On the global behavior of the nonlinear difference equationxn+1=f(pn,xn−m,xn−t(k+1)+1)

Abstract

We consider the following nonlinear difference equation:xn+1=f(pn,xn−m,xn−t(k+1)+1),n=0,1,2,…, wherem∈{0,1,2,…}andk,t∈{1,2,…}with0≤m<t(k+1)−1, the initial valuesx−t(k+1)+1,x−t(k+1)+2,…,x0∈(0,+∞), and{pn}n=0∞is a positive sequence of the periodk+1. We give sufficient conditions under which every positive solution of this equation tends to the periodk+1solution.