Period annuli and multiple solutions for two-point BVD

In this article we consider equations of the type xʺ+g(x) = 0 and xʺ+ƒ(x)xʹ² + g(x) = 0. The Neumann boundary value problem is considered. For polynomials f and g we provide the multiplicity results. These results are based on a thorough analysis of a phase plane. The existence of period annuli is concerned.