Existence and multiplicity of periodic solutions to an indefinite singular equation with two singularities. The degenerate case

Abstract

We analyze the existence of T −periodic solutions to the second-order indefinite singular equation u ′′ = β h(t) cos 2 u which depends on a positive parameter β > 0. Here, h is a sign-changing function with h = 0 and where the nonlinear term of the equation has two singularities. For the first time, the degenerate case is studied, displaying an unexpected feature which contrasts with the results known in the literature for indefinite singular equations.

Existence and multiplicity of periodic solutions to differential equations with attractive singularities

“…The idea how to obtain multiplicity results is similar to the previous works, usually we use homotopy invariance and additivity property of Leray-Schauder degree (see, e.g. [2,6,7,15,19,23,30]). However, the nature of this kind of problems request quite new approach in how to construct the strict lower and upper functions.…”