The method of lower and upper solutions and the stability of periodic oscillations
type: Item
instance of: scholarly article
title: The method of lower and upper solutions and the stability of periodic oscillations
zbMATH Open document ID: 1043.34044
DOI: 10.1016/S0362-546X(01)00888-4
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2002-11-28T00:00:00Z
review text: This paper is devoted to the method of lower and upper solutions to solve the nonlinear periodic problem (1) \(\ddot x+g(t,x)=0\), \(x(0)= x(T)\), \(\dot x(0)= \dot x(T)\), where \(g\in \mathbb{C}^{0,4} (\mathbb{R}/T\mathbb{Z}\times \mathbb{R})\). A function \(\alpha\in \mathbb{C}^2(\mathbb{R}/T\mathbb{Z})\) is said to be a lower solution of (1), if \(\ddot\alpha(t)+ g(t,\alpha(t))\geq 0\) \(\forall\,t\in \mathbb{R}\). An upper solution \(\beta\) is defined in a similar way by reversing the above inequality. C. de Coster and P. Habets showed that if \(\alpha(t)\leq \beta(t)\) \(\forall\,t\in\mathbb{R}\), then (1) has a solution laying between \(\alpha\) and \(\beta\). \textit{E. N. Dancer} and \textit{R. Ortega} [J. Dyn. Differ. Equ. 6, 631--637 (1994; Zbl 0811.34018)] have proved that, when \(\alpha\) and \(\beta\) are strict lower and upper solutions and the region \(\alpha\leq \beta\) contains a unique solution of (1), then this solution is unstable. The main result of this paper gives sufficient conditions for the existence of a unique solution \(\varphi\in [\beta,\alpha]\) that is stable. Significant examples are considered.
reviewed by: Olga Lykova
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34C10
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 37J40
zbMATH DE Number: 1836640
zbMATH Keywords: lower and upper solutions
zbMATH Keywords: stability
zbMATH Keywords: periodic oscillations
author: Daniel E. Nuñez
MaRDI profile type: Publication
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full work available at URL: s0362-546x(01)00888-4
OpenAlex ID: W1998795513
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description: scientific article; zbMATH DE number 1836640