On the rule of trajectory structure for a third-order nonlinear difference equation using semi-cycle analysis method

The paper is concerned with the rule of trajectory structure and global asymptotical stability of the positive equilibrium for the following third-order nonlinear difference equation \(\begin{aligned} w_{m+1}=\frac{w_{m-1}^{\alpha }w_{m-2}+1+c}{w_{m-1}^{\alpha }+w_{m-2}+c},\ \ \ m\in N, \end{aligned}\) $$\begin{array}{c}\hfill w_{m+1}=\frac{w_{m-1}^{\alpha }{w}_{m-2}+1+c}{w_{m-1}^{\alpha }+w_{m-2}+c},m\in N,\end{array}$$ where \(\alpha \in [0,1], c\in [0,\infty )\) $\alpha \in [0,1],c\in [0,\infty )$ , the initial values \(w_{j}\in (0,\infty ), j=0,-1,-2.\) $w_j\in (0,\infty ),j=0,-1,-2.$ By virtue of semi-cycle analysis method, the rules of trajectory structure are tested in prime period 7. The continuous length of positive and negative semi-cycles of any nontrivial solution appears periodically and the rule is \(3^{-},2^{+},1^{-},1^{+}\) $3^{-},2^{+},1^{-},1^{+}$ . Also, the positive equilibrium is globally asymptotically stable. Finally, two examples are given to show effectiveness of our theoretic analysis.