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

Abstract

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.

Keywords

• periodic solutions
• nonlinear neutral differential equations
• large contraction