Global Behavior of xn+1=(α+βxn-k)/(γ+xn) - Chenquan Gan (Discrete Dynamics in Nature and Society, 2013)
This paper aims to investigate the global stability of negative solutions of the difference equation xn+1=(α+βxn-k)/(γ+xn), n=0,1,2,…, where the initial conditions x-k,…,x0∈-∞,0, k is a positive integer, and the parameters β,  γ<0,  α>0. By utilizing the invariant interval and periodic character of solutions, it is found that the unique negative equilibrium is globally asymptotically stable under some parameter conditions. Additionally, two examples are given to illustrate the main results in the end.