Periodic Solutions of Nonlinear Dispersive Wave Equations

Abstract

We prove the existence of generalized solutions 2π-periodic in t and x for nonlinear wave equations of the form

$${u_{tt}} - {u_{xx}} = f(t,x,u)$$

under an asymptotic nonresonance condition of the form

$$\mu < p < {u^{ - 1}}f(t,x,u) \leq q < \nu $$

for a.e. (t,x) ∈ [0,2π]× [0,2π] and large values of |u|, where μ and ν are consecutive elements of the spectrum of the linear part. It is moreover assumed that, for a.e. (t,x), the function

$$sign\;p.f(t,x,u)$$

is nondecreasing in u. The approach is a combination of recent results on Hammerstein equations and Leray-Schauder’s theory.