On the difference equation x n + 1 = a 0 x n + a 1 x n - 1 + ⋯ + a k x n - k b 0 x n + b 1 x n - 1 + ⋯ + b k x n - k

Elmetwally M. Elabbasy; Hamdy El-Metwally; E. M. Elsayed

Mathematica Bohemica (2008)

  • Volume: 133, Issue: 2, page 133-147
  • ISSN: 0862-7959

Abstract

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In this paper we investigate the global convergence result, boundedness and periodicity of solutions of the recursive sequence x n + 1 = a 0 x n + a 1 x n - 1 + ⋯ + a k x n - k b 0 x n + b 1 x n - 1 + ⋯ + b k x n - k , n = 0 , 1 , ⋯ where the parameters a i and b i for i = 0 , 1 , ⋯ , k are positive real numbers and the initial conditions x - k , x - k + 1 , ⋯ , x 0 are arbitrary positive numbers.

How to cite

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Elabbasy, Elmetwally M., El-Metwally, Hamdy, and Elsayed, E. M.. "On the difference equation $x_{n+1}=\dfrac{a_{0}x_{n}+a_{1}x_{n-1}+\dots +a_{k}x_{n-k}}{b_{0}x_{n}+b_{1}x_{n-1}+\dots +b_{k}x_{n-k}} $." Mathematica Bohemica 133.2 (2008): 133-147. <http://eudml.org/doc/250521>.

@article{Elabbasy2008,
abstract = {In this paper we investigate the global convergence result, boundedness and periodicity of solutions of the recursive sequence \[ x\_\{n+1\}=\frac\{a\_\{0\}x\_\{n\}+a\_\{1\}x\_\{n-1\}+\dots +a\_\{k\}x\_\{n-k\}\}\{b\_\{0\}x\_\{n\}+b\_\{1\}x\_\{n-1\}+\dots +b\_\{k\}x\_\{n-k\}\},\,\,\,n=0,1,\dots \,\ \] where the parameters $ a_\{i\}$ and $b_\{i\}$ for $i=0,1,\dots ,k$ are positive real numbers and the initial conditions $x_\{-k\},x_\{-k+1\},\dots ,x_\{0\}$ are arbitrary positive numbers.},
author = {Elabbasy, Elmetwally M., El-Metwally, Hamdy, Elsayed, E. M.},
journal = {Mathematica Bohemica},
keywords = {stability; periodic solution; difference equation; stability; periodic solution; rational difference equation; global convergence; boundedness; recursive sequence},
language = {eng},
number = {2},
pages = {133-147},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the difference equation $x_\{n+1\}=\dfrac\{a_\{0\}x_\{n\}+a_\{1\}x_\{n-1\}+\dots +a_\{k\}x_\{n-k\}\}\{b_\{0\}x_\{n\}+b_\{1\}x_\{n-1\}+\dots +b_\{k\}x_\{n-k\}\} $},
url = {http://eudml.org/doc/250521},
volume = {133},
year = {2008},
}

TY - JOUR
AU - Elabbasy, Elmetwally M.
AU - El-Metwally, Hamdy
AU - Elsayed, E. M.
TI - On the difference equation $x_{n+1}=\dfrac{a_{0}x_{n}+a_{1}x_{n-1}+\dots +a_{k}x_{n-k}}{b_{0}x_{n}+b_{1}x_{n-1}+\dots +b_{k}x_{n-k}} $
JO - Mathematica Bohemica
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 133
IS - 2
SP - 133
EP - 147
AB - In this paper we investigate the global convergence result, boundedness and periodicity of solutions of the recursive sequence \[ x_{n+1}=\frac{a_{0}x_{n}+a_{1}x_{n-1}+\dots +a_{k}x_{n-k}}{b_{0}x_{n}+b_{1}x_{n-1}+\dots +b_{k}x_{n-k}},\,\,\,n=0,1,\dots \,\ \] where the parameters $ a_{i}$ and $b_{i}$ for $i=0,1,\dots ,k$ are positive real numbers and the initial conditions $x_{-k},x_{-k+1},\dots ,x_{0}$ are arbitrary positive numbers.
LA - eng
KW - stability; periodic solution; difference equation; stability; periodic solution; rational difference equation; global convergence; boundedness; recursive sequence
UR - http://eudml.org/doc/250521
ER -

References

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  1. On the periodic nature of some max-type difference equation, Int. J. Math. Math. Sci. 14 (2005), 2227–2239. (2005) MR2177819
  2. On the difference equation x n + 1 = α x n - k β + γ ∏ i = 0 k x n - i , J. Conc. Appl. Math. 5 (2007), 101–113. (2007) MR2292704
  3. 10.1080/10236190108808306, J. Difference Equ. Appl. 7 (2001), 837–850. (2001) MR1870725DOI10.1080/10236190108808306
  4. On the difference equation y n + 1 = ( y n - ( 2 k + 1 ) + p ) ( y n - ( 2 k + 1 ) + q y n - 2 l ) , Proceedings of the 6th ICDE, Taylor and Francis, London, 2004. (2004) MR2092580
  5. 10.1080/10236190008808232, J. Difference Equ. Appl. 6 (2000), 329–335. (2000) Zbl0963.39020MR1785059DOI10.1080/10236190008808232
  6. Convergence of a difference equation via the full limiting sequences method, Diff. Equ. Dyn. Sys. 1 (1993), 289–294. (1993) Zbl0868.39002MR1259169
  7. Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, 1993. (1993) MR1247956
  8. On the recursive sequence y n + 1 = p + y n - 1 q y n + y n - 1 , J. Math. Anal. Appl. 251 (2001), 571–586. (2001) MR1794759
  9. 10.1016/S0898-1221(00)00311-4, Comput. Math. Appl. 41 (2001), 671–678. (2001) MR1822594DOI10.1016/S0898-1221(00)00311-4
  10. On the recursive sequence x n + 1 = ( α x n + β x n - 1 ) / ( A + x n ) , J. Difference Equ. Appl. 6 (2000), 563–576. (2000) MR1802447
  11. 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) MR1623643
  12. 10.1016/j.amc.2004.04.002, Appl. Math. Comp. 163 (2005), 577–591. (2005) MR2121812DOI10.1016/j.amc.2004.04.002

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