On a Class of Difference Equations System of Fifth-Order

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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