Integrodifferential Equations on Time Scales with Henstock-Kurzweil-Pettis Delta Integrals

Abstract

We prove existence theorems for integro-differential equations $x^{\Delta }\left(t\right)=f(t,x(t),{\int }_0^{t}k(t,s,x\left(s\right)\left)\Delta s\right)$, $x\left(0\right)=x_0$, $t\in I_a=[0,a]\cap T$, $a\in R_{+}$, where $T$ denotes a time scale (nonempty closed subset of real numbers $R$), and $I_a$ 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.