Global behavior of the difference equation x n + 1 = a x n - 3 b + c x n - 1 x n - 3

Raafat Abo-Zeid

Archivum Mathematicum (2015)

  • Volume: 051, Issue: 2, page 77-85
  • ISSN: 0044-8753

Abstract

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In this paper, we introduce an explicit formula and discuss the global behavior of solutions of the difference equation x n + 1 = a x n - 3 b + c x n - 1 x n - 3 , n = 0 , 1 , ⋯ where a , b , c are positive real numbers and the initial conditions x - 3 , x - 2 , x - 1 , x 0 are real numbers.

How to cite

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Abo-Zeid, Raafat. "Global behavior of the difference equation $x_{n+1}=\frac{ax_{n-3} }{b+ cx_{n-1}x_{n-3}}$." Archivum Mathematicum 051.2 (2015): 77-85. <http://eudml.org/doc/270130>.

@article{Abo2015,
abstract = {In this paper, we introduce an explicit formula and discuss the global behavior of solutions of the difference equation \[ x\_\{n+1\}=\frac\{ax\_\{n-3\} \}\{b+ cx\_\{n-1\}x\_\{n-3\}\}\,,\qquad n=0,1,\dots \] where $a,b,c$ are positive real numbers and the initial conditions $x_\{-3\}$, $x_\{-2\}$, $x_\{-1\}$, $x_0$ are real numbers.},
author = {Abo-Zeid, Raafat},
journal = {Archivum Mathematicum},
keywords = {difference equation; periodic solution; convergence},
language = {eng},
number = {2},
pages = {77-85},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Global behavior of the difference equation $x_\{n+1\}=\frac\{ax_\{n-3\} \}\{b+ cx_\{n-1\}x_\{n-3\}\}$},
url = {http://eudml.org/doc/270130},
volume = {051},
year = {2015},
}

TY - JOUR
AU - Abo-Zeid, Raafat
TI - Global behavior of the difference equation $x_{n+1}=\frac{ax_{n-3} }{b+ cx_{n-1}x_{n-3}}$
JO - Archivum Mathematicum
PY - 2015
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 051
IS - 2
SP - 77
EP - 85
AB - In this paper, we introduce an explicit formula and discuss the global behavior of solutions of the difference equation \[ x_{n+1}=\frac{ax_{n-3} }{b+ cx_{n-1}x_{n-3}}\,,\qquad n=0,1,\dots \] where $a,b,c$ are positive real numbers and the initial conditions $x_{-3}$, $x_{-2}$, $x_{-1}$, $x_0$ are real numbers.
LA - eng
KW - difference equation; periodic solution; convergence
UR - http://eudml.org/doc/270130
ER -

References

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  1. Abo-Zeid, R., Global asymptotic stability of a higher order difference equation, Bull. Allahabad Math. Soc. 2 (2) (2010), 341–351. (2010) Zbl1227.39015MR2779248
  2. Abo-Zeid, R., Global asymptotic stability of a second order rational difference equation, J. Appl. Math. & Inform. 2 (3) (2010), 797–804. (2010) Zbl1294.39010MR2779248
  3. Agarwal, R.P., Difference Equations and Inequalities, first ed., Marcel Decker, 1992. (1992) Zbl0925.39001MR1155840
  4. Al-Shabi, M.A., Abo-Zeid, R., Global asymptotic stability of a higher order difference equation, Appl. Math. Sci. 4 (17) (2010), 839–847. (2010) Zbl1198.39025MR2595521
  5. Camouzis, E., Ladas, G., Dynamics of Third-Order Rational Difference Equations; With Open Problems and Conjectures, Chapman and Hall/HRC Boca Raton, 2008. (2008) Zbl1129.39002MR2363297
  6. Elsayed, E.M., On the difference equation x n + 1 = x n - 5 - 1 + x n - 2 x n - 5 , Int. J. Contemp. Math. Sciences 3 (33) (2008), 1657–1664. (2008) Zbl1172.39009MR2511022
  7. Elsayed, E.M., On the solution of some difference equations, European J. Pure Appl. Math. 4 (2011), 287–303. (2011) MR2824757
  8. Grove, E.A., Ladas, G., Periodicities in Nonlinear Difference Equations, Chapman and Hall/CRC, 2005. (2005) Zbl1078.39009MR2193366
  9. Karakostas, G., Convergence of a difference equation via the full limiting sequences method, Differential Equations Dynam. Systems 1 (4) (1993), 289–294. (1993) Zbl0868.39002MR1259169
  10. Karatas, R., Cinar, C., Simsek, D., On the positive solution of the difference equation x n + 1 = x n - 5 1 + x n - 2 x n - 5 , Int. J. Contemp. Math. Sciences 1 (10) (2006), 495–500. (2006) MR2287595
  11. Kocic, V.L., Ladas, G., Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic, Dordrecht, 1993. (1993) Zbl0787.39001MR1247956
  12. Kruse, N., Nesemann, T., 10.1006/jmaa.1999.6384, J. Math. Anal. Appl. 235 (1) (1999), 151–158. (1999) Zbl0933.37016MR1758674DOI10.1006/jmaa.1999.6384
  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. (2002) Zbl0981.39011MR1935074
  14. Levy, H., Lessman, F., Finite Difference Equations, Dover, New York, NY, USA, 1992. (1992) MR1217083
  15. Sedaghat, H., 10.1080/10236190802054126, J. Differ. Equations Appl. 15 (3) (2009), 215–224. (2009) Zbl1169.39006MR2498770DOI10.1080/10236190802054126
  16. Simsek, D., Cinar, C., Karatas, R., Yalcinkaya, I., On the recursive sequence x n + 1 = x n - 5 1 + x n - 1 x n - 3 , Int. J. Pure Appl. Math. 28 (1) (2006), 117–124. (2006) Zbl1116.39005MR2227156
  17. Simsek, D., Cinar, C., Yalcinkaya, I., On the recursive sequence x n + 1 = x n - 3 1 + x n - 1 , Int. J. Contemp. Math. Sciences 1 (10) (2006), 475–480. (2006) Zbl1157.39311MR2287592
  18. Stević, S., More on a rational recurrence relation, Appl. Math. E-Notes 4 (2004), 80–84. (2004) Zbl1069.39024MR2077785
  19. Stević, S., 10.1016/j.aml.2005.05.014, Appl. Math. Lett. 19 (5) (2006), 427–431. (2006) Zbl1095.39010MR2213143DOI10.1016/j.aml.2005.05.014

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