Periodic solutions for Lienard equation with an indefinite singularity

In this paper, the problem of periodic solutions is studied for Lienard equations with an indefinite singularity x '' (t) + f(x(t))x' (t) + phi(t)x(m)(t) - alpha(t)/x(mu)(t) = 0, where f : (0, +infinity) -> R is a continuous function which may have a singularity at the origin, the signs of phi and alpha are allowed to change, m is a non-negative constant, and mu is a positive constant. The approach is based on a continuation theorem of Manasevich and Mawhin with techniques of a priori estimates. The main results partly answer the open problem proposed by R. Hakl, P.J. Torres and M. Zamora in the known literature. (C) 2018 Elsevier Ltd. All rights reserved.