On the Solutions of Systems of Difference Equations

Abstract

We show that every solution of the following system of difference equations xn 1 1 xn 2 / xn 2 ? 1 , x 2 n 1 x 3 n / xn 3 ? 1 , . . . , xn k 1 xn 1 / xn 1 ? 1 as well as of the system xn 1 1 xn k / xn k ? 1 , xn 2 1 x 1 n / xn 1 ? 1 , . . . , xn k 1 xn k?1 / xn k?1 ? 1 is periodic with period 2k if k / 0 mod2 , and with period k if k 0 mod 2 where the initial values are nonzero real numbers for x0 1 , x0 2 , . . . , x0 k / 1.