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Efficient conditions guaranteeing the existence and multiplicity of  T -periodic solutions to the second order differential equationu  ′′   =   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, andh  ∈   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 equationu  ′′   =    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.

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