Existence and Multiplicity of Periodic Solutions to Indefinite Singular Equations Having a Non-monotone Term with Two Singularities

Abstract

Abstract Efficient conditions guaranteeing the existence and multiplicity of T-periodic solutions to the second order differential equation u ′′ = h ⁢ ( t ) ⁢ g ⁢ ( u ) {u^{\prime\prime}=h(t)g(u)} are established. Here, g : ( A , B ) → ( 0 , + ∞ ) {g\colon(A,B)\to(0,+\infty)} is a positive function with two singularities, and h ∈ L ⁢ ( ℝ / T ⁢ ℤ ) {h\in L(\mathbb{R}/T\mathbb{Z})} is a general sign-changing function. The obtained results have a form of relation between multiplicities of zeros of the weight function h and orders of singularities of the nonlinear term. Our results have applications in a physical model, where from the equation u ′′ = h ⁢ ( t ) sin 2 ⁡ u {u^{\prime\prime}=\frac{h(t)}{\sin^{2}u}} one can study the existence and multiplicity of periodic motions of a charged particle in an oscillating magnetic field on the sphere. The approach is based on the classical properties of the Leray–Schauder degree.