Periodic Solutions for Some Systems of Difference Equations

Authors : İbrahim YALÇINKAYA, Hamdy ELMETWALLY, Alaa E HAMZA

Pages : 114-121

View : 25 | Download : 16

Publication Date : 2020-04-15

Article Type : Research Paper

Abstract :We will show in this paper that all solutions for the systems $ \varkappa _{n+1}^{insert ignore into journalissuearticles values(1);}=\frac{\varkappa _{n}^{insert ignore into journalissuearticles values(2);}}{\alpha \varkappa _{n}^{insert ignore into journalissuearticles values(2);}-1},\varkappa _{n+1}^{insert ignore into journalissuearticles values(2);}=\frac{\varkappa _{n}^{insert ignore into journalissuearticles values(3);}}{\alpha \varkappa _{n}^{insert ignore into journalissuearticles values(3);}-1},...,\varkappa _{n+1}^{insert ignore into journalissuearticles values(\kappa );}=\frac{\varkappa _{n}^{insert ignore into journalissuearticles values(1);}}{\alpha \varkappa _{n}^{insert ignore into journalissuearticles values(1);}-1},$ and $ \varkappa _{n+1}^{insert ignore into journalissuearticles values(1);}=\frac{\varkappa _{n}^{insert ignore into journalissuearticles values(\kappa );}}{\alpha \varkappa _{n}^{insert ignore into journalissuearticles values(\kappa );}-1},\varkappa _{n+1}^{insert ignore into journalissuearticles values(2);}=\frac{\varkappa _{n}^{insert ignore into journalissuearticles values(1);}}{ \alpha \varkappa _{n}^{insert ignore into journalissuearticles values(1);}-1},...,\varkappa _{n+1}^{insert ignore into journalissuearticles values(\kappa );}=\frac{ \varkappa _{n}^{insert ignore into journalissuearticles values(\kappa -1);}}{\alpha \varkappa _{n}^{insert ignore into journalissuearticles values(\kappa -1);}-1}, $ are periodic with period $p$ where $p$ is given by$p=\left\{ \begin{array}{c} \kappa \text{ \ \ \ \ \ \ \ \ \ \ \ \ \ if \ \ \ \ \ \ \ \ }\kappa =0insert ignore into journalissuearticles values(mod2);, \\ 2\kappa \text{ \ \ \ \ \ \ \ \ \ \ \ if \ \ \ \ \ \ \ \ }\kappa \neq 0insert ignore into journalissuearticles values(mod2);, \end{array} \right\} $ where $\alpha $ and $\varkappa _{0}^{insert ignore into journalissuearticles values(1);},\varkappa _{0}^{insert ignore into journalissuearticles values(2);},...,\varkappa _{0}^{insert ignore into journalissuearticles values(\kappa );}$ are nonzero real numbers with $\varkappa _{0}^{insert ignore into journalissuearticles values(i);}\neq \frac{1}{\alpha },~i=1,2,...,\kappa $, for some $\kappa \in \mathbb{N}$.
Keywords : Asymptotic behavior, difference equations, periodicity, solutions, system