scholarly journals On the Difference Equationxn=anxn-k/(bn+cnxn-1⋯xn-k)

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Stevo Stević ◽  
Josef Diblík ◽  
Bratislav Iričanin ◽  
Zdeněk Šmarda
Keyword(s):  
Difference Equation ◽  
Real Numbers ◽  
Initial Values ◽  
The Difference

The behavior of well-defined solutions of the difference equationxn=anxn-k/(bn+cnxn-1⋯xn-k), n∈ℕ0, wherek∈ℕis fixed, the sequencesan,bnandcnare real,(bn,cn)≠(0,0),n∈ℕ0, and the initial valuesx-k,…,x-1are real numbers, is described.

scholarly journals On the Difference Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Candace M. Kent ◽  
Witold Kosmala ◽  
Stevo Stević
Keyword(s):  
Difference Equation ◽  
Real Numbers ◽  
Initial Values ◽  
The Difference ◽  
Long Term Behavior

The long-term behavior of solutions of the following difference equation: , , where the initial values , , are real numbers, is investigated in the paper.

scholarly journals Long-Term Behavior of Solutions of the Difference Equation

2010 ◽  
Vol 2010 ◽  
pp. 1-17 ◽  
Author(s):  
Candace M. Kent ◽  
Witold Kosmala ◽  
Stevo Stević
Keyword(s):  
Difference Equation ◽  
Real Numbers ◽  
Initial Values ◽  
The Difference ◽  
Long Term Behavior

We investigate the long-term behavior of solutions of the following difference equation: , , where the initial values , , and are real numbers. Numerous fascinating properties of the solutions of the equation are presented.

scholarly journals On the Difference Equation

2008 ◽  
Vol 2008 ◽  
pp. 1-8 ◽  
Author(s):  
İbrahim Yalçinkaya
Keyword(s):  
Difference Equation ◽  
Higher Order ◽  
Real Numbers ◽  
Positive Real ◽  
Global Behaviour ◽  
Initial Values ◽  
The Difference

We investigate the global behaviour of the difference equation of higher order , where the parameters and the initial values and are arbitrary positive real numbers.

scholarly journals On the Difference Equationxn+1=xnxn-k/(xn-k+1a+bxnxn-k)

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Stevo Stević ◽  
Josef Diblík ◽  
Bratislav Iričanin ◽  
Zdenĕk Šmarda
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Closed Form ◽  
Real Numbers ◽  
Initial Values ◽  
The Difference

We show that the difference equationxn+1=xnxn-k/xn-k+1(a+bxnxn-k),n∈ℕ0, wherek∈ℕ, the parametersa,band initial valuesx-i,i=0,k̅are real numbers, can be solved in closed form considerably extending the results in the literature. By using obtained formulae, we investigate asymptotic behavior of well-defined solutions of the equation.

scholarly journals Unbounded Solutions of the Difference Equationxn=xn−lxn−k−1

2011 ◽  
Vol 2011 ◽  
pp. 1-8
Author(s):  
Stevo Stević ◽  
Bratislav Iričanin
Keyword(s):  
Difference Equation ◽  
General Result ◽  
General Character ◽  
Real Numbers ◽  
Unbounded Solutions ◽  
Initial Values ◽  
Easy Problem ◽  
The Difference ◽  
Long Term Behavior

The following difference equationxn=xn−lxn−k−1,n∈ℕ0, wherek,l∈ℕ,k<l,gcd(k,l)=1, and the initial valuesx-l,…,x-2,x-1are real numbers, has been investigated so far only for some particular values ofkandl. To get any general result on the equation is turned out as a not so easy problem. In this paper, we give the first result on the behaviour of solutions of the difference equation of general character, by describing the long-term behavior of the solutions of the equation for all values of parameterskandl, where the initial values satisfy the following conditionmin{x-l,…,x-2,x-1}.

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 the Difference Equationxn+1=xn-1g(xn)

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Hongjian Xi ◽  
Taixiang Sun ◽  
Bin Qin ◽  
Hui Wu
Keyword(s):  
Continuous Function ◽  
Difference Equation ◽  
Positive Solution ◽  
Positive Solutions ◽  
Initial Conditions ◽  
Global Behavior ◽  
Initial Values ◽  
The Difference ◽  
Surjective Function

We consider the following difference equationxn+1=xn-1g(xn),n=0,1,…,where initial valuesx-1,x0∈[0,+∞)andg:[0,+∞)→(0,1]is a strictly decreasing continuous surjective function. We show the following. (1) Every positive solution of this equation converges toa,0,a,0,…,or0,a,0,a,…for somea∈[0,+∞). (2) Assumea∈(0,+∞). Then the set of initial conditions(x-1,x0)∈(0,+∞)×(0,+∞)such that the positive solutions of this equation converge toa,0,a,0,…,or0,a,0,a,…is a unique strictly increasing continuous function or an empty set.

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.

Sign in / Sign up

Export Citation Format

Share Document