Global asymptotic behavior and boundedness of positive solutions to an odd-order rational difference equation

In this note we consider the following high-order rational difference equation xn=1+∏i=1k(1−xn−i)∑i=1kxn−i,n=0,1,…, where k≥3k≥3 is odd number, x−k,x−k+1,x−k+2,…,x−1x−k,x−k+1,x−k+2,…,x−1 is positive numbers. We obtain the boundedness of positive solutions for the above equation, and with the perturbation of initial values, we mainly use the transformation method to prove that the positive equilibrium point of this equation is globally asymptotically stable.