Global asymptotic stability for a fourth-order rational difference equation (Q1040147)

Our aim is to investigate the global behavior of the following fourth-order rational difference equation: \[ x_{n+1}=\frac{x_nx_{n-2}x_{n-3}+x_n+x_{n-2}+x_{n-3}+a}{x_nx_{n-2}+x_nx_{n-3}+x_{n-2}x_{n-3}+1+a},\quad n=0,1,2,\dots \] where \(a\in[0,\infty)\) and the initial values \(x_{-3},x_{-2},x_{-1},x_0\in (0,\infty)\). To verify that the positive equilibrium point of the equation is globally asymptotically stable, we use the rule of the successive lengths of positive and negative semicycles of nontrivial solutions of the aforementioned equation.