Boundedness of solutions for a class of retarded Liénard equation
type: Item
instance of: scholarly article
title: Boundedness of solutions for a class of retarded Liénard equation
zbMATH Open document ID: 1044.34023
DOI: 10.1016/S0022-247X(03)00455-4
author: Bingwen Liu
author: Lihong Huang
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-11-20T00:00:00Z
review text: The authors consider the retarded Liénard equation \[ x''+f_1(x) x+f_2(x) x^2+g(x(t-h))=e(t), \] where \(h\) is a nonnegative constant, \(f_1, f_2\), and \(g\) are continuous functions on \(\mathbb{R}=(-\infty, +\infty)\), and \(e(t)\) is a continuous function on \(\mathbb{R}^+=[0, +\infty)\). They obtain some new sufficient conditions, as well as some new necessary and sufficient conditions for all solutions and their derivatives to be bounded avoiding the following traditional conditions: (H1) \(f_1(x)\geq hL\) for all \(x\in \mathbb{R}\) or when \(| x| \) is large; (H2)~ there exists a constant \(N>1\) such that \[ g(x)\left(\int^x_0 f_1(u)\,du- Nhg(x)\right)\geq 0\quad \text{for \( x\in \mathbb{R}\) or when \(x\) is large}; \] (H3) \(xg(x)\geq 0\) for all \(x\in \mathbb{R}\) or when \(| x| \) is large. Two illustrative examples are also included.
Mathematics Subject Classification ID: 34K12
zbMATH DE Number: 2006336
zbMATH Keywords: Retarded Liénard equation
zbMATH Keywords: boundedness
reviewed by: Qiru Wang
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2006336