Electron. J. Differential Equations, Vol. 2023 (2023), No. 29, pp. 120.
Integrodifferential equations of mixed type on time scales with DeltaHK and DeltaHKP integrals
Aneta SikorskaNowak
Abstract:
In this article we prove the existence of solutions to the integrodifferential
equation of mixed type
$$ \displaylines{
x^\Delta (t)=f \Big( t,x(t), \int_0^t k_1 (t,s)g(s,x(s))
\Delta s, \int_0^a k_2(t,s)h(s,x(s)) \Delta s \Big),\\
x(0)=x_0, \quad x_0 \in E,\; t \in I_a=[0,a] \cap \mathbb{T},\; a>0,
}$$
where
\(\mathbb{T}\)
denotes a time scale (nonempty closed subset of real
numbers
\(\mathbb{R}\)), I_{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 HenstockKurzweil delta integrals, which generalizes the HenstockKurzweil
integrals. In the second part f, g, h, x are weaklyweakly sequentially
continuous functions and integrals are taken in the sense of HenstockKurzweilPettis
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, HenstockKurzweil delta integral, HL delta integral; Banach space;
HenstockKurzweilPettis delta integral; fixed point; measure of noncompactness;
Caratheodory solutions; pseudosolution.
DOI: https://doi.org/10.58997/ejde.2023.29
Show me the PDF file (380 KB),
TEX file for this article.

Aneta SikorskaNowak
Faculty of Mathematics and Computer Science
Adam Mickiewicz University
Uniwersytetu Poznanskiego 4
61614 Poznan, Poland
email: anetas@amu.edu.pl

Return to the EJDE web page