On a Class of Difference Equations System of Fifth-Order

Year 2024, Volume: 7 Issue: 3, 186 - 202, 30.09.2024

Merve Kara , Yasin Yazlik

https://doi.org/10.33401/fujma.1492703

Abstract

In the current paper, we investigate the following new class of system of difference equations \begin{align} u_{n+1}=&f^{-1}\left( g\left( v_{n-1}\right) \frac{A_{1}f\left( u_{n-2}\right)+B_{1}g\left( v_{n-4}\right) }{C_{1}f\left( u_{n-2}\right)+D_{1}g\left( v_{n-4}\right)}\right), \nonumber \\ v_{n+1}=&g^{-1}\left( f\left( u_{n-1}\right) \frac{A_{2}g\left( v_{n-2}\right)+B_{2}f\left( u_{n-4}\right) }{C_{2}g\left( v_{n-2}\right)+D_{2}f\left( u_{n-4}\right)}\right) ,\ n\in \mathbb{N}_{0}, \nonumber \end{align} where the initial conditions $u_{-p}$, $v_{-p}$, for $p=\overline{0,4}$ are real numbers, the parameters $A_{r}$, $B_{r}$, $C_{r}$, $D_{r}$, for $r\in\{1,2\}$ are real numbers, $A_{r}^{2}+B_{r}^{2}\neq 0\neq C_{r}^{2}+D_{r}^{2}$, for $r\in\{1,2\}$, $f$ and $g$ are continuous and strictly monotone functions, $f\left( \mathbb{R}\right) =\mathbb{R}$, $g\left( \mathbb{R}\right) =\mathbb{R}$, $f\left( 0\right) =0$, $g\left( 0\right) =0$. In addition, we solve aforementioned general two dimensional system of difference equations of fifth-order in explicit form. Moreover, we obtain the solutions of mentioned system according to whether the parameters being zeros or not. Finally, we present an interesting application.

Keywords

Difference equations systems, Solution, Stability

References

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