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''' + k(2) x' = epsilonf (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''' + k(2)x' = f(x, x', x") also is considered. (C) 2001 Academic Press.