Existence of periodic solutions for a semilinear ordinary differential equation

Dancer [3] found a necessary and sufficient condition for the existence of periodic solutions to the equation $$ ddot x +g_1(dot x) + g_0(x) = f(t),.$$ His condition is based on a functional that depends on the solution to the above equation with $g_0=0$. However, that solution is not always explicitly known which makes the condition unverifiable in practical situations. As an alternative, we find computable bounds for the functional that provide a sufficient condition and a necessary condition for the existence of solutions.