On the rational recursive sequence x n + 1 = A + i = 0 k α i x n - i / i = 0 k β i x n - i

E. M. E. Zayed; M. A. El-Moneam

Mathematica Bohemica (2008)

  • Volume: 133, Issue: 3, page 225-239
  • ISSN: 0862-7959

Abstract

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The main objective of this paper is to study the boundedness character, the periodic character, the convergence and the global stability of positive solutions of the difference equation x n + 1 = A + i = 0 k α i x n - i / i = 0 k β i x n - i , n = 0 , 1 , 2 , where the coefficients A , α i , β i and the initial conditions x - k , x - k + 1 , , x - 1 , x 0 are positive real numbers, while k is a positive integer number.

How to cite

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Zayed, E. M. E., and El-Moneam, M. A.. "On the rational recursive sequence $ \ x_{n+1}=\Big ( A+\sum _{i=0}^k\alpha _ix_{n-i}\Big ) \Big / \sum _{i=0}^k\beta _ix_{n-i} $." Mathematica Bohemica 133.3 (2008): 225-239. <http://eudml.org/doc/250538>.

@article{Zayed2008,
abstract = {The main objective of this paper is to study the boundedness character, the periodic character, the convergence and the global stability of positive solutions of the difference equation \[ x\_\{n+1\}=\bigg ( A+\sum \_\{i=0\}^k\alpha \_ix\_\{n-i\}\bigg ) \Big / \sum \_\{i=0\}^k\beta \_ix\_\{n-i\},\ \ n=0,1,2,\dots \] where the coefficients $A$, $\alpha _i$, $\beta _i$ and the initial conditions $x_\{-k\},x_\{-k+1\},\dots ,x_\{-1\},x_0$ are positive real numbers, while $k$ is a positive integer number.},
author = {Zayed, E. M. E., El-Moneam, M. A.},
journal = {Mathematica Bohemica},
keywords = {difference equations; boundedness character; period two solution; convergence; global stability; boundedness character; period two solution; convergence; global stability; rational difference equation; positive solutions},
language = {eng},
number = {3},
pages = {225-239},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the rational recursive sequence $ \ x_\{n+1\}=\Big ( A+\sum _\{i=0\}^k\alpha _ix_\{n-i\}\Big ) \Big / \sum _\{i=0\}^k\beta _ix_\{n-i\} $},
url = {http://eudml.org/doc/250538},
volume = {133},
year = {2008},
}

TY - JOUR
AU - Zayed, E. M. E.
AU - El-Moneam, M. A.
TI - On the rational recursive sequence $ \ x_{n+1}=\Big ( A+\sum _{i=0}^k\alpha _ix_{n-i}\Big ) \Big / \sum _{i=0}^k\beta _ix_{n-i} $
JO - Mathematica Bohemica
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 133
IS - 3
SP - 225
EP - 239
AB - The main objective of this paper is to study the boundedness character, the periodic character, the convergence and the global stability of positive solutions of the difference equation \[ x_{n+1}=\bigg ( A+\sum _{i=0}^k\alpha _ix_{n-i}\bigg ) \Big / \sum _{i=0}^k\beta _ix_{n-i},\ \ n=0,1,2,\dots \] where the coefficients $A$, $\alpha _i$, $\beta _i$ and the initial conditions $x_{-k},x_{-k+1},\dots ,x_{-1},x_0$ are positive real numbers, while $k$ is a positive integer number.
LA - eng
KW - difference equations; boundedness character; period two solution; convergence; global stability; boundedness character; period two solution; convergence; global stability; rational difference equation; positive solutions
UR - http://eudml.org/doc/250538
ER -

