Some systems of nonlinear difference equations of higher order with periodic solutions

Abstract

We show that every positive solution of the system of difference equations x n + 1(1) = 1 + x n(2) /x n - 1(3) , x n + 1(2) = 1 + x n(3) /x n - 1(4) ,..., x n + 1 (k) = 1 + x n(1) /x n - 1(2) , n ε ℕ 0 , where k ε ℕ is fixed, is periodic with period equal to 5k if k ≢ 0 (mod 5), and with period k if k ≡ 0 (mod 5). It is shown also that every positive solution of the system of difference equations x n + 1(1) = 1 + x n(2) + x n - 1(3) /x n - 2(4) , x n + 1(2) = 1 + x n(3) + x n - 1(4) /x n - 2(5) ,..., x n + 1(k) = 1 + x n(1) + x n - 1(2) ,/x n - 2(3) n ε ℕ 0 , is periodic with period equal to 2 3-i k, if GCD(k, 8) = 2 i , i ε {0, 1, 2, 3} (the greatest common divisor of k and 8). Two more systems are considered. These results generalize the well-known periodicity of the corresponding scalar equations.