Global stability of a rational difference equation

In this paper, we study the global stability of the difference equation xn+1=xna+a0xn+⋯+akxn-k,n=0,1,…, where the parameters a,ai∈(0,∞)a,ai∈(0,∞) for i=0,…,ki=0,…,k, x-k,…x-k,…, x-1∈[0,∞)x-1∈[0,∞) and x0∈(0,∞)x0∈(0,∞). We prove that the unique positive equilibrium is globally asymptotically stable if and only if it is locally asymptotically. Also we provide sufficient condition for it to be globally asymptotically stable and our results solve the open problem proposed by Kulenović and Ladas (Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, 2002).