Multiple existence of periodic solutions for Liénard system
type: Item
instance of: scholarly article
title: Multiple existence of periodic solutions for Liénard system
zbMATH Open document ID: 0824.34041
author: Yanyan Li
published in: Differential and Integral Equations
publication date: 1995-11-16T00:00:00Z
review text: The authors consider the existence of multiple solutions, to the Liénard system (E) \(x''(t)- {d\over dt} G(x(t))+ f(t, x(t))= e\), (B) \(x(0)= x(2\pi)\); \(x'(0)= x'(2\pi)\), where \(e\in \mathbb{R}^ N\), \(G: \mathbb{R}^ N\to \mathbb{R}^ N\), \(f: \mathbb{R}\times \mathbb{R}^ N\to \mathbb{R}^ N\) are all continuous functions. The authors show that for a sufficiently large constant vector \(R_ 0\in \mathbb{R}^ N\) the equation (E) will have at least \(2^ N\) solutions if \(e> R_ 0\), meaning that each component of \(e\) is greater than the corresponding component of \(R_ 0\). The technique of proof uses the invariance of degree under homotopy argument introduced by J. Mawhin in the late 1970-s. The reviewer comments that the assumption of \(e\) being ``large'' permits one to offer fairly elementary conclusions based on non-standard analysis arguments, if some assumptions are made about \(G'\) and \(f\) for large values of \(t\), or for all values of \(x\).
reviewed by: Vadim Komkov
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 34B15
zbMATH DE Number: 775648
zbMATH Keywords: multiple periodic solutions
zbMATH Keywords: Liénard system
zbMATH Keywords: invariance of degree under homotopy
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 775648