Global Asymptotic Stability for Linear Fractional Difference Equation

Consider the difference equation xn+1=(α+∑i=0kaixn-i)/(β+∑i=0kbixn-i),  n=0,1,…, where all parameters α,β,ai,bi,  i=0,1,…,k, and the initial conditions xi,  i∈{-k,…,0} are nonnegative real numbers. We investigate the asymptotic behavior of the solutions of the considered equation. We give easy-to-check conditions for the global stability and global asymptotic stability of the zero or positive equilibrium of this equation