The rule of cycle length and global asymptotic stability for a third-order nonlinear difference equation (Q1042622)

The paper deals with third-order nonlinear difference equation \[ x_{n+1} = \frac{x_{n-1}^b x_{n-2} + 1}{x_{n-1}^b + x_{n-2}},\quad n = 0,1,\dots, \tag{1} \] where \(b \in [0,1]\) and the initial values are nonnegative. A solution \(\{x_n\}\) of (1) is said to be eventually trivial if \(x_n\) is eventually equal to \(\bar x = 1.\) A solution is said to be eventually positive (negative) if \(x_n\) is eventually higher (smaller) then \(\bar x = 1.\) The authors prove that the positive equilibrium \(\bar x = 1\) of (1) is globally asymptotically stable and every positive solution of (1) is either eventually trivial or non-oscillatory with respect to \(\bar x = 1\) and eventually positive or strictly oscillatory with the lengths of positive and negative semi-cycles periodically successively occurring with prime period \(7\) and the rule to be \(3^-, 2^+, 1^-, 1^+\) in a period.