Periodic solutions to second-order indefinite singular equations

The efficient conditions guaranteeing the existence of a T-periodic solution to the second order differential equationu″=h(t)g(u) are established in the paper. Here, g is a positive and decreasing function which has a strong singularity at the origin, and the weight h∈L(R/TZ) is a sign-changing function. The obtained results have the form of relation between the multiplicities of the zeroes of the weight function h and the order of the singularity of the nonlinear term. The approach is based on Leray-Schauder degree theory, proving that no T-periodic solution of a certain homotopy appears on the boundary of an unbounded open set during the deformation to an autonomous problem.