Qualitative properties for a fourth-order rational difference equation (VII) (Q2908907)

The dynamical behavior of the fourth-order rational difference equation NEWLINE\[NEWLINE x_{n+1}=\frac{x_{n}x_{n}-2x_{n-3}^{b}+x_{n}+x_{n-2}+x_{n-3}^{b}+a}{ x_{n}x_{n-2}+x_{n-2}x_{n-3}^{b}+x_{n}x_{n-3}^{b}+1+a}\;\;(n=0,1,2,\dots) \tag{1}NEWLINE\]NEWLINE where \(a,b\in \left[ 0,\infty \right) \) and the initial values \( x_{-3},x_{-2},x_{-1},x_{0}\in \left( 0,\infty \right)\), is investigated with the methods of \textit{X. Li} [J. Math. Anal. Appl. 311, No. 1, 103--111 (2005; Zbl 1082.39004)] and of \textit{X. Li} and \textit{R. P. Agarwal} [J. Korean Math. Soc. 44, No. 4, 787--797 (2007; Zbl 1127.39018)]. It is found that the successive lengths of positive and negative semicycles of nontrivial solutions of the above equation occur periodically and can be expressed in the form: \(\dots,3^{-},3^{+},3^{-},3^{+},3^{-},3^{+},\dots\) or \( \dots,1^{+},1^{-},1^{+},1^{-},1^{+},1^{-},\dots\) or \( \dots,2^{-},1^{+},2^{-},1^{+},2^{-},1^{+},\dots\) or \( \dots,3^{+},3^{-},3^{+},3^{-},3^{+},3^{-},\dots\). When \(b=0\), (1) is trivial, and it is assumed that \(b>0\) in the sequel.NEWLINENEWLINEThe definitions of positive and negative semicycles and the length of a semicycle are given. Further, the properties of the solution are examined. As a main result,\ the ``rule for the trajectory structure'' of the nontrivial solution of (1) is given when \(\{x_{n}\}_{n=-3}^{\infty }\) is a strictly oscillatory solution of (1). Also, as a second main result, it is mentioned that the positive equilibrium of (1) is globally stable when \( a,b\in \left[ 0,\infty \right) \).