SOLVABILITY OF A FOUR DIMENSIONAL SYSTEM OF DIFFERENCE EQUATIONS

Yıl 2024, , 62 - 74, 27.02.2024

Şule Devecioğlu , Merve Kara

https://doi.org/10.20290/estubtdb.1403701

Öz

In this study, we investigate the following four-dimensional difference equations system
{█(u_n=(αu_(n-3) t_(n-2)+β)/(γv_(n-1) t_(n-2) u_(n-3) ), @v_n=(αv_(n-3) u_(n-2)+β)/(γw_(n-1) u_(n-2) v_(n-3) ),n∈N_0,@w_n=(αw_(n-3) v_(n-2)+β)/(γt_(n-1) v_(n-2) w_(n-3) ), @t_n=(αt_(n-3) w_(n-2)+β)/(γu_(n-1) w_(n-2) t_(n-3) ), )┤
where the initial values u_(-d),v_(-d),w_(-d),t_(-d), d∈{1,2,3} and the parameters α,β,γ are real numbers. Then, we obtain the solutions of system of third-order difference equations in explicit form. In addition, the solutions according to some special cases of the parameters are examined. Finally, numerical examples are given to demonstrate the theoretical results.

Anahtar Kelimeler

Periodicity, System of difference equation, Solution

Kaynakça

• [1] Abo-Zeid R., Kamal H. Global behavior of two rational third order difference equations. Univers J Math Appl 2019; 2 (4): 212-217.
• [2] Abo-Zeid R. Behavior of solutions of a second order rational difference equation. Math Morav 2019; 23(1): 11-25.
• [3] Akrour Y, Kara M, Touafek N, Yazlik Y. Solutions formulas for some general systems of difference equations. Miskolc Math Notes 2021; 22(2): 529-555.
• [4] Elaydi S. An Introduction to Difference Equations. Springer, New York, 1996.
• [5] Elsayed EM. Solution for systems of difference equations of rational form of order two. Comput Appl Math 2014; 33(3): 751-765.
• [6] Elsayed EM. Expression and behavior of the solutions of some rational recursive sequences. Math Metheod Appl Sci 2016; 39(18): 5682-5694.
• [7] Halim Y, Touafek N, Yazlik Y. Dynamic behavior of a second-order non-linear rational difference equation. Turkish J Math 2015; 39(6): 1004-1018.
• [8] Halim Y, Rabago JFT. On the solutions of a second-order difference equation in terms of generalized Padovan sequences. Math Slovaca 2018; 68(3): 625-638.
• [9] Kara M, Yazlik Y. Solvability of a nonlinear three-dimensional system of difference equations with constant coefficients. Math Slovaca 2021; 71(5): 1133–1148.
• [10] Kara M, Yazlik Y. Solutions formulas for three-dimensional difference equations system with constant coefficients. Turk J Math Comput Sci 2022; 14(1): 107–116.
• [11] Stević S. Representation of solutions of bilinear difference equations in terms of generalized Fibonacci sequences. Electron J Qual Theory Differ Equ 2014; 67: 1–15.
• [12] Taskara N, Tollu DT, Yazlik Y. Solutions of rational difference system of order three in terms of Padovan numbers. J Adv Res Appl Math 2015; 7(3): 18–29.
• [13] Tollu DT, Yazlik Y, Taskara N. On a solvable nonlinear difference equation of higher order. Turkish J Math 2018; 42: 1765–1778.
• [14] Touafek N, Elsayed EM. On a second order rational systems of difference equations. Hokkaido Math J 2015; 44: 29–45.
• [15] Yazlik Y, Tollu DT, Taskara N. On the solutions of difference equation systems with Padovan numbers. Appl. Math 2013; 4: 15-20.