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

Abstract EN

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.