Solvability of a Three-Dimensional System of Nonlinear Difference Equations

Year 2022, Volume: 10 Issue: 1, 1 - 15, 01.03.2022

Merve Kara

https://doi.org/10.36753/mathenot.992987

Cited By: 2

Abstract

In this paper, we solve the following three-dimensional system of difference equations

$$\begin{array}{c}{x}_n\end{array}$$

=yn−4zn−5yn−1(an+bnzn−2xn−3yn−4zn−5)

,

yn

=zn−4xn−5zn−1(αn+βnxn−2yn−3zn−4xn−5)

,

zn

=xn−4yn−5xn−1(An+Bnyn−2zn−3xn−4yn−5)

, n∈N0,

xn=yn−4zn−5yn−1(an+bnzn−2xn−3yn−4zn−5),yn=zn−4xn−5zn−1(αn+βnxn−2yn−3zn−4xn−5),zn=xn−4yn−5xn−1(An+Bnyn−2zn−3xn−4yn−5), n∈N0,

where the sequences ${\left(a_n\right)}_{n\in N_0}$, ${\left(b_n\right)}_{n\in N_0}$, ${\left({\alpha }_n\right)}_{n\in N_0}$, ${\left({\beta }_n\right)}_{n\in N_0}$, ${\left(A_n\right)}_{n\in N_0}$, ${\left(B_n\right)}_{n\in N_0}$ and the initial values $x_{-j},y_{-j}$, $j=\overline{1,5}$, are real numbers. In addition, the constant coefficient of the mentioned system is solved in closed form. Finally, we also describe the forbidden set of solutions of the system of difference equations.

Keywords

system of difference equations, closed-form, forbidden set

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