A Global Convergence Result for a Higher Order Difference Equation

Letf(z1,…,zk)∈C(Ik,I)be a given function, whereIis (bounded or unbounded) subinterval ofℝ, andk∈ℕ. Assume thatf(y1,y2,…,yk)≥f(y2,…,yk,y1)ify1≥max{y2,…,yk},f(y1,y2,…,yk)≤f(y2,…,yk,y1)ify1≤min{y2,…,yk}, andfis non- decreasing in the last variablezk. We then prove that every bounded solution of an autonomous difference equation of orderk, namely,xn=f(xn−1,…,xn−k),n=0,1,2,…,with initial valuesx−k,…,x−1∈I, is convergent, and every unbounded solution tends either to+∞or to−∞.