A Global Convergence Result for a Higher Order Difference Equation

Abstract

Let f(z1,…,zk)∈C(Ik,I) be a given function, where I is (bounded or unbounded) subinterval of ℝ, and k∈ℕ. Assume that f(y1,y2,…,yk)≥f(y2,…,yk,y1) if y1≥max{y2, …,yk}, f(y1,y2,…,yk)≤f(y2,…,yk,y1) if y1≤min{y2,…,yk}, and f is non- decreasing in the last variable zk. We then prove that every bounded solution of an autonomous difference equation of order k, namely, xn=f(xn−1,…,xn−k), n=0,1,2,…, with initial values x−k,…,x−1∈I, is convergent, and every unbounded solution tends either to +∞ or to −∞.