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Nonresonance with respect to the Fučik spectrum for periodic solutions of second order ordinary differential equations. (English) Zbl 0709.34037
This paper is concerned with the existence of periodic solutions for the equation \(x''(t)+g(t,x(t))=e(t)\) in the case where the nonlinearity g(t,s) lies asymptotically between 0 and one point \((Q_+,Q_ -)\) of the first branch of the Fučik spectrum, or between two points \((q_+,q_ -)\) and \((Q_+,Q_ -)\) of two consecutive branches of that spectrum. The emphasis is put on cases where the inferior or superior limits of the quotient g(t,s)/s as \(s\to +\infty\) or -\(\infty\) are constantly (i.e. for all t) equal to one spectral value. Conditions of positive density at infinity (which were introduced in D. G. de Figueiredo, J.-P. Gossez [C. R. Acad. Sci. Paris, Ser. I. 302, 543-545 (1986; Zbl 0596.35049)]) are considered. Liénard equations are also treated.
Reviewer: J.-P.Gossez

MSC:
34C25 Periodic solutions to ordinary differential equations
Keywords:
Fučik spectrum
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