The existence of periodic solutions for a second order nonlinear neutral differential equation with functional delay

In this article we study the existence of periodic solutions of the second order nonlinear neutral differential equation with functional delay \[ \frac{d^{2}}{dt^{2}}x\left( t\right) +p\left( t\right) \frac{d}{dt}x\left( t\right) +q\left( t\right) x^{3}\left( t\right) = \frac{d}{dt}g\left( t,x\left( t-\tau\left( t\right) \right) \right) +f\left( t,x^{3}\left( t\right) ,x^{3}\left( t-\tau\left( t\right) \right)\right). \] The main tool employed here is the Burton-Krasnoselskii's hybrid fixed point theorem dealing with a sum of two mappings, one is a large contraction and the other is compact.