Periodic solutions of periodic difference equations

Abstract

In this paper, we discuss the existence of periodic solutions of the periodic difference equation $$ x(n + 1) = f(n, x(n)),\ \ n \in \mathbf{Z} $$ and the periodic difference equation with finite delay $$ x(n + 1) = f(n, x_n),\ \ n \in \mathbf{Z}, $$ where $x$ and $f$ are $d$-vectors, and $\mathbf{Z}$ denotes the set of integers. We show the existence of periodic solutions by using Browder's fixed point theorem, and illustrate an example by using a boundedness result due to Shunian Zhang.