Global behavior of a third order rational difference equation

Raafat Abo-Zeid

Mathematica Bohemica (2014)

  • Volume: 139, Issue: 1, page 25-37
  • ISSN: 0862-7959

Abstract

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In this paper, we determine the forbidden set and give an explicit formula for the solutions of the difference equation x n + 1 = a x n x n - 1 - b x n + c x n - 2 , n ∈ ℕ 0 where a , b , c are positive real numbers and the initial conditions x - 2 , x - 1 , x 0 are real numbers. We show that every admissible solution of that equation converges to zero if either a < c or a > c with ( a - c ) / b < 1 . When a > c with ( a - c ) / b > 1 , we prove that every admissible solution is unbounded. Finally, when a = c , we prove that every admissible solution converges to zero.

How to cite

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Abo-Zeid, Raafat. "Global behavior of a third order rational difference equation." Mathematica Bohemica 139.1 (2014): 25-37. <http://eudml.org/doc/261090>.

@article{Abo2014,
abstract = {In this paper, we determine the forbidden set and give an explicit formula for the solutions of the difference equation \[x\_\{n+1\}=\frac\{ax\_\{n\}x\_\{n-1\}\}\{-bx\_\{n\}+ cx\_\{n-2\}\},\quad n\in \mathbb \{N\}\_0 \] where $a$, $b$, $c$ are positive real numbers and the initial conditions $x_\{-2\}$, $x_\{-1\}$, $x_0$ are real numbers. We show that every admissible solution of that equation converges to zero if either $a<c$ or $a>c$ with $\{(a-c)\}/\{b\}<1$. When $a>c$ with $\{(a-c)\}/\{b\}>1$, we prove that every admissible solution is unbounded. Finally, when $a=c$, we prove that every admissible solution converges to zero.},
author = {Abo-Zeid, Raafat},
journal = {Mathematica Bohemica},
keywords = {difference equation; forbidden set; periodic solution; unbounded solution; existence of solution; convergence to zero; unbounded solution; periodicity; forbidden set; rational difference equation; asymptotic periodicity},
language = {eng},
number = {1},
pages = {25-37},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Global behavior of a third order rational difference equation},
url = {http://eudml.org/doc/261090},
volume = {139},
year = {2014},
}

TY - JOUR
AU - Abo-Zeid, Raafat
TI - Global behavior of a third order rational difference equation
JO - Mathematica Bohemica
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 139
IS - 1
SP - 25
EP - 37
AB - In this paper, we determine the forbidden set and give an explicit formula for the solutions of the difference equation \[x_{n+1}=\frac{ax_{n}x_{n-1}}{-bx_{n}+ cx_{n-2}},\quad n\in \mathbb {N}_0 \] where $a$, $b$, $c$ are positive real numbers and the initial conditions $x_{-2}$, $x_{-1}$, $x_0$ are real numbers. We show that every admissible solution of that equation converges to zero if either $a<c$ or $a>c$ with ${(a-c)}/{b}<1$. When $a>c$ with ${(a-c)}/{b}>1$, we prove that every admissible solution is unbounded. Finally, when $a=c$, we prove that every admissible solution converges to zero.
LA - eng
KW - difference equation; forbidden set; periodic solution; unbounded solution; existence of solution; convergence to zero; unbounded solution; periodicity; forbidden set; rational difference equation; asymptotic periodicity
UR - http://eudml.org/doc/261090
ER -

References

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