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 2 pi-periodic bouncing solutions having their maximum less than K and at least one 2 pi-periodic classical solution having its minimum greater than K. The proofs are based on the ideas of the Poincare-Birkhoff Twist Map Theorem and approximation principles.