Behaviors of the solutions via their closed-form formulas for two rational third-order difference equations (Q2065433)

Summary: In this work, we derive the solution formulas and study their behaviors for the difference equations \(x_{n + 1}=(\alpha x_n x_{n - 3} / (- \beta x_{n - 3} + \gamma x_{n - 2}))\), \(n\in \mathbb{N}_0\) and \(x_{n + 1}=(\alpha x_n x_{n - 3} / (\beta x_{n - 3} - \gamma x_{n - 2}))\), \(n\in \mathbb{N}_0\) with real initials and positive parameters. We show that there exist periodic solutions for the second equation under certain conditions when \(\beta^2<4\alpha\gamma\). Finally, we give some illustrative examples.