Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem
type: Item
instance of: scholarly article
title: Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem
zbMATH Open document ID: 1032.34040
DOI: 10.1016/S0022-0396(02)00152-3
author: Pedro J. Torres
published in: Journal of Differential Equations
publication date: 2003-06-17T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34B16
Mathematics Subject Classification ID: 34B15
zbMATH DE Number: 1929527
zbMATH Keywords: periodic solutions
zbMATH Keywords: Krasnoselskii fixed-point theorem
zbMATH Keywords: jumping nonlinearity
zbMATH Keywords: singular equation
reviewed by: Irena Rachůnková
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1929527