References

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  1. Aboutaleb, M. T., El-Sayed, M. A., Hamza, A. E., Stability of the recursive sequence x n + 1 = ( α - β x n ) / ( γ + x n - 1 ) , J. Math. Anal. Appl. 261 (2001), 126-133. (2001) Zbl0990.39009MR1850961
  2. Agarwal, R., Difference Equations and Inequalities, Theory, Methods and Applications, Marcel Dekker, New York (1992). (1992) Zbl0925.39001MR1155840
  3. Amleh, A. M., Grove, E. A., Ladas, G., Georgiou, D. A., On the recursive sequence x n + 1 = α + ( x n - 1 / x n ) , J. Math. Anal. Appl. 233 (1999), 790-798. (1999) MR1689579
  4. Vault, R. De, Kosmala, W., Ladas, G., Schultz, S. W., Global behavior of y n + 1 = ( p + y n - k ) / ( q y n + y n - k ) , Nonlinear Analysis 47 (2001), 4743-4751. (2001) MR1975867
  5. Vault, R. De, Ladas, G., Schultz, S. W., On the recursive sequence x n + 1 = A / x n + 1 / x n - 2 , Proc. Amer. Math. Soc. 126 (1998), 3257-3261. (1998) MR1473661
  6. Vault, R. De, Schultz, S. W., On the dynamics of x n + 1 = ( β x n + γ x n - 1 ) / ( B x n + D x n - 2 ) , Comm. Appl. Nonlinear Analysis 12 (2005), 35-39. (2005) MR2129054
  7. El-Metwally, H., Grove, E. A., Ladas, G., A global convergence result with applications to periodic solutions, J. Math. Anal. Appl. 245 (2000), 161-170. (2000) Zbl0971.39004MR1756582
  8. El-Metwally, H., Ladas, G., Grove, E. A., Voulov, H. D., On the global attractivity and the periodic character of some difference equations, J. Difference Equ. Appl. 7 (2001), 837-850. (2001) Zbl0993.39008MR1870725
  9. EL-Owaidy, H. M., Ahmed, A. M., Mousa, M. S., On asymptotic behavior of the difference equation x n + 1 = α + ( x n - 1 p / x n p ) , J. Appl. Math. & Comput. 12 (2003), 31-37. (2003) MR1976801
  10. EL-Owaidy, H. M., Ahmed, A. M., Elsady, Z., Global attractivity of the recursive sequence x n + 1 = ( α - β x n - k ) / ( γ + x n ) , J. Appl. Math. & Comput. 16 (2004), 243-249. (2004) MR2080567
  11. Karakostas, G., Convergence of a difference equation via the full limiting sequences method, Diff. Equations and Dynamical. System 1 (1993), 289-294. (1993) Zbl0868.39002MR1259169
  12. Karakostas, G., Stević, S., On the recursive sequences x n + 1 = A + f ( x n , , x n - k + 1 ) / x n - 1 , Commun. Appl. Nonlin. Anal. 11 (2004), 87-99. (2004) MR2069821
  13. Kocic, V. L., Ladas, G., Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht (1993). (1993) Zbl0787.39001MR1247956
  14. Kulenovic, M. R. S., Ladas, G., Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman & Hall/CRC Press (2002). (2002) Zbl0981.39011MR1935074
  15. Kulenovic, M. R. S., Ladas, G., Sizer, W. S., On the recursive sequence x n + 1 = ( α x n + β x n - 1 ) / ( γ x n + δ x n - 1 ) , Math. Sci. Res. Hot-Line 2 (1998), 1-16. (1998) Zbl0960.39502MR1623643
  16. Kuruklis, S. A., The asymptotic stability of x n + 1 - a x n + b x n - k = 0 , J. Math. Anal. Appl. 188 (1994), 719-731. (1994) MR1305480
  17. Ladas, G., Gibbons, C. H., Kulenovic, M. R. S., Voulov, H. D., On the trichotomy character of x n + 1 = ( α + β x n + γ x n - 1 ) / ( A + x n ) , J. Difference Equations and Appl. 8 (2002), 75-92. (2002) Zbl1005.39017MR1884593
  18. Ladas, G., Gibbons, C. H., Kulenovic, M. R. S., On the dynamics of x n + 1 = ( α + β x n + γ x n - 1 ) / ( A + B x n ) , Proceeding of the Fifth International Conference on Difference Equations and Applications, Temuco, Chile, Jan. 3-7, 2000, Taylor and Francis, London (2002), 141-158. (2002) MR2016061
  19. Ladas, G., Camouzis, E., Voulov, H. D., On the dynamic of x n + 1 = ( α + γ x n - 1 + δ x n - 2 ) / ( A + x n - 2 ) , J. Difference Equ. Appl. 9 (2003), 731-738. (2003) MR1992906
  20. Ladas, G., On the recursive sequence x n + 1 = ( α + β x n + γ x n - 1 ) / ( A + B x n + C x n - 1 ) , J. Difference Equ. Appl. 1 (1995), 317-321. (1995) MR1350447
  21. Li, W. T., Sun, H. R., Global attractivity in a rational recursive sequence, Dyn. Syst. Appl. 11 (2002), 339-346. (2002) Zbl1019.39007MR1941754
  22. Lin, Yi-Zhong, Common domain of asymptotic stability of a family of difference equations, Appl. Math. E-Notes 1 (2001), 31-33. (2001) MR1833834
  23. Stevi'c, S., On the recursive sequences x n + 1 = x n - 1 / g ( x n ) , Taiwanese J. Math. 6 (2002), 405-414. (2002) 
  24. Stevi'c, S., On the recursive sequences x n + 1 = g ( x n , x n - 1 ) / ( A + x n ) , Appl. Math. Letter 15 (2002), 305-308. (2002) MR1891551
  25. Stevi'c, S., On the recursive sequences x n + 1 = α + ( x n - 1 p / x n p ) , J. Appl. Math. Comput. 18 (2005), 229-234. (2005) MR2137703
  26. Zayed, E. M. E., El-Moneam, M. A., On the rational recursive sequence x n + 1 = ( D + α x n + β x n - 1 + γ x n - 2 ) / ( A x n + B x n - 1 + C x n - 2 ) , Commun. Appl. Nonlin. Anal. 12 (2005), 15-28. (2005) MR2163175
  27. Zayed, E. M. E., El-Moneam, M. A., On the rational recursive sequence x n + 1 = ( α x n + β x n - 1 + γ x n - 2 + δ x n - 3 ) / ( A x n + B x n - 1 + C x n - 2 + D x n - 3 ) , J. Appl. Math. Comput. 22 (2006), 247-262. (2006) MR2248455

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