Periodic solutions for a Liénard type \(p\)-Laplacian differential equation
type: Item
instance of: scholarly article
title: Periodic solutions for a Liénard type \(p\)-Laplacian differential equation
zbMATH Open document ID: 1166.34314
author: Hua Meng
author: Fei Long
published in: Journal of Computational and Applied Mathematics
publication date: 2009-02-25T00:00:00Z
review text: The authors investigate existence and uniqueness of a \(T\)-periodic solution of the Liénard type \(p\)-Laplacian differential equation: \[ (\varphi_p(x'(t)))'+f(x(t)) x'(t)+ g(t,x(t))=e(t)\,, \] where \(p>0\), \(\varphi:\mathbb R\rightarrow \mathbb R\) is given by \(\varphi_p(s)=|s|^{p-2}s\) for \(s\neq 0\) and \(\varphi(0)=0\), \(f:\mathbb R\rightarrow \mathbb R\) and \(g:\mathbb R^2\rightarrow \mathbb R\) are continuous functions, \(e:\mathbb R\rightarrow \mathbb R\) is a continuous periodic with period \(T>0\) function, \(g\) is \(T\)-periodic in the first argument. As the authors underline, recently this matter is studied mainly under the assumptions \((H_1)\): \(g(t,u)=\beta(t)g(u)\text{ or } g(t,u)=g(u)\;\text{ for all } t,u\in \mathbb R,\) but there are scarce results these conditions. Using topological degree theory the authors prove: Theorem. Let the following conditions hold: \((A_1)\): The functions \({\partial\over\partial t}g(t,x)\) and \({\partial\over\partial x}g(t,x)\) are continuous, and \({\partial\over\partial x}g(t,x)<0\) for all \(t,\,x\in \mathbb R\). Suppose that there exists a positive constant \(d\) such that \((A_2)\): \(x(g(t,x)-e(t))<0\) for all \(|x|>d\), \(t,\,x\in \mathbb R\). Then the considered equation has a unique \(T\)-periodic solution.
reviewed by: Ivan Ginchev
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 5518732
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: periodic solutions
zbMATH Keywords: Liénard equation
zbMATH Keywords: topological degree
Wikidata QID: Q115359855
MaRDI profile type: Publication
full work available at URL: j.cam.2008.06.001
OpenAlex ID: W1978252996
cites work: Existence and uniqueness of periodic solutions for a kind of Duffing type \(p\)-Laplacian equation
cites work: Existence and uniqueness of periodic solutions for a kind of Liénard equation with two deviating arguments
cites work: Existence and uniqueness of periodic solutions for a kind of Duffing equation with two deviating arguments
cites work: Existence and uniqueness of periodic solutions for a kind of Liénard type \(p\)-Laplacian equation
cites work: Periodic solution for nonlinear systems with \(p\)-Laplacian-like operators
DOI: 10.1016/J.CAM.2008.06.001
description: scientific article; zbMATH DE number 5518732