Global dynamics of a non-autonomous rational difference equation (Q2927985)

The author investigates the boundedness of all positive solutions and the global stability of the period-two solutions of the non-autonomous second-order rational difference equation NEWLINE\[NEWLINE x_{n+1}=p_n +\frac{x_n}{x_{n-1}},\quad n=0,1,2,\dots, NEWLINE\]NEWLINE where \(\{p_n\}\) is a period-two sequence of positive real numbers and the initial conditions \(x_{-1}\), \(x_0\) are positive real numbers.NEWLINENEWLINEAlthough the results established are valid, it appears to me that the author's work is incomplete. Furthermore, some of the conditions imposed are too restrictive. For example, to prove the boundedness, the author assumes \(p_n>1\) for all \(n\). Yet, as can be seen below, boundedness can be established for \(p_n>0\) for all \(n\).NEWLINENEWLINELet \(p_{2n}=\alpha\), \(p_{2n+1}=\beta\), \(u_n=x_{2n-1}\), and \(v_n=x_{2n}\). Then NEWLINE\[NEWLINE\begin{aligned} u_{n+1}&= \alpha +\frac{v_n}{u_n}\\ v_{n+1}&=\beta+\frac{u_{n+1}}{v_n}=\beta+\frac{\alpha +\frac{v_n}{u_n}}{v_n}=\beta+\frac{\alpha}{v_n}+\frac{1}{u_n}. \end{aligned}NEWLINE\]NEWLINE Therefore, \(u_{n+1}>\alpha\) and \(v_{n+1}>\beta.\) Furthermore, NEWLINE\[NEWLINE v_{n+2}=\beta+\frac{\alpha}{v_{n+1}}+\frac{1}{u_{n+1}}<\beta+\frac{\alpha}{\beta}+\frac{1}{\alpha}, NEWLINE\]NEWLINE and NEWLINE\[NEWLINE u_{n+3}=\alpha +\frac{v_{n+2}}{u_{n+2}} <\alpha + \frac{\beta+\frac{\alpha}{\beta}+\frac{1}{\alpha}}{\alpha}=\alpha+\frac{\beta}{\alpha}+\frac{1}{\alpha}+\frac{1}{\alpha^2}. NEWLINE\]