Two-point boundary value problems for planar systems: a lower and upper solutions approach
type: Item
instance of: scholarly article
title: Two-point boundary value problems for planar systems: a lower and upper solutions approach
zbMATH Open document ID: 1492.34023
author: Alessandro Fonda
author: Andrea Sfecci
author: Rodica Toader
published in: Journal of Differential Equations
publication date: 2021-11-30T00:00:00Z
review text: The paper is devoted to ensure the existence of solutions of the two dimensional system \[ (P) \begin{cases}x^{\prime}=f(t, x, y), & y^{\prime}=g(t, x, y) \\ (x(a), y(a)) \in \ell_{S}, & (x(b), y(b)) \in \ell_{A},\end{cases} \] where \(f:[a, b] \times \mathbb{R}^{2} \rightarrow \mathbb{R}\) and \(g:[a, b] \times \mathbb{R}^{2} \rightarrow \mathbb{R}\) are continuous functions, and \(\ell_{S}\) and \(\ell_{A}\) are two lines in the plane, the ``starting line'' and the ``arrival line'', respectively. They define a general concept of lower solution \(\alpha\) and upper solution \(\beta\), and consider both the well-ordered case \(\alpha \leq \beta\) and the non-well-ordered case \(\alpha \not \leq \beta\). In the well-ordered case they consider different situations, in the first one they deal with the case when both lines \(\ell_{S}\) and \(\ell_{A}\) are not vertical, in the second one, only one of the two lines is allowed to be vertical, in the third one, the case when both lines are vertical is studied. Moreover, some applications to second order equations are given, in particular for an equation involving the mean curvature operator. In the non well-ordered, they consider nonlinearities that are bounded perturbations of linear ones, and homogeneous Dirichlet boundary conditions. Moreover, they assume that the associated linear system is at resonance and avoid the interaction with higher order eigenvalues. An application of the obtained results is given when a condition of Landesman-Lazer type is assumed. In the last section some possible extensions to higher order dimensional systems are suggested. The existence results are deduced by means of topological degree arguments.
reviewed by: Alberto Cabada
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 7436220
zbMATH Keywords: upper and lower solutions
zbMATH Keywords: Sturm-Liouville boundary value problems
zbMATH Keywords: degree theory
zbMATH Keywords: Mean curvature equation
MaRDI profile type: Publication
full work available at URL: j.jde.2021.11.002
OpenAlex ID: W3215771974
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DOI: 10.1016/J.JDE.2021.11.002
description: scientific article; zbMATH DE number 7436220