**Journal of Prime Research in Mathematics**

Vol. 19 (2023), Issue 2, pp. 37 – 59

ISSN: 1817-3462E (Online) 1818-5495 (Print)

ISSN: 1817-3462E (Online) 1818-5495 (Print)

# A solvable three dimensional system of difference equations of second order with arbitrary powers

**M. C. Benkara Mostefa\(^{a,*}\), A. C¸ete\(^b\), N. Touafek\(^c\), Y. Yazlik\(^d\)**

\(^a\)Department of Mathematics, Faculty of Exact Sciences, University of Mentouri Constantine 1, Constantine, Algeria

\(^b\)Nevsehir Haci Bektas Veli University, Faculty of Science and Art, Department of Mathematics, Nevsehir, Turkey

\(^c\)LMAM Laboratory, Faculty of Exact Sciences and Informatics, University of Jijel, 18000 Jijel, Algeria.

\(^c\)Nevsehir Haci Bektas Veli University, Faculty of Science and Art, Department of Mathematics, Nevsehir, Turkey

Correspondence should be addressed to: che.benkara@gmail.com, ayferdoganc@gmail.com,ntouafek@gmail.com, yyazlik@gmail.com

Copyright © 2023 M. C. Benkara Mostefa, A. C¸ete, N. Touafek, Y. Yazlik . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Published:**Received: 10 May 2023; Accepted: 28 November 2023; Published Online: 07 December 2023.

### Abstract

The solvability in a closed form of the following three-dimensional system of difference equations of second order with arbitrary powers

$$x_{n+1}=\frac{y_{n}y_{n-1}^{q}}{x_{n}^{p}(a+by_{n}y_{n-1}^{q})},\,y_{n+1}=\frac{z_{n}z_{n-1}^{r}}{y_{n}^{q}(c+dz_{n}z_{n-1}^{r})}

,\,z_{n+1}=\frac{x_{n}x_{n-1}^{p}}{z_{n}^{r}(h+kx_{n}x_{n-1}^{p})},\,n,\,p,\,q,\,r\in\mathbb{N}_{0}$$

where the initial values x

$$x_{n+1}=\frac{y_{n}y_{n-1}^{q}}{x_{n}^{p}(a+by_{n}y_{n-1}^{q})},\,y_{n+1}=\frac{z_{n}z_{n-1}^{r}}{y_{n}^{q}(c+dz_{n}z_{n-1}^{r})}

,\,z_{n+1}=\frac{x_{n}x_{n-1}^{p}}{z_{n}^{r}(h+kx_{n}x_{n-1}^{p})},\,n,\,p,\,q,\,r\in\mathbb{N}_{0}$$

where the initial values x

_{_i}, y_{_i}, z_{_i}, i=0,1 are non-zero real numbers and the parameters a,b,c,d,h, are real numbers, will be the subject of the present work. We will also provide the behavior of the solutions of some particular cases of our system.#### Keywords:

Systems of difference equations, form of the solutions, limiting behavior of the solutions,periodic solutions.