A Global Convergence Result for a Higher Order Difference Equation

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 nondecreasing 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 −∞.