Existence of a periodic solution for superlinear second order ODEs

TL;DR

This paper establishes a precise condition for the existence of periodic solutions in superlinear second-order differential equations with time-periodic coefficients, using a fixed-point approach based on rotational dynamics.

Contribution

It provides a necessary and sufficient condition for periodic solutions in superlinear second-order ODEs with bounded perturbations, advancing the theoretical understanding of such equations.

Findings

Derived a condition for the existence of T-periodic solutions.

Applied a fixed-point theorem leveraging rotational properties.

Extended the theory to equations with superlinear growth in x.

Abstract

We prove a necessary and sufficient condition for the existence of a $T$-periodic solution for the time-periodic second order differential equation $\ddot{x}+f(t,x)+p(t,x,\dot x)=0$, where $f$ grows superlinearly in $x$ uniformly in time, while $p$ is bounded. Our method is based on a fixed-point theorem which uses the rotational properties of the dynamics.