Communications of the
Korean Mathematical Society
CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

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Commun. Korean Math. Soc. 2020; 35(1): 301-319

Online first article September 19, 2019      Printed January 31, 2020

https://doi.org/10.4134/CKMS.c180472

Copyright © The Korean Mathematical Society.

A solvable system of difference equations

Necati Taskara, Durhasan T. Tollu, Nouressadat Touafek, Yasin Yazlik

Selcuk University; Necmettin Erbakan University; Mohamed Seddik Ben Yahia University; Nevsehir Haci Bektas Veli University

Abstract

In this paper, we show that the system of difference equations \begin{equation*} x_{n}=\frac{ay_{n-1}^{p}+b\left( x_{n-2}y_{n-1}\right) ^{p-1}}{ cy_{n-1}+dx_{n-2}^{p-1}},\ y_{n}=\frac{\alpha x_{n-1}^{p}+\beta \left( y_{n-2}x_{n-1}\right) ^{p-1}}{\gamma x_{n-1}+\delta y_{n-2}^{p-1}}, \end{equation*} $n \in \mathbb{N}_{0}$ where the parameters $a,b,c,d,\alpha ,\beta ,\gamma ,\delta,p$ and the initial values $x_{-2}$, $x_{-1}$, $y_{-2}$, $y_{-1}$ are real numbers, can be solved. Also, by using obtained formulas, we study the asymptotic behaviour of well-defined solutions of aforementioned system and describe the forbidden set of the initial values. Our obtained results significantly extend and develop some recent results in the literature.

Keywords: Difference equations, solution in closed-form, forbidden set, asymptotic behaviour

MSC numbers: Primary 39A10, 39A20, 39A23

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