The Periodic Character of the Difference Equation xn+1=f(xn−l+1,xn−2k+1)

Abstract

In this paper, we consider the nonlinear difference equation xn+1=f(xn−l+1,xn−2k+1), n=0,1,…, where k,l∈{1,2,…} with 2k≠l and gcd(2k,l)=1 and the initial values x−α,x−α+1,…,x0∈(0,+∞) with α=max{l−1,2k−1}. We give sufficient conditions under which every positive solution of this equation converges to a ( not necessarily prime ) 2-periodic solution, which extends and includes corresponding results obtained in the recent literature.