Electron. J. Differential Equations, Vol. 2023 (2023), No. 29, pp. 1-20.

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

Aneta Sikorska-Nowak

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 \(\mathbb{R}\)), Ia 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.
DOI: https://doi.org/10.58997/ejde.2023.29

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Aneta Sikorska-Nowak
Faculty of Mathematics and Computer Science
Adam Mickiewicz University
Uniwersytetu Poznanskiego 4
61-614 Poznan, Poland
email: anetas@amu.edu.pl

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