Periodic behaviour of nonlinear second order discrete dynamical systems

Abstract

In this work we provide conditions for the existence of periodic solutions to nonlinear, second-order difference equations of the form \begin{equation*} y(t+2)+by(t+1)+cy(t)=g(t,y(t)) \end{equation*} where $c\ne 0$, and $g:{\mathbb{Z}}^{+}\times \mathbb{R}\to \mathbb{R}$ is continuous and periodic in $t$. Our analysis uses the Lyapunov-Schmidt reduction in combination with fixed point methods and topological degree theory.