On Two Nonlinear Difference Equations

Abstract

The behavior of solutions of the following nonlinear difference equations \[ x_{n+1}=\displaystyle\frac{q}{p+x_n^{\nu}} \quad \text{and} \quad y_{n+1}=\displaystyle\frac{q}{-p+y_n^{\nu}}, \] where $p, q \in\mathbb{R}^+$ and $\nu\in \mathbb{N}$ are studied. The solution form of these two equations when $\nu =1$ are expressed in terms of Horadam numbers. Furthermore, the behavior of their solutions are investigated for all integer $\nu > 0$ and several numerical examples are presented to illustrate the results exhibited. The present work generalizes those seen in [{\it Adv. Differ. Equ.}, {\bf 2013}:174 (2013), 7 pages].