0\right)=x_0$, $t\in I_a=\left[0,a\right]$, $a>0$, 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.