Existence and uniqueness results for Lienard's equations

A new set of conditions for the existence and uniqueness of the periodic solution of the equation
\ddot{x} + f(x)\dot{x} + g(x) = 0
(superscript dot =
d/dt
) is given. Lyapunov-like functions are used in the derivation of the results. This permits the formulation of the existence and uniqueness conditions in a simple yet general way. The present conditions include those of Libnard and Levinson and Smith when particularized to the above equation. They also permit the inclusion of most of the cases of two-stroke oscillators covered by the author's previous theorems for equations of the above type. Periodicity results for the forced equation
\ddot{x} + f(x)x + g(\dot{x}) = e(t)
with
e(t)
periodic and bounded are also given.