Solvability of a Three-Dimensional System of Nonlinear Difference Equations

Kara, Merve

doi:10.36753/mathenot.992987

Mathematical Sciences and Applications E-Notes

2022

DOI: 10.36753/mathenot.992987

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Solvability of a Three-Dimensional System of Nonlinear Difference Equations

Merve Kara

Abstract: In this paper, we solve the following three-dimensional system of difference equationswhere the sequencesand the initial values x −j , y −j , j = 1, 5, are real numbers. In addition, the constant coefficients of the mentioned system is solved in closed form. Finally, we also describe the forbidden set of solutions of the system of difference equations.

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Cited by 3 publications

References 28 publications

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On solutions of three-dimensional system of difference equations with constant coefficients

Kara

Aktaş

2023

Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat.

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In this study, we show that the system of difference equations \begin{align} x_{n}=\frac{x_{n-2}y_{n-3}}{y_{n-1}\left(a+bx_{n-2}y_{n-3} \right) }, \nonumber \\ y_{n}=\frac{y_{n-2}z_{n-3}}{z_{n-1}\left(c+dy_{n-2}z_{n-3} \right) },~n\in\mathbb{N}_{0}, ~ \nonumber \\ z_{n}=\frac{z_{n-2}x_{n-3}}{x_{n-1}\left(e+fz_{n-2}x_{n-3} \right) }, \nonumber \\ \end{align} where the initial values $x_{-i}, y_{-i}, z_{-i}$, $i=\overline{1,3}$ and the parameters $a$, $b$, $c$, $d$, $e$, $f$ are non-zero real numbers, can be solved in closed form. Moreover, we obtain the solutions of above system in explicit form according to the parameters $a$, $c$ and $e$ are equal $1$ or not equal $1$. In addition, we get periodic solutions of aforementioned system. Finally, we define the forbidden set of the initial conditions by using the acquired formulas.

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On solutions of three-dimensional system of difference equations with constant coefficients

Kara

Aktaş

2023

Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat.

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On a Class of Difference Equations System of Fifth-Order

Kara,

Yazlik

2024

Fundamental Journal of Mathematics and Applications

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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.

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On Some Solvable Systems of Some Rational Difference Equations of Third Order

Al-Basyouni

Elsayed

2023

Mathematics

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Our aim in this paper is to obtain formulas for solutions of rational difference equations such as xn+1=1±xn−1yn/1−yn,yn+1=1±yn−1xn/1−xn, and xn+1=1±xn−1yn−2/1−yn,yn+1=1±yn−1xn−2/1−xn, where the initial conditions x−2, x−1, x0, y−2, y−1, y0 are non-zero real numbers. In addition, we show that the some of these systems are periodic with different periods. We also verify our theoretical outcomes at the end with some numerical applications and draw it by using some mathematical programs to illustrate the results.

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Solvability of a Three-Dimensional System of Nonlinear Difference Equations

Cited by 3 publications

References 28 publications

On solutions of three-dimensional system of difference equations with constant coefficients

On solutions of three-dimensional system of difference equations with constant coefficients

On a Class of Difference Equations System of Fifth-Order

On Some Solvable Systems of Some Rational Difference Equations of Third Order

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