American Institute of Mathematical Sciences

July  2011, 29(3): 757-767. doi: 10.3934/dcds.2011.29.757

Non-asymptotic Lazer-Leach type conditions for a nonlinear oscillator

 1 Departamento de Matemática, Universidad de Buenos Aires and CONICET, Ciudad Universitaria, Pabellón I, (1428) Buenos Aires, Argentina, Argentina

Received  January 2010 Revised  August 2010 Published  November 2010

A well-known result by Lazer and Leach establishes that if $g:\R\to \R$ is continuous and bounded with limits at infinity and $m\in \mathbb{N}$, then the resonant periodic problem

$u'' + m^2 u + g(u)=p(t),\qquad u(0)-u(2\pi)=u'(0)-u'(2\pi)=0$

admits at least one solution, provided that

$(\a_m(p)^2+$β$_m(p)^2$$)^\frac 1\2$< $\frac 2\pi |g(+\infty)-g(-\infty)|,$

where $\a_m(p)$ and β$_m(p)$ denote the $m$-th Fourier coefficients of the forcing term $p$.
In this article we prove that, as it occurs in the case $m=0$, the condition on $g$ may be relaxed. In particular, no specific behavior at infinity is assumed.

Citation: Pablo Amster, Pablo De Nápoli. Non-asymptotic Lazer-Leach type conditions for a nonlinear oscillator. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 757-767. doi: 10.3934/dcds.2011.29.757
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