Integro-differential equations on time scales with Henstock-Kurzweil delta integrals
type: Item
instance of: scholarly article
title: Integro-differential equations on time scales with Henstock-Kurzweil delta integrals
zbMATH Open document ID: 1257.34076
author: Aneta Sikorska-Nowak
published in: Discussiones Mathematicae. Differential Inclusions, Control and Optimization
publication date: 2012-09-05T00:00:00Z
Mathematics Subject Classification ID: 34N05
Mathematics Subject Classification ID: 34K30
Mathematics Subject Classification ID: 47N20
Mathematics Subject Classification ID: 47H08
zbMATH DE Number: 6077295
zbMATH Keywords: integro-differential equations
zbMATH Keywords: nonlinear Volterra integral equation
zbMATH Keywords: time scale
zbMATH Keywords: Henstock-Kurzweil delta integral
zbMATH Keywords: Henstock-Lebesgue delta integral
zbMATH Keywords: measure of noncompactness
MaRDI profile type: Publication
full work available at URL: dmdico.1128
OpenAlex ID: W2327313550
DOI: 10.7151/DMDICO.1128
review text: The article is concerned with the solvability of integro-differential equations on time scales, of the form NEWLINE\[NEWLINEx^\Delta(t)=f\left(t,x(t),\int_0^t k(t,s,x(s))\Delta s\right),\;\;\;t\in[0,a]\cap{\mathbb T},\tag{1}NEWLINE\]NEWLINE where \(\mathbb T\) is a given time scale such that \(0\in\mathbb T\), and both \(f\) and \(x\) take values in a Banach space.NEWLINENEWLINEA solution is assumed to be a uniformly generalized absolutely continuous in the restricted sense (\(ACG_*\)) function for which the equality in (1) holds \(\mu_\Delta\)-almost everywhere in \([0,a]\cap{\mathbb T}\). Under these conditions, equation (1) can be written in the equivalent integral form NEWLINE\[NEWLINEx(t)=x(0)+\int_0^t f\left(z,x(z),\int_0^z k(z,s,x(s))\Delta s\right)\Delta z,\;\;\;t\in[0,a]\cap{\mathbb T},\tag{2}NEWLINE\]NEWLINE where the integral on the right-hand side is the Henstock-Lebesgue \(\Delta\)-integral (also known as the strong Henstock-Kurzweil \(\Delta\)-integral).NEWLINENEWLINEThe author uses the Mönch fixed-point theorem to obtain the local existence of a solution of (2) satisfying a~given initial condition \(x(0)=x_0\). The conditions on \(f\) and \(k\), which guarantee that the assumptions of the fixed point theorem are satisfied, are based on Kuratowski's measure of noncompactness.
reviewed by: Antonín Slavík
description: scientific article; zbMATH DE number 6077295