Well-ordered and non-well-ordered lower and upper solutions for periodic planar systems
type: Item
instance of: scholarly article
title: Well-ordered and non-well-ordered lower and upper solutions for periodic planar systems
zbMATH Open document ID: 1471.34074
author: Alessandro Fonda
author: Giuliano Klun
author: Andrea Sfecci
published in: Advanced Nonlinear Studies
publication date: 2021-07-26T00:00:00Z
review text: The authors consider the following periodic problem \[ (P) \left\{ \begin{array}{ll} x'=f(t,x,y), & y'=g(t,x,y),\\ \\ x(0)=x(T), & y(0)=y(T), \end{array} \right. \] where \(f: {\mathbb R}^3\to {\mathbb R}\) and \(g: {\mathbb R}^3\to {\mathbb R}\) are continuous functions, \(T\)-periodic in their first variable. The authors introduce a general definition of a lower and an upper solution, that improves previous ones given in the literature, and prove the existence of a solution of problem \((P)\). The new definition allows to the lower and upper solutions to be not differentiable at some points of its domain, moreover some discontinuities at related functions on the definition may be also considered. They deduce the existence of solutions of problem \((P)\) on the classical well order case and in the non well ordered one by assuming, in this last case, some growth conditions on \(f\) and \(g\) in order to avoid resonance. A direct application is given for the \(\phi\)-Laplacian equation \[ (\phi(x'(t))'=h(t,x(t),x'(t)), \] with \(\phi:I \to J\) is an increasing homeomorphism between two intervals \(I\) and \(J\) containing \(0\), and \(\phi(0)=0\).
reviewed by: Alberto Cabada
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 7374861
zbMATH Keywords: lower and upper solutions
zbMATH Keywords: periodic systems
zbMATH Keywords: degree theory
MaRDI profile type: Publication
full work available at URL: ans-2021-2117
OpenAlex ID: W3131805333
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DOI: 10.1515/ANS-2021-2117
description: scientific article; zbMATH DE number 7374861