Periodic Solutions for Some Systems of Difference Equations

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

References

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