On the behaviour of solutions of the difference equation \(x_{n+1}=\alpha+ \frac{x_{n-2}}{x_{n-1}} + \frac{x_{n-1}}{x_n}\) (Q2905835)

The authors consider the third order nonlinear difference equation NEWLINE\[NEWLINE x_{n+1} = \alpha + {{x_{n-2}}\over{x_{n-1}}} + {{x_{n-1}}\over{x_n}} NEWLINE\]NEWLINE with \(\alpha\geq 0\), \(x_{-2},\;x_{-1},\;x_0\in(0,\infty)\). By setting \(y_{n-1}= x_{n-1}/x_n\) the following difference equation is associated NEWLINE\[NEWLINE \displaystyle{y_{n+1} = {{\alpha+y_{n-1}+y_{n-1}}\over{\alpha+y_n+y_{n-1}}}} NEWLINE\]NEWLINE with the unique equilibrium \(\bar{y}=1\). By setting NEWLINE\[NEWLINE z_{n+1} = {{1}\over{\alpha}}(\alpha/2+y_{n+1}) NEWLINE\]NEWLINE there is associated the normalized equation NEWLINE\[NEWLINE \displaystyle{z_{n+1} = {{z_n+\gamma z_{n-1} + \delta z_{n-2}}\over {2(z_n+cz_{n-1})}}}. NEWLINE\]NEWLINE The paper considers various qualitative behavior of the solutions of these equations (stability, monotonicity, boundedness).