Coexistence of bouncing and classical periodic solutions of generalized Lazer–Solimini equation. (July 2020) - Tomeček, Jan; Rachůnková, Irena; Burkotová, Jana; Stryja, Jakub
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.