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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  9. A rational difference equation, Comput. Math. Appl. 41 (2001), 671–678. (2001) MR1822594
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  12. Dynamics of a rational difference equation, Appl. Math. Comp. 163 (2005), 577–591. (2005) MR2121812

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