scholarly journals Global behavior of the difference equation $x_{n+1}=\frac{Ax_{n-1}} {B-Cx_{n}x_{n-2}}$

2011 ◽  
Vol 31 (1) ◽  
pp. 43 ◽  
Author(s):  
R. Abo-Zeid ◽  
Cengiz Cinar
Keyword(s):  
Difference Equation ◽  
Global Stability ◽  
Global Behavior ◽  
Real Numbers ◽  
Positive Real ◽  
Admissible Solutions ◽  
The Difference

The aim of this work is to investigate the global stability, periodic nature, oscillation and the boundedness of all admissible solutions of the difference equation $x_{n+1}=\frac{Ax_{n-1}} {B-Cx_{n}x_{n-2}}$, n=0,1,2,... where A, B, C are positive real numbers.

scholarly journals Global Attractivity of a Higher-Order Difference Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
R. Abo-Zeid
Keyword(s):  
Difference Equation ◽  
Global Stability ◽  
Global Attractivity ◽  
Higher Order ◽  
Real Numbers ◽  
Positive Real ◽  
Order Difference ◽  
Order Difference Equation ◽  
Admissible Solutions ◽  
The Difference

The aim of this work is to investigate the global stability, periodic nature, oscillation, and the boundedness of all admissible solutions of the difference equationxn+1=Axn-2r-1/(B-C∏i=lkxn-2i), n=0,1,2,…whereA,B,Care positive real numbers andl,r,kare nonnegative integers, such thatl≤k.

On the solutions of a higher order difference equation

2020 ◽  
Vol 27 (2) ◽  
pp. 165-175 ◽  
Author(s):  
Raafat Abo-Zeid
Keyword(s):  
Difference Equation ◽  
Explicit Formula ◽  
Initial Conditions ◽  
Global Behavior ◽  
Real Numbers ◽  
Positive Real ◽  
Order Difference ◽  
Forbidden Set ◽  
Order Difference Equation ◽  
The Difference

AbstractIn this paper, we determine the forbidden set, introduce an explicit formula for the solutions and discuss the global behavior of solutions of the difference equationx_{n+1}=\frac{ax_{n}x_{n-k}}{bx_{n}-cx_{n-k-1}},\quad n=0,1,\ldots,where{a,b,c}are positive real numbers and the initial conditions{x_{-k-1},x_{-k},\ldots,x_{-1},x_{0}}are real numbers. We show that when{a=b=c}, the behavior of the solutions depends on whetherkis even or odd.

On the difference equation $y_{n + 1} = \frac{{\alpha + y_n^p }} {{\beta y_{n - 1}^p }} - \frac{{\gamma + y_{n - 1}^p }} {{\beta y_n^p }} $

2011 ◽  
Vol 61 (6) ◽  
Author(s):  
İlhan Öztürk ◽  
Saime Zengin
Keyword(s):  
Difference Equation ◽  
Global Stability ◽  
Equilibrium Point ◽  
Global Attractor ◽  
Initial Conditions ◽  
Real Numbers ◽  
Positive Real ◽  
The Difference

AbstractIn this paper, we investigate the global stability and the periodic nature of solutions of the difference equation $y_{n + 1} = \frac{{\alpha + y_n^p }} {{\beta y_{n - 1}^p }} - \frac{{\gamma + y_{n - 1}^p }} {{\beta y_n^p }},n = 0,1,2,... $ where α, β, γ ∈ (0,∞), α(1 − p) − γ > 0, 0 < p < 1, every y n ≠ 0 for n = −1, 0, 1, 2, … and the initial conditions y−1, y0 are arbitrary positive real numbers. We show that the equilibrium point of the difference equation is a global attractor with a basin that depends on the conditions of the coefficients.

scholarly journals Global behavior of a higher order rational difference equation

Filomat ◽  
2016 ◽  
Vol 30 (12) ◽  
pp. 3265-3276 ◽  
Author(s):  
R. Abo-Zeida
Keyword(s):  
Difference Equation ◽  
Initial Conditions ◽  
Higher Order ◽  
Global Behavior ◽  
Real Numbers ◽  
Positive Real ◽  
All Solutions ◽  
Forbidden Set ◽  
Rational Difference Equation ◽  
The Difference

In this paper, we derive the forbidden set and discuss the global behavior of all solutions of the difference equation xn+1=Axn-k/B-C ?k,i=0 xn-i, n = 0,1,... where A,B,C are positive real numbers and the initial conditions x-k,..., x-1, x0 are real numbers.

scholarly journals Global behavior of a third order difference equation with quadratic term

2021 ◽  
Vol 27 (1) ◽  
Author(s):  
R. Abo-Zeid ◽  
H. Kamal
Keyword(s):  
Difference Equation ◽  
Global Behavior ◽  
Quadratic Term ◽  
Real Numbers ◽  
Third Order ◽  
Order Difference ◽  
Initial Values ◽  
Order Difference Equation ◽  
Admissible Solutions ◽  
The Difference

