On the Existence of Periodic Solutions for the Quasi-Linear Third-Order Differential Equation

In this paper we consider the nonlinear third-order quasi-linear differential equation x + k2x = f x x x and obtain some simple conditions for the existence of a periodic solution for it. In so doing we use the implicit function theorem to prove a theorem about the existence of periodic solutions and consider one example to show the realizability of the conditions.The validity of the conditions for the parameter-free problem x + k2x = f x x x also is considered.