Thompson’s metric and global stability of difference equations

Abstract

We apply the Thompson’s metric to study the global stability of the equilibium of the following difference equation $$ y_{n} = \frac{f_{2m+1}^{2m+1} (y_{n-k_{1}}^r, y_{n-k_{2}}^r, \dots, y_{n-k_{2m+1}}^r)}{f_{2m}^{2m+1} (y_{n-k_{1}}^r, y_{n-k_{2}}^r, \dots, y_{n-k_{2m+1}}^r)}, \;\;\;\; n = 0,1,2, \ldots, $$ where \({f_{2m+1}^{2m+1}}\) and \({f_{2m}^{2m+1}}\) are polynomials in 2m + 1 variables which satisfy certain conditions. Our results include Stevic’s (Appl Math Comput 216:179–186, 2010), Zhang et al.’s (Bull Aust Math Soc 81:251–259, 2010) and other related results as special cases.