Non-asymptotic lazer-leach type conditions for a nonlinear oscillator

Abstract

A well-known result by Lazer and Leach establishes that if g: R → R is continuous and bounded with limits at infinity and m ε2 ℕ, then the resonant periodic problem u" + m2u + g(u) = p(t), u(0)-u(2π) = u'(0)-u'(2π) = 0 admits at least one solution, provided that αm(p)2+β(p)2 < 2/π|g(+∞)-g(- ∞)|, where αm(p) and βm(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.