Integrodifferential equations of mixed type on time scales with Delta-HK and Delta-HKP integrals

Abstract

In this article we prove the existence of solutions to the integrodifferential equation of mixed type \begin{gather*}x^\Delta (t)=f \Big( t,x(t), \int_0^t k_1 (t,s)g(s,x(s)) \Delta s, \int_0^a k_2(t,s)h(s,x(s)) \Delta s \Big),\cr x(0)=x_0, \quad x_0 \in E,\; t \in I_a=[0,a] \cap \mathbb{T},\; a>0, \end{gather*} where \(\mathbb{T}\) denotes a time scale (nonempty closed subset of real numbers \(\mathbb{R}\)), \(I_a\) is a time scale interval. In the first part of this paper functions \(f,g,h\) are Caratheodory functions with values in a Banach space E and integrals are taken in the sense of Henstock-Kurzweil delta integrals, which generalizes the Henstock-Kurzweil integrals. In the second part f, g, h, x are weakly-weakly sequentially continuous functions and integrals are taken in the sense of Henstock-Kurzweil-Pettis delta integrals. Additionally, functions f, g, h satisfy some boundary conditions and conditions expressed in terms of measures of noncompactness.