Global stability and asymptotics of some classes of rational difference equations (Q819690)

The author proves that the equilibrium solution \(\bar{x}=1\) is globally asymptotically stable for the difference equations \[ x_{n+3}=\frac{x_{n+j}+x_{n+i}~x_{n+k}+a}{x_{n+i}+x_{n+j}~x_{n+k}+a},\quad n=0,1,2,\dots \] where the initial values \(x_{-2},x_{-1},x_0\) are positive, the parameter \(a\) is nonnegative, \(i,j\in\{0,1,2\}\) but different from each other, and \(k=3-i-j\). In his proof he utilizes a global convergence result due to \textit{N. Kruse and T. Nesemann} [J. Math. Anal. Appl. 235, 151--158 (1999; Zbl 0933.37016)]. In addition, using an inclusion theorem due to \textit{L. Berg} [J. Difference Equ. Appl. 10, 399--408 (2004; Zbl 1056.39003)], he finds asymptotics of some solutions of the above difference equations.