AbstractIn this paper, we solve and study the global behavior of the admissible solutions of the difference equation $$\begin{aligned} x_{n+1}=\frac{x_{n}x_{n-2}}{-ax_{n-1}+bx_{n-2}}, \quad n=0,1,\ldots , \end{aligned}$$ x n + 1 = x n x n - 2 - a x n - 1 + b x n - 2 , n = 0 , 1 , … , where $$a, b>0$$ a , b > 0 and the initial values $$x_{-2}$$ x - 2 , $$x_{-1}$$ x - 1 , $$x_{0}$$ x 0 are real numbers.

scholarly journals Global behavior of a third order difference equation

2012 ◽  
Vol 43 (3) ◽  
pp. 375-384
Author(s):  
Raafat Abo-zeid
Keyword(s):  
Difference Equation ◽  
Global Stability ◽  
Positive Solutions ◽  
Global Behavior ◽  
Real Numbers ◽  
Third Order ◽  
Order Difference ◽  
Order Difference Equation ◽  
The Difference

The aim of this paper is to investigate the global stability and periodic nature of the positive solutions of the difference equation\[x_{n+1}=\frac{A+Bx_{n-1}}{C+Dx_{n}x_{n-2}},\qquad n=0,1,2,\ldots\] where $A,B$ are nonnegative real numbers and$C, D>0$.

scholarly journals On the Rational Recursive Sequencexn+1=(A+∑i=0kαixn−i)/(B+∑i=0kβixn−i)

2007 ◽  
Vol 2007 ◽  
pp. 1-12 ◽  
Author(s):  
E. M. E. Zayed ◽  
M. A. El-Moneam
Keyword(s):  
Positive Integer ◽  
Difference Equation ◽  
Global Stability ◽  
Positive Solutions ◽  
Initial Conditions ◽  
Integer Number ◽  
Real Numbers ◽  
Positive Real ◽  
Boundedness Character ◽  
The Difference

The main objective of this paper is to study the boundedness character, the periodic character, the convergence, and the global stability of the positive solutions of the difference equationxn+1=(A+∑i=0kαixn−i)/(B+∑i=0kβixn−i), n=0,1,2,…,whereA, B, αi, βiand the initial conditionsx−k,...,x−1,x0are arbitrary positive real numbers, whilekis a positive integer number.

Global behavior of a higher order difference equation

2014 ◽  
Vol 64 (4) ◽  
Author(s):  
R. Abo-Zeid
Keyword(s):  
Difference Equation ◽  
Global Stability ◽  
Positive Solutions ◽  
Higher Order ◽  
Global Behavior ◽  
Real Numbers ◽  
Order Difference ◽  
Order Difference Equation ◽  
The Difference

AbstractThe aim of this paper is to investigate the global stability and periodic nature of the positive solutions of the difference equation $$x_{n + 1} = \frac{{A + Bx_{n - 2k - 1} }} {{C + D\prod\limits_{i = 1}^k {x_{n - 2i} } }}, n = 0,1,2, \ldots ,$$ where A, B are nonnegative real numbers, C,D > 0 and l, k are nonnegative integers such that l ≤ k.

On the global attractivity and the solution of recursive sequence

2010 ◽  
Vol 47 (3) ◽  
pp. 401-418 ◽  
Author(s):  
Elsayed Elsayed
Keyword(s):  
Difference Equation ◽  
Global Attractivity ◽  
Initial Conditions ◽  
Real Numbers ◽  
Positive Real ◽  
Special Cases ◽  
Recursive Sequence ◽  
The Difference

In this paper we study the behavior of the difference equation \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$x_{n + 1} = ax_{n - 2} + \frac{{bx_n x_{n - 2} }}{{cx_n + dx_{n - 3} }},n = 0,1,...$$ \end{document} where the initial conditions x−3 , x−2 , x−1 , x0 are arbitrary positive real numbers and a, b, c, d are positive constants. Also, we give the solution of some special cases of this equation.

scholarly journals On the Recursive Sequencexn=A+xn−kp/xn−1r

2009 ◽  
Vol 2009 ◽  
pp. 1-8 ◽  
Author(s):  
Fangkuan Sun ◽  
Xiaofan Yang ◽  
Chunming Zhang
Keyword(s):  
Difference Equation ◽  
Dynamic Behavior ◽  
Positive Solutions ◽  
Initial Conditions ◽  
Real Numbers ◽  
Positive Real ◽  
The Difference ◽  
The Stability

This paper studies the dynamic behavior of the positive solutions to the difference equationxn=A+xn−kp/xn−1r,n=1,2,…, whereA,p, andrare positive real numbers, and the initial conditions are arbitrary positive numbers. We establish some results regarding the stability and oscillation character of this equation forp∈(0,1).

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