Periodic solutions of difference equations

Abstract

In this paper, we discuss the existence of periodic solutions of periodic difference equations including x ( n + 1 ) = f ( n , x ( n ) ) , n ∈ Z , where x and f are d -vectors, and Z denotes the set of integers, as well as scalar autonomous difference equations including x ( n ) = − f ( x ( n − 1 ) ) , n ∈ Z . First we state some recent results concerning periodic solutions of periodic difference equations. Then, we show the existence of periodic solutions of scalar autonomous difference equations by using Schauder’s fixed point theorem. We illustrate the results with some examples.