A global convergence result for a higher order difference equation

Let f (z(1),..., z(k)) is an element of C(I-k, I) be a given function, where I is (bounded or unbounded) subinterval of R, and k is an element of N. Assume that f (y(1), y(2),..., y(k)) >= f ( y(2),..., y(k), y(1)) if y(1) >= max {y(2),...,yk}, f (y(1), y(2),..., y(k)) <= f (y(2),..., y(k), y(1)) if y(1) <= min {y(2),..., y(k)}, 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, x(n) = f (x(n-1),..., x(n-k)), n = 0, 1, 2,..., with initial values x(-k),..., x(-1) is an element of I, is convergent, and every unbounded solution tends either to +infinity or to -infinity. Copyright (c) 2007 Bratislav D. Iricanin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.