Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 29, pp. 1-20.

Title: Integrodifferential equations of mixed type on time scales with Delta-HK and Delta-HKP integrals
       
Author: Aneta Sikorska-Nowak (Adam Mickiewicz Univ., Poznan, Poland)

Abstract: 
In this article we prove the existence of solutions to the integrodifferential
equation of mixed type
$$\displaylines{
x^\Delta (t)=f \Big( t,x(t), \int_0^t k_1 (t,s)g(s,x(s))
\Delta s, \int_0^a k_2(t,s)h(s,x(s)) \Delta s \Big),\cr
x(0)=x_0, \quad x_0 \in E,\; t \in I_a=[0,a] \cap \mathbb{T},\; a>0,
}$$
where $\mathbb{T}$ denotes a time scale (nonempty closed subset of real 
numbers $\mathbb{R}$), I<sub>a</sub> 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.

Submitted February 15, 2023. Published March 14, 2023.
Math Subject Classifications: 35A06, 34A12, 34A34, 34B15, 34G20, 34N99.
Key Words: Integrodifferential equations; nonlinear Volterra integral equation;
    time scales, Henstock-Kurzweil delta integral, HL delta integral; Banach space;
    Henstock-Kurzweil-Pettis delta integral; fixed point; measure of noncompactness;
    Caratheodory solutions; pseudo-solution.