A variational approach to nonresonance with respect to the Fučik spectrum. (English) Zbl 0768.34025

The paper is concerned with the periodic problem (1) \(-u''(t) = f(t,u(t))\) in \([0,2\pi]\), \(u(0) = u(2 \pi)\), \(u'(0) = u'(2\pi)\). The authors give sufficient conditions for the existence of at least one solution \(u\) of (1) in \(H^ 2(0,2\pi)\). The nonlinearity \(f(t,s)\) interferes in a certain sense with the associated Fučík spectrum. The variational approach with a variant of the saddle point theorem of Rabinowitz and a version of the Wirtinger inequality are used for the proof. The result improves some results of S. Invernizi [Commentat. Math. Univ. Carol. 27, 285-291 (1986; Zbl 0603.34016)] and J.-P. Gossez and P. Omari [Nonlinear Anal., Theory Methods Appl. 14, 1079-1104 (1990; Zbl 0724.34048)]. The authors give also the characterization of the first branch \(C_ 1\) of the Fučík spectrum. The approach can be easily adapted to the Dirichlet problem \(- u'' = f(t,u(t))\) in \([0,\pi]\), \(u(0) = u(\pi) = 0\).
Reviewer: J.Kalas (Brno)


34C25 Periodic solutions to ordinary differential equations
34L05 General spectral theory of ordinary differential operators
Full Text: DOI


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