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

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)sin2⁡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.

Keywords

• singular differential equation
• indefinite weight
• periodic solution
• 34c25
• 34b16
• 47h11
• 70f05