On the rational recursive sequence x n + 1 = α 0 x n + α 1 x n - l + α 2 x n - k β 0 x n + β 1 x n - l + β 2 x n - k

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

Mathematica Bohemica (2010)

  • Volume: 135, Issue: 3, page 319-336
  • ISSN: 0862-7959

Abstract

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The main objective of this paper is to study the boundedness character, the periodicity character, the convergence and the global stability of positive solutions of the difference equation x n + 1 = α 0 x n + α 1 x n - l + α 2 x n - k β 0 x n + β 1 x n - l + β 2 x n - k , n = 0 , 1 , 2 , ⋯ where the coefficients α i , β i ∈ ( 0 , ∞ ) for i = 0 , 1 , 2 , and l , k are positive integers. The initial conditions x - k , ⋯ , x - l , ⋯ , x - 1 , x 0 are arbitrary positive real numbers such that l < k . Some numerical experiments are presented.

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Zayed, E. M. E., and El-Moneam, M. A.. "On the rational recursive sequence $ x_{n+1}=\dfrac{\alpha _0x_n+\alpha _1x_{n-l}+\alpha _2x_{n-k}}{\beta _0x_n+\beta _1x_{n-l}+\beta _2x_{n-k}}$." Mathematica Bohemica 135.3 (2010): 319-336. <http://eudml.org/doc/38133>.

@article{Zayed2010,
abstract = {The main objective of this paper is to study the boundedness character, the periodicity character, the convergence and the global stability of positive solutions of the difference equation \[ x\_\{n+1\}=\frac\{\alpha \_0x\_n+\alpha \_1x\_\{n-l\}+\alpha \_2x\_\{n-k\}\}\{\beta \_0x\_n+\beta \_1x\_\{n-l\}+\beta \_2x\_\{n-k\}\}, \quad n=0,1,2,\dots \] where the coefficients $\alpha _i,\beta _i\in (0,\infty )$ for $ i=0,1,2,$ and $l$, $k$ are positive integers. The initial conditions $ x_\{-k\}, \dots , x_\{-l\}, \dots , x_\{-1\}, x_0 $ are arbitrary positive real numbers such that $l<k$. Some numerical experiments are presented.},
author = {Zayed, E. M. E., El-Moneam, M. A.},
journal = {Mathematica Bohemica},
keywords = {difference equation; boundedness; period two solution; convergence; global stability; rational difference equation; boundedness; period two solution; convergence; global stability; positive solutions; numerical experiments},
language = {eng},
number = {3},
pages = {319-336},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the rational recursive sequence $ x_\{n+1\}=\dfrac\{\alpha _0x_n+\alpha _1x_\{n-l\}+\alpha _2x_\{n-k\}\}\{\beta _0x_n+\beta _1x_\{n-l\}+\beta _2x_\{n-k\}\}$},
url = {http://eudml.org/doc/38133},
volume = {135},
year = {2010},
}

TY - JOUR
AU - Zayed, E. M. E.
AU - El-Moneam, M. A.
TI - On the rational recursive sequence $ x_{n+1}=\dfrac{\alpha _0x_n+\alpha _1x_{n-l}+\alpha _2x_{n-k}}{\beta _0x_n+\beta _1x_{n-l}+\beta _2x_{n-k}}$
JO - Mathematica Bohemica
PY - 2010
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 135
IS - 3
SP - 319
EP - 336
AB - The main objective of this paper is to study the boundedness character, the periodicity character, the convergence and the global stability of positive solutions of the difference equation \[ x_{n+1}=\frac{\alpha _0x_n+\alpha _1x_{n-l}+\alpha _2x_{n-k}}{\beta _0x_n+\beta _1x_{n-l}+\beta _2x_{n-k}}, \quad n=0,1,2,\dots \] where the coefficients $\alpha _i,\beta _i\in (0,\infty )$ for $ i=0,1,2,$ and $l$, $k$ are positive integers. The initial conditions $ x_{-k}, \dots , x_{-l}, \dots , x_{-1}, x_0 $ are arbitrary positive real numbers such that $l<k$. Some numerical experiments are presented.
LA - eng
KW - difference equation; boundedness; period two solution; convergence; global stability; rational difference equation; boundedness; period two solution; convergence; global stability; positive solutions; numerical experiments
UR - http://eudml.org/doc/38133
ER -

References

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