A singular periodic Ambrosetti–Prodi problem of Rayleigh equations without coercivity conditions

In this paper, we use the Leray–Schauder degree theory to study the following singular periodic problems: [Formula: see text], [Formula: see text], where [Formula: see text] is a continuous function with [Formula: see text], function [Formula: see text] is continuous with an attractive singularity at the origin, and [Formula: see text] is a constant. We consider the case where the friction term [Formula: see text] satisfies a local superlinear growth condition but not necessarily of the Nagumo type, and function [Formula: see text] does not need to satisfy coercivity conditions. An Ambrosetti–Prodi type result is obtained.

Bouncing solutions of an equation with attractive singularity

For any n, m ∈ N, we prove the existence of 2mπ-periodic solutions, with n bouncings in each period, for a second-order forced equation with attractive singularity by using the approach of successor map and Poincaré-Birkhoff twist theorem.