A variational approach to resonance for asymmetric oscillators

Pages: 163 - 181, Volume 6, Issue 1, March 2007

doi:10.3934/cpaa.2007.6.163       Abstract        Full Text (197.8K)       Related Articles

D. Bonheure - Université Catholique de Louvain, Institut de Mathématique Pure et Appliquée, Chemin du Cyclotron, 2 , B-1348 Louvain-la-Neuve, Belgium (email)
C. Fabry - Université Catholique de Louvain, Institut de Mathématique Pure et Appliquée, Chemin du Cyclotron, 2 , B-1348 Louvain-la-Neuve, Belgium (email)

Abstract: We consider in this note the equation

$x'' + \alpha x^+ - \beta x^- + g(x) =p(t),$

where $x^+ =$ max{$x,0$} is the positive part of $x$, $x^- $ =max{$-x,0$} its negative part and $\alpha,\beta$ are positive parameters. We assume that $g :\mathbb R \to \mathbb R$ is continuous and bounded on $\mathbb R$, $p:\mathbb R \to \mathbb R$ is continuous and $2\pi$-periodic. We provide some sufficient conditions of Ahmad, Lazer and Paul type for the existence of $2\pi$-periodic solutions when $(\alpha,\beta)$ belongs to one of the curves of the Fučík spectrum corresponding to $2\pi$-periodic boundary conditions.

Keywords:  Asymmetric oscillators, periodic solutions, nonlinear resonance,Ahmad-Lazer-Paul type conditions.
Mathematics Subject Classification:  Primary: 34B15, 34C25; Secondary: 70K30.

Received: January 2006;      Revised: June 2006;      Published: December 2006.