On monotonic solutions of systems of nonlinear second order differential equations
type: Item
instance of: scholarly article
title: On monotonic solutions of systems of nonlinear second order differential equations
zbMATH Open document ID: 1181.34047
author: Lianwen Wang
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2009-03-25T00:00:00Z
review text: The author discusses the bounded, monotone and asymptotic behavior of solutions for the second order nonlinear differential equation of the form \[ p(t)f (x'(t))'=q(t)g(x(t)),\quad t\geq a,\tag{*} \] where the functions \(p,q:[a,\infty)\to\mathbb{R}\) and \(f,g:\mathbb{R}\to\mathbb{R}\) are continuous, \(p(t)> 0\), and \(q(t)\geq 0\) is not eventually vanishing on \([a,\infty),rf(r)>0\) and \(rg(r)>0\) for \(r\neq 0\) and \(f\) is increasing on \(\mathbb{R}\). The author establishes some necessary and sufficient conditions for boundedness of all solutions of system (*), and proves that the asymptotic properties of the solutions of (*) are characterized by means of the convergence or divergence of some of the following integrals \[ \int^\infty_0 f^{-1}\left( \pm\frac{1}{p(t)}\int^t_aq(s)\,ds\right)\,dt, \] \[ \int^\infty_0f^{-1} \left(\pm\frac{1}{p(t)}\int^\infty_tq(s)\,ds\right)\,dt, \] and \[ \int^\infty_0f^{-1} \left(\frac{1}{p(t)}\right)dt. \] The obtained results extend and improve many known results of various authors.
reviewed by: He Chungyou
Mathematics Subject Classification ID: 34C11
Mathematics Subject Classification ID: 34D05
zbMATH DE Number: 5532902
zbMATH Keywords: nonlinear differential equations
zbMATH Keywords: second order
zbMATH Keywords: boundedness
zbMATH Keywords: monotonicity
zbMATH Keywords: asymptotic properties
zbMATH Keywords: fixed-point theorem
MaRDI profile type: Publication
full work available at URL: j.na.2008.03.042
OpenAlex ID: W2024063681
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DOI: 10.1016/J.NA.2008.03.042
description: scientific article; zbMATH DE number 5532902