On two nonlinear difference equations.(English)Zbl 1380.39007
Summary: The behavior of solutions of the following nonlinear difference equations \[x_{n+1}=\frac{q}{p+x^\nu_n}\text{ and }y_{n+1}\frac{q}{-q+y^\nu_n},\] with real nonzero initial conditions \(x_0\) and \(y_0\), where \(p,q\in\mathbb R^+\) and \(\nu\in\mathbb N\), is studied. The solution forms of these two equations when \(\nu=1\) are expressed in terms of Horadam numbers. Meanwhile, the behavior of their solutions is investigated for all integers \(\nu>1\) and several numerical examples are presented to illustrate the results exhibited. The present work generalizes those seen in [T. T. Durhasan et al., “On the solutions of two special types of Riccati difference equation via Fibonacci numbers”, Adv. Difference Equ. 2013, No. 1, Paper No. 174, 7 p. (2013;doi:10.1186/1687-1847-2013-174)].
MSC:
| 39A10 | Additive difference equations |
| 11B39 | Fibonacci and Lucas numbers and polynomials and generalizations |
