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.

scholarly journals On the Rational Recursive Sequence

2008 ◽  
Vol 2008 ◽  
pp. 1-15 ◽  
Author(s):  
E. M. E. Zayed ◽  
A. B. Shamardan ◽  
T. A. Nofal
Keyword(s):  
Difference Equation ◽  
Global Stability ◽  
Global Attractor ◽  
Positive Solutions ◽  
Initial Conditions ◽  
Positive Equilibrium ◽  
Real Numbers ◽  
Recursive Sequence ◽  
Boundedness Character ◽  
The Difference

We study the global stability, the periodic character, and the boundedness character of the positive solutions of the difference equation , , in the two cases: (i) ; (ii) , where the coefficients and, and the initial conditions are real numbers. We show that the positive equilibrium of this equation is a global attractor with a basin that depends on certain conditions posed on the coefficients 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).

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 On the Recursive Sequencexn+1=A+xnp/xn−1r

2007 ◽  
Vol 2007 ◽  
pp. 1-9 ◽  
Author(s):  
Stevo Stevic
Keyword(s):  
Difference Equation ◽  
Positive Solutions ◽  
Analogous Result ◽  
Real Numbers ◽  
Positive Real ◽  
Unbounded Solutions ◽  
Boundedness Character ◽  
The Difference

The paper considers the boundedness character of positive solutions of the difference equationxn+1=A+xnp/xn−1r,n∈ℕ0, whereA,p, andrare positive real numbers. It is shown that (a) Ifp2≥4r>4, orp≥1+r,r≤1, then this equation has positive unbounded solutions; (b) ifp2<4r, or2r≤p<1+r,r∈(0,1), then all positive solutions of the equation are bounded. Also, an analogous result is proved regarding positive solutions of the max type difference equationxn+1=max{A,xnp/xn−1r}, whereA,p,q∈(0,∞).

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.

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.

scholarly journals Dynamics of a Higher-Order System of Difference Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
Qi Wang ◽  
Qinqin Zhang ◽  
Qirui Li
Keyword(s):  
Positive Integer ◽  
Positive Solutions ◽  
Difference Equations ◽  
Initial Conditions ◽  
Higher Order ◽  
Order System ◽  
Real Numbers ◽  
Precise Description ◽  
System Of Difference Equations ◽  
Positive Real

Consider the following system of difference equations:xn+1(i)=xn-m+1(i)/Ai∏j=0m-1xn-j(i+j+1)+αi,xn+1(i+m)=xn+1(i),x1-l(i+l)=ai,l,Ai+m=Ai,αi+m=αi,i,l=1,2,…,m;n=0,1,2,…,wheremis a positive integer,Ai,αi,i=1,2,…,m, and the initial conditionsai,l,i,l=1,2,…,m,are positive real numbers. We obtain the expressions of the positive solutions of the system and then give a precise description of the convergence of the positive solutions. Finally, we give some numerical results.

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 On the Dynamics of the Recursive Sequence

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Mehmet Gümüş ◽  
Özkan Öcalan ◽  
Nilüfer B. Felah
Keyword(s):  
Periodic Solution ◽  
Difference Equation ◽  
Periodic Solutions ◽  
Positive Solutions ◽  
Initial Conditions ◽  
Recursive Sequence ◽  
Boundedness Character ◽  
The Difference

We investigate the boundedness character, the oscillatory, and the periodic character of positive solutions of the difference equation , where , , and the initial conditions are arbitrary positive numbers. We investigate the boundedness character for . Also, we investigate the existence of a prime two periodic solution for is odd. Moreover, when is even, we prove that there are no prime two periodic solutions of the equation above.

scholarly journals Global Behavior ofxn+1=(α+βxn-k)/(γ+xn)

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Chenquan Gan ◽  
Xiaofan Yang ◽  
Wanping Liu
Keyword(s):  
Positive Integer ◽  
Difference Equation ◽  
Global Stability ◽  
Initial Conditions ◽  
Global Behavior ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
The Difference

This paper aims to investigate the global stability of negative solutions of the difference equationxn+1=(α+βxn-k)/(γ+xn),n=0,1,2,…, where the initial conditionsx-k,…,x0∈-∞,0,kis a positive integer, and the parametersβ,  γ<0,  α>0. By utilizing the invariant interval and periodic character of solutions, it is found that the unique negative equilibrium is globally asymptotically stable under some parameter conditions. Additionally, two examples are given to illustrate the main results in the end.

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