Behaviors of the Solutions via Their Closed-Form Formulas for Two Rational Third-Order Difference Equations

In this work, we derive the solution formulas and study their behaviors for the difference equations x n + 1 = α x n x n − 3 / − β x n − 3 + γ x n − 2 , n ∈ ℕ 0 and x n + 1 = α x n x n − 3 / β x n − 3 − γ x n − 2 , n ∈ ℕ 0 with real initials and positive parameters. We show that there exist periodic solutions for the second equation under certain conditions when β 2 < 4 α γ . Finally, we give some illustrative examples.