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


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Brézis, H., and L. Nirenberg: Characterizations of the ranges of some nonlinear operators and applications to boundary value problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978) 225–326.Google Scholar
  2. 2.
    De Simon, L., and G. Torelli: Soluzione periodiche di equazioni a derivate parziali di tipo iperbolicc nonlineari, Rend. Sem. Mat. Univ. Padova 40 (1968) 380–401.Google Scholar
  3. 3.
    Leray, J., and J. Schauder: Topologie et équations fonctionnelles, Ann. Sci. Ecole Norm. Sup. (3) 51 (1934) 45–78.Google Scholar
  4. 4.
    Mancini, G.: Periodic solutions of semilinear wave equations via generalized Leray-Schauder degree, to appear.Google Scholar
  5. 5.
    Mancini, G.: Periodic solutions of some semilinear autonomous wave equations, to appear.Google Scholar
  6. 6.
    Mawhin, J.: Recent trends in nonlinear boundary value problems, in VII. Internationale Konferenz über nichtlineare Schwingungen, Band I.2, Berlin, Akademie-Verlag, 1977, 52–70.Google Scholar
  7. 7.
    Mawhin, J.: Solutions périodiques d’équations aux dérivées partielles hyperboliques non linéaires, in Mélanges “Théodore Vogel”, Bruxelles, Université Libre de Bruxelles, 1978, 301–315.Google Scholar
  8. 8.
    Mawhin, J., and M. Willem: Compact perturbations of some nonlinear Hammerstein equations, Riv. Mat. Univ. Parma, to appear.Google Scholar
  9. 9.
    McKenna, P.J.: On the reductions of a semilinear hyperbolic problem to a Landesman-Lazer type problem, to appear.Google Scholar
  10. 10.
    Minty, G.: Monotone (nonlinear) operators in a Hilbert space, Duke Math. J. 29 (1962) 341–346.CrossRefGoogle Scholar
  11. 11.
    Rabinowitz, P.: Periodic solutions of nonlinear hyperbolic partial differential equations, Comm. Pure Appl. Math. 20 (1967) 145–205.CrossRefGoogle Scholar
  12. 12.
    Rabinowitz, P.: Some minimax theorems and applications to nonlinear partial differential equations, in Nonlinear Analysis, New York, Academic Press, 1978, 161–178.Google Scholar

Copyright information

© Springer Basel AG 1979

Authors and Affiliations

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

Personalised recommendations