On the Solutions of a Fourth Order Difference Equation

Year 2021, Volume: 4 Issue: 2, 76 - 81, 30.06.2021

R Abo-zeıd

https://doi.org/10.32323/ujma.917838

Cited By: 2

https://izlik.org/JA56WR79NZ

Abstract

In this paper, we solve and study the global behavior of the well defined solutions of the difference equation $$x_{n+1}=\frac{x_{n}x_{n-3}}{Ax_{n-2}+Bx_{n-3}}, \quad n=0,1,...,$$ where $A, B>0$ and the initial values $x_{-i}$, $i\in\{0,1,2,3\}$ are real numbers.

Keywords

difference equation , invariant set , forbidden set , convergence

References

• [1] R. Abo-Zeid, On a fourth order rational difference equation, Tbilisi Math. J., 12 (4) (2019), 71-79.
• [2] R. Abo-Zeid, Global behavior of a fourth order difference equation with quadratic term, Bol. Soc. Mat. Mexicana, 25 (2019), 187-194.
• [3] R. Abo-Zeid, Global behavior of two third order rational difference equations with quadratic terms, Math. Slovaca, 69 (1) (2019), 147-158.
• [4] R. Abo-Zeid, Behavior of solutions of a higher order difference equation, Alabama J. Math., 42 (2018), 1-10.
• [5] R. Abo-Zeid, On the solutions of a higher order difference equation, Georgian Math. J., DOI:10.1515/gmj-2018-0008.
• [6] R. Abo-Zeid, Forbidden sets and stability in some rational difference equations, J. Difference Equ. Appl., 24 (2) (2018), 220-239.
• [7] R. Abo-Zeid, Global behavior of a higher order rational difference equation, Filomat 30(12) (2016), 3265􀀀3276.
• [8] R. Abo-Zeid, Global behavior of a fourth order difference equation, Acta Comment. Univ. Tartu. Math., 18(2) (2014), 211-220.
• [9] R. P. Agarwal and E. M. Elsayed, Periodicity and stability of solutions of higher order rational difference equation, Adv. Stud. Contemp. Math., 17 (2) (2008), 181–201.
• [10] H. S. Alayachi, M. S. M. Noorani and E. M. Elsayed, Qualitative analysis of a fourth order difference equation, J. Appl. Anal. Comput., 10 (4) (2020), 1343–1354.
• [11] A.M. Amleh, E. Camouzis and G. Ladas On the dynamics of a rational difference equation, Part 2, Int. J. Difference Equ., 3(2) (2008), 195-225.
• [12] A.M. Amleh, E. Camouzis and G. Ladas On the dynamics of a rational difference equation, Part 1, Int. J. Difference Equ., 3(1) (2008), 1-35.
• [13] F. Balibrea and A. Cascales, On forbidden sets, J. Difference Equ. Appl. 21(10) (2015), 974􀀀996.
• [14] E. Camouzis and G. Ladas, Dynamics of Third Order Rational Difference Equations: With Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, 2008.
• [15] H. El-Metwally and E. M. Elsayed, Qualitative study of solutions of some difference equations, Abstr. Appl. Anal., Volume 2012, Article ID 248291, 16 pages, 2012.
• [16] M. G¨um¨us¸, The global asymptotic stability of a system of difference equations, J. Difference Equ. Appl., 24 (6) (2018), 976-991.
• [17] M. Gu¨mu¨s¸ and O¨ . O¨ calan, Global asymptotic stability of a nonautonomous difference equation, J. Appl. Math., Volume 2014, Article ID 395954, 5 pages, 2014.
• [18] E.A. Jankowski and M.R.S. Kulenovi´c, Attractivity and global stability for linearizable difference equations, Comput. Math. Appl. 57 (2009), 1592􀀀1607.
• [19] C.M. Kent and H. Sedaghat, Global attractivity in a quadratic-linear rational difference equation with delay, J. Difference Equ. Appl., 15 (10) (2009), 913􀀀925.
• [20] R. Khalaf-Allah, Asymptotic behavior and periodic nature of two difference equations, Ukrainian Math. J., 61 (6) (2009), 988-993.
• [21] V. L. Kocic, G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with applications, Kluwer Academic, Dordrecht, 1993.
• [22] M. R. S. Kulenovi´c and M. Mehulji´c, Global behavior of some rational second order difference equations, Int. J. Difference Equ., 7 (2) (2012), 153–162.
• [23] M.R.S. Kulenovic and G. Ladas, Dynamics of Second Order Rational Difference Equations: With Open Problems and Conjectures, Chapman and Hall/HRC, Boca Raton, 2002.
• [24] S. Stevic, Boundedness character of a fourth order nonlinear difference equation, Chaos, Sol. Frac., 40 (2009), 2364–2369.