| Title:
|
Global behavior of a third order rational difference equation (English) |
| Author:
|
Abo-Zeid, Raafat |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
139 |
| Issue:
|
1 |
| Year:
|
2014 |
| Pages:
|
25-37 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper, we determine the forbidden set and give an explicit formula for the solutions of the difference equation $$x_{n+1}=\frac {ax_{n}x_{n-1}}{-bx_{n}+ cx_{n-2}},\quad n\in \mathbb {N}_0 $$ where $a$, $b$, $c$ are positive real numbers and the initial conditions $x_{-2}$, $x_{-1}$, $x_0$ are real numbers. We show that every admissible solution of that equation converges to zero if either $a<c$ or $a>c$ with ${(a-c)}/{b}<1$. \endgraf When $a>c$ with ${(a-c)}/{b}>1$, we prove that every admissible solution is unbounded. Finally, when $a=c$, we prove that every admissible solution converges to zero. (English) |
| Keyword:
|
difference equation |
| Keyword:
|
forbidden set |
| Keyword:
|
periodic solution |
| Keyword:
|
unbounded solution |
| MSC:
|
39A20 |
| MSC:
|
39A21 |
| MSC:
|
39A23 |
| MSC:
|
39A30 |
| idZBL:
|
Zbl 06362241 |
| idMR:
|
MR3231428 |
| DOI:
|
10.21136/MB.2014.143635 |
| . |
| Date available:
|
2014-03-20T08:26:42Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143635 |
| . |
| Reference:
|
[1] Agarwal, R. P.: Difference Equations and Inequalities. Theory, Methods, and Applications.Pure and Applied Mathematics 155 Marcel Dekker, New York (1992). Zbl 0925.39001, MR 1155840 |
| Reference:
|
[2] Aloqeili, M.: Dynamics of a rational difference equation.Appl. Math. Comput. 176 (2006), 768-774. Zbl 1100.39002, MR 2232069, 10.1016/j.amc.2005.10.024 |
| Reference:
|
[3] Andruch-Sobiło, A., Migda, M.: Further properties of the rational recursive sequence $x_{n+1}=\frac{ax_{n-1}}{b+cx_{n}x_{n-1}}$.Opusc. Math. 26 (2006), 387-394. Zbl 1131.39003, MR 2280266 |
| Reference:
|
[4] Andruch-Sobiło, A., Migda, M.: On the rational recursive sequence $x_{n+1}=\frac{ax_{n-1}}{b+cx_{n}x_{n-1}}$.Tatra Mt. Math. Publ. 43 (2009), 1-9. MR 2588871 |
| Reference:
|
[5] Berg, L., Stević, S.: On difference equations with powers as solutions and their connection with invariant curves.Appl. Math. Comput. 217 (2011), 7191-7196. Zbl 1260.39002, MR 2781112, 10.1016/j.amc.2011.02.005 |
| Reference:
|
[6] Berg, L., Stević, S.: On some systems of difference equations.Appl. Math. Comput. 218 (2011), 1713-1718. Zbl 1243.39009, MR 2831394, 10.1016/j.amc.2011.06.050 |
| Reference:
|
[7] Camouzis, E., Ladas, G.: Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures.Advances in Discrete Mathematics and Applications 5 Chapman and Hall/HRC, Boca Raton (2008). Zbl 1129.39002, MR 2363297 |
| Reference:
|
[8] Grove, E. A., Ladas, G.: Periodicities in Nonlinear Difference Equations.Advances in Discrete Mathematics and Applications 4 Chapman and Hall/CRC, Boca Raton (2005). Zbl 1078.39009, MR 2193366 |
| Reference:
|
[9] Iričanin, B., Stević, S.: On some rational difference equations.Ars Comb. 92 (2009), 67-72. Zbl 1224.39014, MR 2532566 |
| Reference:
|
[10] Karakostas, G.: Convergence of a difference equation via the full limiting sequences method.Differential Equations Dynam. Systems 1 (1993), 289-294. Zbl 0868.39002, MR 1259169 |
| Reference:
|
[11] Kocić, V. L., Ladas, G.: Global Behavior of Nonlinear Difference Equations of Higher Order with Applications.Mathematics and Its Applications 256 Kluwer Academic Publishers, Dordrecht (1993). MR 1247956 |
| Reference:
|
[12] Kruse, N., Nesemann, T.: Global asymptotic stability in some discrete dynamical systems.J. Math. Anal. Appl. 235 (1999), 151-158. Zbl 0933.37016, MR 1758674, 10.1006/jmaa.1999.6384 |
| Reference:
|
[13] Kulenović, M. R. S., Ladas, G.: Dynamics of Second Order Rational Difference Equations.With Open Problems and Conjectures Chapman and Hall/HRC, Boca Raton (2002). Zbl 0981.39011, MR 1935074 |
| Reference:
|
[14] Levy, H., Lessman, F.: Finite Difference Equations.Reprint of the 1961 edition. Dover Publications New York (1992). MR 1217083 |
| Reference:
|
[15] Sedaghat, H.: Global behaviours of rational difference equations of orders two and three with quadratic terms.J. Difference Equ. Appl. 15 (2009), 215-224. Zbl 1169.39006, MR 2498770, 10.1080/10236190802054126 |
| Reference:
|
[16] Stević, S.: On a system of difference equations.Appl. Math. Comput. 218 (2011), 3372-3378. Zbl 1256.39008, MR 2851439, 10.1016/j.amc.2011.08.079 |
| Reference:
|
[17] Stević, S.: On a system of difference equations with period two coefficients.Appl. Math. Comput. 218 (2011), 4317-4324. Zbl 1256.39008, MR 2862101, 10.1016/j.amc.2011.10.005 |
| Reference:
|
[18] Stević, S.: On a third-order system of difference equations.Appl. Math. Comput. 218 (2012), 7649-7654. Zbl 1242.39011, MR 2892731, 10.1016/j.amc.2012.01.034 |
| Reference:
|
[19] Stević, S.: On the difference equation $x_n=x_{n-2}/(b_n+c_nx_{n-1}x_{n-2})$.Appl. Math. Comput. 218 (2011), 4507-4513. MR 2862122 |
| Reference:
|
[20] Stević, S.: On some solvable systems of difference equations.Appl. Math. Comput. 218 (2012), 5010-5018. Zbl 1253.39011, MR 2870025, 10.1016/j.amc.2011.10.068 |
| Reference:
|
[21] Stević, S.: On positive solutions of a $(k +1)$th order difference equation.Appl. Math. Lett. 19 (2006), 427-431. Zbl 1095.39010, MR 2213143, 10.1016/j.aml.2005.05.014 |
| Reference:
|
[22] Stević, S.: More on a rational recurrence relation.Appl. Math. E-Notes (electronic only) 4 (2004), 80-85. Zbl 1069.39024, MR 2077785 |
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