- Konuralp Journal of Mathematics
- Vol: 8 Issue: 1
- Periodic Solutions for Some Systems of Difference Equations
Periodic Solutions for Some Systems of Difference Equations
Authors : Ibrahim Yalçinkaya, Hamdy El-metwally, Alaa E. Hamza
Pages : 114-121
View : 17 | Download : 5
Publication Date : 2020-04-15
Article Type : Research
Abstract :We will show in this paper that all solutions for the systems $ \varkappa _{n+1}^{(1)}=\frac{\varkappa _{n}^{(2)}}{\alpha \varkappa _{n}^{(2)}-1},\varkappa _{n+1}^{(2)}=\frac{\varkappa _{n}^{(3)}}{\alpha \varkappa _{n}^{(3)}-1},...,\varkappa _{n+1}^{(\kappa )}=\frac{\varkappa _{n}^{(1)}}{\alpha \varkappa _{n}^{(1)}-1},$ and $ \varkappa _{n+1}^{(1)}=\frac{\varkappa _{n}^{(\kappa )}}{\alpha \varkappa _{n}^{(\kappa )}-1},\varkappa _{n+1}^{(2)}=\frac{\varkappa _{n}^{(1)}}{ \alpha \varkappa _{n}^{(1)}-1},...,\varkappa _{n+1}^{(\kappa )}=\frac{ \varkappa _{n}^{(\kappa -1)}}{\alpha \varkappa _{n}^{(\kappa -1)}-1}, $ are periodic with period $p$ where $p$ is given by$p=\left\{ \begin{array}{c} \kappa \text{ \ \ \ \ \ \ \ \ \ \ \ \ \ if \ \ \ \ \ \ \ \ }\kappa =0(mod2), \\ 2\kappa \text{ \ \ \ \ \ \ \ \ \ \ \ if \ \ \ \ \ \ \ \ }\kappa \neq 0(mod2), \end{array} \right\} $ where $\alpha $ and $\varkappa _{0}^{(1)},\varkappa _{0}^{(2)},...,\varkappa _{0}^{(\kappa )}$ are nonzero real numbers with $\varkappa _{0}^{(i)}\neq \frac{1}{\alpha },~i=1,2,...,\kappa $, for some $\kappa \in \mathbb{N}$.Keywords : Asymptotic behavior, difference equations, periodicity, solutions, system