Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem
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. 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. 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.
scientific article; zbMATH DE number 1929527