# Expressions and dynamical behavior of solutions of eighteenth-order of a class of rational difference equations

Volume 28, Issue 3, pp 258--269
Publication Date: June 26, 2022 Submission Date: December 21, 2021 Revision Date: April 11, 2022 Accteptance Date: May 21, 2022
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### Authors

L. Sh. Aljoufi - Department of Mathematics, College of Science, Jouf University, P.O. Box 2014, Sakaka, Jouf , Saudi Arabia. S. A. Mohammady - Department of Mathematics, College of Science, Jouf University, P.O. Box 2014, Sakaka, Jouf , Saudi Arabia. - Department of Mathematics, Faculty of Science, Helwan University, Helwan 11795, Jouf, Egypt. A. M. Ahmed - Department of Mathematics, Faculty of Science, Al Azhar University, Nasr City 11884, Cairo, Egypt.

### Abstract

The aim of this work is to obtain the forms of the solutions of the following nonlinear eighteenth-order difference equations $x_{n+1}=\frac{x_{n-17}}{\pm 1\pm x_{n-2}x_{n-5}x_{n-8}x_{n-11}x_{n-14}x_{n-17}},\ \ \ \ n=0,1,2,\ldots,$ where the initial conditions $x_{-17},x_{-16},\ldots,x_{0}$ are arbitrary real numbers. Moreover, we investigate stability, boundedness, oscillation, and the periodic character of these solutions. Finally, we confirm the results with some numerical examples and graphs by using Matlab program.

### Share and Cite

##### ISRP Style

L. Sh. Aljoufi, S. A. Mohammady, A. M. Ahmed, Expressions and dynamical behavior of solutions of eighteenth-order of a class of rational difference equations, Journal of Mathematics and Computer Science, 28 (2023), no. 3, 258--269

##### AMA Style

Aljoufi L. Sh., Mohammady S. A., Ahmed A. M., Expressions and dynamical behavior of solutions of eighteenth-order of a class of rational difference equations. J Math Comput SCI-JM. (2023); 28(3):258--269

##### Chicago/Turabian Style

Aljoufi, L. Sh., Mohammady, S. A., Ahmed, A. M.. "Expressions and dynamical behavior of solutions of eighteenth-order of a class of rational difference equations." Journal of Mathematics and Computer Science, 28, no. 3 (2023): 258--269

### Keywords

• Recursive sequence
• oscillation
• semicycles
• stability
• periodicity
• solutions of difference equations

•  39A10
•  39A22
•  39A23

### References

• [1] R. P. Agarwal, Difference equations and inequalities, Marcel Dekker, New York (2000)

• [2] R. P. Agarwal, E. M. Elsayed, Periodicity and stability of solutions of higher order rational difference equation, Adv. Stud. Contemp. Math. (Kyungshang), 17 (2008), 181--201

• [3] A. M. Ahmed, On the dynamics of a higher-order rational difference equation, Discrete Dyn. Nat. Soc., 2011 (2011), 8 pages

• [4] A. Ahmed, S. Al Mohammady, L. Sh. Aljoufi, Expressions and Dynamical Behavior of Solutions of a Class of Rational Difference Equations of Fifteenth-Order, J. Math. Comput. Sci., 25 (2022), 10--22

• [5] A. M. Ahmed, H. M. El-Owaidy, A. Hamza, A. M. Youssef, On the recursive sequence $x_{n+1}=\dfrac{a+bx_{n-1}}{A+Bx_{n}^{k}}$, J. Appl. Math. Inform., 27 (2009), 275--289

• [6] A. M. Ahmed, A. M. Youssef, A solution form of a class of higher-order rational difference equations, J. Egyptian Math. Soc., 21 (2013), 248--253

• [7] M. Aloqeili, Dynamics of a rational difference equation, Appl. Math. Comput., 176 (2006), 768--774

• [8] A. M. Amleh, J. Hoag, G. Ladas, A difference equation with eventually periodic solutions, II, Comput. Math. Appl., 36 (1998), 401--404

• [9] C. Cinar, On the Positive Solutions of the Difference Equation $x_{n+1}=\dfrac{x_{n-1}}{1+x_{n}x_{n-1}}$, Appl. Math. Comput., 150 (2004), 21--24

• [10] C. Cinar, On the Difference Equation $x_{n+1}=\dfrac{x_{n-1}}{-1+x_{n}x_{n-1}}$, Appl. Math. Comput., 158 (2004), 813--816

• [11] C. Cinar, On the Positive Solutions of the Difference Equation $x_{n+1}=\dfrac{ax_{n-1}}{1+bx_{n}x_{n-1}}$, Appl. Math. Comput., 156 (2004), 587--590

• [12] R. Devault, V. L. Kocic, D. Stutson, Global behavior of solutions of the nonlinear difference equation $x_{n+1}=\frac{p_{n}+x_{n-1}}{x_{n}}$, J. Difference Equ. Appl., 11 (2005), 707--719

• [13] E. M. Elsayed, On the difference equation $x_{n+1}=\frac{x_{n-5}}{-1+x_{n-2}x_{n-5}}$, Int. J. Contemp. Math. Sci., 3 (2008), 1657--1664

• [14] E. M. Elsayed, M. M. Alzubaidi, Expressions and dynamical behavior of rational recursive sequences, J. Comput. Anal. Appl., 28 (2020), 67--78

• [15] E. A. Grove, G. Ladas, Periodicities in nonlinear difference equations, Chapman & Hall/CRC Press, Boca Raton (2005)

• [16] R. Karatas, C. Cinar, D. Simsek, On positive solutions of the difference equation $x_{n+1}=\frac{x_{n-5}}{1+x_{n-2}x_{n-5}}$, Int. J. Contemp. Math. Sci., 1 (2006), 495--500

• [17] V. L. Kocic, G. Ladas, Global behavior of nonlinear difference equations of higher order with applications, Kluwer Academic Publishers Group, Dordrecht (1993)

• [18] M. R. S. Kulenovic, G. Ladas, Dynamics of second order rational difference equations with open problems and conjectures, Chapman and Hall/CRC Press, New York (2001)