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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.

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Copyright information

© Springer Basel AG 1979

Authors and Affiliations

  • Jean Mawhin
    • 1
  1. 1.Institut MathématiqueUniversité de LouvainLouvain-la-NeuveBelgique

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