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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)

MSC:

34C25 Periodic solutions to ordinary differential equations
34L05 General spectral theory of ordinary differential operators
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References:

[1] Fučik, S., Boundary value problems with jumping nonlinearities, Čas. pǒst. mat., 101, 69-87, (1976) · Zbl 0332.34016
[2] Fučik, S., Solvability of nonlinear equationd and boundary value problems, (1980), Reidel Dordrecht
[3] Mawhin, J.; Ward, J., Periodic solutions of some forced Liénard differential equations at resonance, Arch. math., 41, 337-351, (1943) · Zbl 0537.34037
[4] Invernizi, S., A note on nonuniform nonresonance for jumping nonlinearities, Commentat. math. univ. carol., 27, 285-291, (1986) · Zbl 0603.34016
[5] Gossez, J.-P.; Omari, P., Nonresonance with respect to the fučik spectrum for periodic solutions of second order ordinary differential equations, Nonlinear analysis, 14, 1079-1104, (1990) · Zbl 0724.34048
[6] Gossez, J.-P.; Omari, P., Periodic solutions of a second order ordinary differential equation: a necessary and sufficient condition for nonresonance, J. diff. eqns, 94, 67-82, (1991) · Zbl 0743.34045
[7] Costa, D.; Oliveira, A., Existence of solution for a class of semilinear problems at double resonance, Boll. soc. bras. mat., 19, 21-37, (1988) · Zbl 0704.35048
[8] Berestycki, H.; Figueiredo, D.G.de, Double resonance in semilinear elliptic problems, Communs partial diff. eqns, 6, 91-120, (1981) · Zbl 0468.35043
[9] Rabinowitz, P., Minimax methods in critical point theory with applications to differential equations, C.B.M.S. reg. conf., 65, (1986), Am. math. Soc
[10] Mawhin, J.; Willem, M., Critical point theory and Hamiltonian systems, (1989), Springer New York · Zbl 0676.58017
[11] Clarke, F., Optimization and nonsmooth analysis, (1983), Wiley New York · Zbl 0582.49001
[12] {\scKavian} O., Quelques remarques sur le spectre demi-linéaire de certains opérateurs auto-adjoints (preprint).
[13] {\scRamos} M., Private communication.
[14] {\scCosta} D., Private communication.
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