Non-asymptotic Lazer-Leach type conditions for a nonlinear oscillator
Pablo Amster, Pablo L. De Nápoli · Discrete and Continuous Dynamical Systems · 2010
A well-known result by Lazer and Leach establishesthat if $g:\R\to \R$ is continuous and boundedwith limits at infinity and$m\in \mathbb{N}$, then the resonant periodic problem$u'' + m^2 u + g(u)=p(t),\qquad u(0)-u(2\pi)=u'(0)-u'(2\pi)=0$admitsat least one solution,provided that$(\a_m(p)^2+$β$_m(p)^2$$)^\frac 1\2$where $\a_m(p)$ and β$_m(p)$ denote the $m$-th Fourier coefficients ofthe forcingterm $p$. 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.