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Zbl 1032.34040
Torres, Pedro J.
Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem. (English)
[J] J. Differ. Equations 190, No.2, 643-662 (2003). ISSN 0022-0396

The author provides new existence results for the periodic boundary value problem $$ x''=f(t,x), \quad x(0)=x(T),\ x'(0)=x'(T), \leqno (1) $$ where $f$ is a Carathéodory function. The proofs are based on the Krasnoselskii fixed-point theorem for completely continuous operators in a Banach space that exhibits a cone compression and expansion, and on the sign behaviour of Green's function of the linearized equation. \par The main results are contained in two theorems which give conditions guaranteeing the existence of a positive solution to (1). Modified assertions for negative solutions are shown. \par As applications of these general results, the author obtains new existence results for equations with jumping nonlinearities and for equations with a repulsive or attractive singularity in the origin. Weak singularities are considered here, too.
[Irena Rachuunková (Olomouc)]

MSC 2000:: *34C25 Periodic solutions of ODE
34B16 Singular nonlinear boundary value problems
34B15 Nonlinear boundary value problems of ODE

Keywords: periodic solutions; Krasnoselskii fixed-point theorem; jumping nonlinearity; singular equation

Cited in: Zbl 1219.34055

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