On Two Nonlinear Difference Equations

Abstract

The behavior of solutions of the following nonlinear difference equations \[ x_{n+1}=\displaystyle\frac{q}{p+x_n^ν} \quad \text{and} \quad y_{n+1}=\displaystyle\frac{q}{-p+y_n^ν}, \] 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].

The first version of the paper has been drafted on March 24, 2014