Abstract
In this paper, we solve open problem 1 in Gumuş (Differ Equ Appl 24(6):976–991 2018). We investigate the oscillatory behavior, boundedness, persistence of positive solutions and the global asymptotic stability of the unique positive equilibrium point of the system of two rational difference equations:
where the parameters \(A> 0\) and \(B> 0\), the initial conditions \(x_{i}, y_{i}\) are arbitrary positive numbers for \(i = -k,-k + 1, \ldots , 0\) and \(k \in \mathbb {Z}^{+}\). Moreover, we give some numerical examples that support our analytical results.
Similar content being viewed by others
Notes
The infinite semi-cycle in the solution is a positive semi-cycle with respect to x and a negative semi-cycle with respect to y.
References
Camouzis, E., Papaschinopoulos, G.: Global asymptotic behavior of positive solutions on the system of rational difference equations \(x_{n+1} = 1+\frac{x_{n}}{y_{n- m}}, y_{n+1} = 1+\frac{y_{n}}{x_{n- m}}\), Appl. Math. Lett. 17(6), 733–737 (2004)
Clark, D., Kulenovic, M.R.S.: Global asymptotic behavior of a two-dimensional difference equation modelling competition. Nonlinear Anal. 52, 1765–1776 (2003)
Din, Q.: On the system of rational dierence equations. Demonstratio Mathematica 47(2), 324–335 (2014)
Elaydi, S.: An Introduction to Difference Equations. Springer, New York (1996)
Elaydi, S.: Discrete Chaos: With Applications in Science and Engineering. Chapman and Hall/CRC, Boka Raton (2007)
Gumuş, M.: The global asymptotic stability of a system of difference equations. J. Differ. Equ. Appl. 24(6), 976–991 (2018)
Papaschinopoulos, G.: On a system of two nonlinear difference equations. J. Math. Anal. Appl. 219(2), 415–426 (1998)
Papaschinopoulos, G.: On the system of two difference equations \(x_{n+1} = A + \frac{x_{n-1}}{y_{n}}, y_{n+1} = A + \frac{y_{n-1}}{x_{n}}\). Int. J. Math. Math. Sci. 23, 839–848 (2000)
Zhang, D., Ji, W., Wang, L., Li, X.: On the symmetrical system of rational difference equations \(x_{n+1} = A + \frac{y_{n-k}}{y_{n}}, y_{n+1} = A + \frac{x_{n-k}}{x_{n}}\). Appl. Math. 4, 834–837 (2013)
Zhang, Q., Zhang, W., Shao, Y., Liu, J.: On the System of High Order Rational Difference Equations. Int. Scholarly Res. Not. 1–5 (2014)
Zhang, Q., Yang, L., Liu, J.: On the Recursive System \(x_{n+1} = A + \frac{x_{n-m}}{y_{n}}, y_{n+1} = A + \frac{y_{n-m}}{x_{n}}\). Acta Math. Univ. Comenianae 82(2), 201–208 (2013)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Abualrub, S., Aloqeili, M. Dynamics of the System of Difference Equations \(x_{n+1} = A + \frac{y_{n-k}}{y_{n}}, \,\,\, y_{n+1} = B + \frac{x_{n-k}}{x_{n}}\). Qual. Theory Dyn. Syst. 19, 69 (2020). https://doi.org/10.1007/s12346-020-00408-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12346-020-00408-y