Periodic bouncing solutions of the Lazer-Solimini equation with weak repulsive singularity

TL;DR

This paper establishes the existence and multiplicity of periodic bouncing solutions for a second-order differential equation with a weak repulsive singularity, using the Poincaré-Birkhoff Theorem.

Contribution

It introduces new methods to prove multiple periodic solutions for equations with singularities, expanding understanding of bouncing dynamics.

Findings

Existence of periodic bouncing solutions proven.

Multiple solutions classified by period and collision count.

Application of Poincaré-Birkhoff Theorem to singular equations.

Abstract

We prove the existence and multiplicity of periodic solutions of bouncing type for a second-order differential equation with a weak repulsive singularity. Such solutions can be catalogued according to the minimal period and the number of elastic collisions with the singularity in each period. The proof relies on the Poincar\'e-Birkhoff Theorem.