Unbounded solutions of a class of planar systems
type: Item
instance of: scholarly article
title: Unbounded solutions of a class of planar systems
zbMATH Open document ID: 1062.34038
author: Xiaojing Yang
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-08-19T00:00:00Z
review text: The author obtains some sufficient conditions for the existence of unbounded solutions for the following planar system \[ x'= a^+ y^+- a^- y^-+ f(t),\quad y'= -b^+ x^++ b^- x^-+ g(t),\tag{1} \] where \(a^\pm\), \(b^\pm\) are positive constants satisfying \[ {1\over\sqrt{a^+ b^+}}+ {1\over\sqrt{a^+ b^-}}+ {1\over\sqrt{a^- b^+}}+ {1\over\sqrt{a^- b^-}}= {4\over\omega}, \] and \(x^\pm= \max\{\pm x,0\}\), \(y^\pm= \max\{\pm y,0\}\), \(\omega\in \mathbb{R}^+/Q\), and \(f,g\in L^\infty[0, 2\pi]\) are \(2\pi\)-periodic functions. It is shown, by using some auxiliary functions, that there exists \(R_0> 0\) such that every solutions \(x(t)\) of (1) with the initial value \((x(0),x'(0))\) satisfying \[ x^2(0)+ (x'(0))^2\geq R^2_0 \] goes to infinity in the future or in the past. This result is proved by applying the well-known Birkhoff ergodic theorem. A suggestive example is given.
Mathematics Subject Classification ID: 34C11
zbMATH DE Number: 2092958
zbMATH Keywords: periodic solutions
zbMATH Keywords: planar system
zbMATH Keywords: Fučik spectrum
reviewed by: Marian Ivanovici
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.04.044
OpenAlex ID: W2083326959
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DOI: 10.1016/J.JMAA.2004.04.044
description: scientific article; zbMATH DE number 2092958