Some results about the global attractivity of bounded solutions of difference equations with applications to periodic solutions

摘要

We obtain some results about the global attractivity of bounded solutions of difference equation Xn+1 = f (X-n,Xn-1,...,Xn-k), n = 0,1,... where f is non-increasing or non-decreasing in each argument and every point in I is an equilibrium point of above equation where I is an invariant interval for this equation. By our results we prove that when k is an odd positive integer and p >= 1 is a real number, every positive solution of Xn+1 = P=x(n-k)/1+x(n) , n = 0,1,.. converges to a period-two solution of this equation. We also apply our results to the rational difference equation Xn+1 = 1 + Xn-2k+1/x(n-2l) , n = 0,1,... where k,l is an element of {0,1,...}, and we show that every positive solution of this equation converges to a period-two solution of this equation. (c) 2005 Elsevier Ltd. All rights reserved.