Coexistence of bouncing and classical periodic solutions of generalized Lazer–Solimini equation

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 2π-periodic bouncing solutions having their maximum less than K and at least one 2π-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.