Existence and multiplicity results for systems of first-order differential equations via the method of solution-regions
type: Item
instance of: scholarly article
title: Existence and multiplicity results for systems of first-order differential equations via the method of solution-regions
zbMATH Open document ID: 1403.34023
author: Marlène Frigon
published in: Advanced Nonlinear Studies
publication date: 2018-08-22T00:00:00Z
review text: This excellent paper is devoted to prove the existence and multiplicity of solutions of the following system of first order differential equations \[ u'(t)=f(t,u(t))\text{ for a.e }t \in [0,T], \] coupled to either initial, \(u(0)=r\), or periodic conditions, \(u(0)=u(T)\). Here \(f:[0,T] \times {\mathbb R}^N \to {\mathbb R}^N\) is a Carathéodory function. The main novelty of this paper is the introduction of the new concepts of solution-regions and strict solution-regions. These new methods generalize the previous ones of ordered lower and upper solutions and solution-tube. In the paper, it is shown that this method covers more general regions and, as a consequence, it is possible to deduce existence and multiplicity results under weaker assumptions on the function \(f\) that defines the equation. The results are deduced from fixed point and index theory. Several examples are shown along the paper that point out the applicability of the obtained results.
reviewed by: Alberto Cabada
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34A12
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 6921845
zbMATH Keywords: system of first order differential equation
zbMATH Keywords: multiple solution
zbMATH Keywords: initial value and periodic boundary value problem
zbMATH Keywords: fixed point index
zbMATH Keywords: solution-region
zbMATH Keywords: upper and lower solution
MaRDI profile type: Publication
full work available at URL: ans-2017-6049
OpenAlex ID: W2790400756
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DOI: 10.1515/ANS-2017-6049
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description: scientific article; zbMATH DE number 6921845