2022 Affine-periodic solutions for generalized ODEs and other equations
Márcia Federson, Rogelio Grau, Carolina Mesquita
Topol. Methods Nonlinear Anal. 60(2): 725-760 (2022). DOI: 10.12775/TMNA.2022.027

Abstract

It is known that the concept of affine-periodicity encompasses classic notions of symmetries as the classic periodicity, anti-periodicity and rotating symmetries (in particular, quasi-periodicity). The aim of this paper is to establish the basis of affine-periodic solutions of generalized ODEs. Thus, for a given real number $T> 0$ and an invertible $n\times n$ matrix $Q$, with entries in $\mathbb C$, we establish conditions for the existence of a $(Q,T)$-affine-periodic solution within the framework of nonautonomous generalized ODEs, whose integral form displays the nonabsolute Kurzweil integral, which encompasses many types of integrals, such as the Riemann, the Lebesgue integral, among others. The main tools employed here are the fixed point theorems of Banach and of Krasnosel'skiĭ. We apply our main results to measure differential equations with Henstock-Kurzweil-Stiejtes righthand sides as well as to impulsive differential equations and dynamic equations on time scales which are particular cases of the former.

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Márcia Federson. Rogelio Grau. Carolina Mesquita. "Affine-periodic solutions for generalized ODEs and other equations." Topol. Methods Nonlinear Anal. 60 (2) 725 - 760, 2022. https://doi.org/10.12775/TMNA.2022.027

Information

Published: 2022
First available in Project Euclid: 31 July 2022

MathSciNet: MR4563255
Digital Object Identifier: 10.12775/TMNA.2022.027

Keywords: Affine-periodic solutions , Banach fixed point theorem , Henstock-Kurzweil-Stieltjes integral , Krasnosel'skiĭ fixed point theorem

Rights: Copyright © 2022 Juliusz P. Schauder Centre for Nonlinear Studies

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Vol.60 • No. 2 • 2022
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