Well-defined solutions of the difference equation xn = xn−3kxn−4kxn−5k xn−kxn−2k(±1±xn−3kxn−4kxn−5k)

2019 ◽  
Vol 12 (06) ◽  
pp. 2040016
Author(s):  
Güven Çi̇nar ◽  
Ali̇ Geli̇şken ◽  
Ozan Özkan
Keyword(s):  
Difference Equation ◽  
Initial Conditions ◽  
Numerical Results ◽  
Real Numbers ◽  
The Difference

We investigate the behavior of well-defined solutions of the difference equation [Formula: see text] where the initial conditions [Formula: see text], [Formula: see text] are arbitrary nonzero real numbers. Also, we give some special results and numerical results.

scholarly journals Existence of a Period-Two Solution in Linearizable Difference Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
E. J. Janowski ◽  
M. R. S. Kulenović
Keyword(s):  
Difference Equation ◽  
Linear Equation ◽  
Difference Equations ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Minimal Period ◽  
Real Numbers ◽  
Existence And Nonexistence ◽  
The Difference ◽  
Necessary And Sufficient

Consider the difference equationxn+1=f(xn,…,xn−k),n=0,1,…,wherek∈{1,2,…}and the initial conditions are real numbers. We investigate the existence and nonexistence of the minimal period-two solution of this equation when it can be rewritten as the nonautonomous linear equationxn+l=∑i=1−lkgixn−i,n=0,1,…,wherel,k∈{1,2,…}and the functionsgi:ℝk+l→ℝ. We give some necessary and sufficient conditions for the equation to have a minimal period-two solution whenl=1.

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 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 Global Asymptotic Stability for Linear Fractional Difference Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
A. Brett ◽  
E. J. Janowski ◽  
M. R. S. Kulenović
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Stability ◽  
Difference Equation ◽  
Global Asymptotic Stability ◽  
Initial Conditions ◽  
Positive Equilibrium ◽  
Real Numbers ◽  
Fractional Difference ◽  
Fractional Difference Equation ◽  
The Difference

Consider the difference equation xn+1=(α+∑i=0kaixn-i)/(β+∑i=0kbixn-i),  n=0,1,…, where all parameters α,β,ai,bi,  i=0,1,…,k, and the initial conditions xi,  i∈{-k,…,0} are nonnegative real numbers. We investigate the asymptotic behavior of the solutions of the considered equation. We give easy-to-check conditions for the global stability and global asymptotic stability of the zero or positive equilibrium of this equation.

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 Solutions of the Difference Equation

2010 ◽  
Vol 2010 ◽  
pp. 1-13 ◽  
Author(s):  
Candace M. Kent ◽  
Witold Kosmala ◽  
Michael A. Radin ◽  
Stevo Stević
Keyword(s):  
Difference Equation ◽  
Initial Conditions ◽  
Boundedness Of Solutions ◽  
Real Numbers ◽  
Unbounded Solutions ◽  
The Difference ◽  
Long Term Behavior

Our goal in this paper is to investigate the long-term behavior of solutions of the following difference equation: , where the initial conditions and are real numbers. We examine the boundedness of solutions, periodicity of solutions, and existence of unbounded solutions and how these behaviors depend on initial conditions.

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 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.

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