Existence and multiplicity of periodic solutions for a Liénard differential system
type: Item
instance of: scholarly article
title: Existence and multiplicity of periodic solutions for a Liénard differential system
zbMATH Open document ID: 0878.34015
author: O. Diekmann
published in: Differential Equations and Dynamical Systems
publication date: 1997-11-10T00:00:00Z
review text: The author proves existence and multiplicity results for the periodic boundary value problem (Liénard type system) \[ x''(t) +{{d}\over{dt}}\text{ grad} F(x(t)) + g(t,x(t)) = s,\;x(0)= x(2 \pi).\tag{*} \] Here \(s\) is a real bifurcation parameter, \(F:\mathbb{R}^n \to \mathbb{R}\) is of \(C^2\) and \(g: \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n\) is continuous and \(2\pi\)-periodic in \(t\). Using the techniques of a priori estimates and the coincidence degree theory, the author establishes that, under some additional conditions imposed on \(F, g\), there exist \(s_0 \leq s_1 \in \mathbb{R}^n\) such that BVP \((*)\) 1) has no solution for \(s < s_0\); 2) has at least one solution for \(s = s_1\); 3) has at least two solutions for \(s > s_1\).
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 1011018
zbMATH Keywords: Liénard type equation
zbMATH Keywords: periodic solution
zbMATH Keywords: multiplicity
zbMATH Keywords: coincidence degree
zbMATH Keywords: a priori estimate
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1011018