Asymptotic periodicity of a higher-order difference equation

We give a complete picture regarding the asymptotic periodicity of positive solutions of the following di. erence equation: x(n) = f (x(n-p1) ,..., x(n-pk), x(n-q1) ,..., x(n-qm)), n epsilon N-0, where p(i), i epsilon {1 ,..., k}, and q(j), j epsilon {1 ,..., m}, are natural numbers such that p1 < p2 < . . . < p(k), q(1) < q(2) < . . . < q(m) and gcd(p(1) ,..., p(k), q(1) ,..., q(m)) = 1, the function f epsilon C[(0, infinity)(k+m), (alpha, infinity)], alpha > 0, is increasing in the first k arguments and decreasing in other m arguments, there is a decreasing function g epsilon C[(alpha, infinity), (alpha, infinity)] such that g(g(x)) = x, x epsilon (alpha, infinity), [GRAPHICS] x epsilon (alpha, infinity), lim(x ->alpha+g)(x) = +infinity, and lim(x ->+infinity)g(x) = alpha. It is proved that if all p(i), i epsilon {1 ,..., k}, are even and all q(j), j. {1 ,..., m} are odd, every positive solution of the equation converges to (not necessarily prime) a periodic solution of period two, otherwise, every positive solution of the equation converges to a unique positive equilibrium. Copyright (c) 2007.