## Existence of periodic solutions for a semilinear ordinary differential equation

### Summary

Summary: 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.

### Mathematics Subject Classification

34B15, 34C15, 34C25, 34C99

### Keywords/Phrases

ordinary differential equation, periodic solutions