Thompson's metric and global stability of difference equations (Q427266)

The author investigates the global stability of the equilibrium of the difference equation \[ y_n=\frac{f^{2m+1}_{2m+1}(y_{n-k_1}^r,y_{n-k_2}^r,\dots,y^r_{n-k_{2m+1}})} {f^{2m+1}_{2m}(y_{n-k_1}^r,y_{n-k_2}^r,\dots,y^r_{n-k_{2m+1}})}, \tag{*} \] where \(f^{2m+1}_{2m+1}\), \(f^{2m+1}_{2m}\) are polynomials of \(2n+1\) variables, \(k_1,\dots k_{2m+1}\) are integers and \(r\in (0,1]\). The main results of the paper present conditions on coefficients of these polynomial which guarantee that \(y=1\) is the equilibrium point of (*) and solutions of this equation satisfy \(\lim_{n\to\infty} y_n=1.\) The proofs are based on a convergence theorem in the Thompson metric space proved in the recent author's paper [J. Difference Equ. Appl. 17, No. 5, 737--752 (2011; Zbl 1244.39003)].