On a class of concrete general system of difference equations

Abstract

It is demonstrated that the following class of difference equations system
\begin{align*}
\begin{cases}
u_{n+1}=\Phi^{-1}\left( \Psi\left( v_{n}\right) \frac{a_{1}\Phi\left( u_{n}\right)+b_{1}\Psi\left( v_{n-1}\right) }{c_{1}\Phi\left( u_{n}\right)+d_{1}\Psi\left( v_{n-1}\right)}\right), & \\
v_{n+1}=\Psi^{-1}\left( \Phi\left( u_{n}\right) \frac{a_{2}\Psi\left( v_{n}\right)+b_{2}\Phi\left( u_{n-1}\right) }{c_{2}\Psi\left( v_{n}\right)+d_{2}\Phi\left( u_{n-1}\right)}\right) ,&
\end{cases} n\in \mathbb{N}_{0},
\end{align*}
where the initial conditions $u_{-i}$, $v_{-i}$, for $i\in\{0,1\},$ are real numbers, the parameters $c_{j}^2+d_{j}^2\neq0$, $a_{j}$, $b_{j}$, $c_{j}$, $d_{j}$, for $j\in\{1,2\},$ are real numbers, $\Phi$ and $\Psi$ are continuous and strictly monotone functions such that $\Phi\left( \mathbb{R}\right) =\mathbb{R}$, $\Psi\left( \mathbb{R}\right) =\mathbb{R}$, $\Phi\left( 0\right) =0$, $\Psi\left( 0\right) =0$, can be solved in all cases.