**Journal of Prime Research in Mathematics**

Vol. 20 (2024), Issue 1, pp. 65 – 80

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

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

# Formulas of the solutions of a solvable system of nonlinear difference equations

**Hamida Hamioud\(^a\), Nouressadat Touafek\(^{a,*}\), Imane Dekkar\(^a\), Mohammed B. Almatrafi\(^b\)**

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

\(^b\)Department of Mathematics, College of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia.

Correspondence should be addressed to: hamida.hamioud2018@gmail.com, ntouafek@gmail.com, imane_dek@hotmail.fr, mmutrafi@taibahu.edu.sa

Copyright © 2024 Hamida Hamioud, Nouressadat Touafek, Imane Dekkar, Mohammed B. Almatrafi . 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: 19 June 2024; Accepted: 13 August 2024; Published Online: 22 August 2024.

### Abstract

Consider the following general three dimensional system of difference equations

\begin{cases}

x_{n+1}=f^{-1}\left(\frac{g(y_{n} )g(y_{n-1} )(f(x_{n-1} ))^{p}}{f(x_{n} )[a_{n} (g(y_{n-2} ))^{q} +b_{n} g(y_{n} )g(y_{n-1} )]}\right),\\

y_{n+1}=g^{-1} \ \left(\frac{h(z_{n} )h(z_{n−1} )(g(y_{n−1} ))^{q}}{g(y_{n} )[c_{n} (h(z_{n−2} ))^{r} +d_{n} h(z_{n} )h(z_{n−1} )]}\right),\\

\ z_{n+1} =h^{−1} \ \ \left(\frac{f(x_{n} )f(x_{n−1} )(h(z_{n−1} ))^{r}}{h(z_{n} )[s_{n} (f(x_{n−2} ))^{p} +t_{n} f(x_{n} )f(x_{n−1} )]}\right),\end{cases}

\begin{cases}

x_{n+1}=f^{-1}\left(\frac{g(y_{n} )g(y_{n-1} )(f(x_{n-1} ))^{p}}{f(x_{n} )[a_{n} (g(y_{n-2} ))^{q} +b_{n} g(y_{n} )g(y_{n-1} )]}\right),\\

y_{n+1}=g^{-1} \ \left(\frac{h(z_{n} )h(z_{n−1} )(g(y_{n−1} ))^{q}}{g(y_{n} )[c_{n} (h(z_{n−2} ))^{r} +d_{n} h(z_{n} )h(z_{n−1} )]}\right),\\

\ z_{n+1} =h^{−1} \ \ \left(\frac{f(x_{n} )f(x_{n−1} )(h(z_{n−1} ))^{r}}{h(z_{n} )[s_{n} (f(x_{n−2} ))^{p} +t_{n} f(x_{n} )f(x_{n−1} )]}\right),\end{cases}

where 𝑛∈ℕ0,𝑝,𝑞,𝑟∈ℕ,𝑓,𝑔,ℎ :D→ℝ are continuous one-to-one functions on D⊆ℝ, the coefficients (𝑎𝑛)𝑛∈ℕ0,(𝑏𝑛)𝑛∈ℕ0(𝑐𝑛)𝑛∈ℕ0,(𝑑𝑛)𝑛∈ℕ0,(𝑠𝑛)𝑛∈ℕ0,(𝑡𝑛)𝑛∈ℕ0 are non-zero real numbers and the initial values 𝑥−𝑖,𝑦−𝑖,𝑧−𝑖,𝑖=0,1,2, are real numbers. We will give explicit formulas for well-defined solutions of the aforementioned system in both variable and constant cases of the coefficients. As an application, we will deduce the formulas of the solutions of the particular system obtained from the general one by taking 𝑓(𝑥)=𝑔(𝑥)=ℎ(𝑥)=𝑥.

#### Keywords:

System of difference equations, Well-defined solutions, Closed-form of the solutions