Coexistence of bouncing and classical periodic solutions of generalized Lazer–Solimini equation. (July 2020)

• Record Type: Journal Article
• Authors: Tomeček, Jan; Rachůnková, Irena; Burkotová, Jana; Stryja, Jakub
• Is Part Of: Nonlinear analysis. Volume 196(2020)
• Publication Date: 2020-07
• Subjects: 34A37 -- 34B18 -- 34C25; Nonnegative periodic solution -- Singular IVP -- Impulsive differential equation -- Generalized Lazer–Solimini equation -- Coexistence -- Twist map theorem; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2020.111783
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: The paper deals with the singular differential equation x ′ ′ + g ( x ) = p ( t ), where g has a weak singularity at x = 0 . Sufficient conditions for a coexistence of two types of periodic solutions are presented. The first type is a classical periodic solution which is strictly positive on R and does not reach the singularity. The second type is a bouncing periodic solution which reaches the singularity at isolated points. In particular, we state a constant K > 0 such that there exist at least two 2 π -periodic bouncing solutions having their maximum less than K and at least one 2 π -periodic classical solution having its minimum greater than K . The proofs are based on the ideas of the Poincaré–Birkhoff Twist Map Theorem and approximation principles.