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.

Periodic and asymptotic behavior of a difference equation

2019 ◽  
Vol 12 (06) ◽  
pp. 2040004
Author(s):  
Murat Ari ◽  
Ali Geli̇şken
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Closed Form ◽  
Real Numbers ◽  
Numerical Examples ◽  
Initial Values

In this paper, the closed form of solutions given for the following difference equation [Formula: see text] where [Formula: see text] and the initial values [Formula: see text] are real numbers. We investigate periodic and asymptotic behavior of this equation. Also, some numerical examples are given.

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 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 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 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 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 a class of solvable higher-order difference equations

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 461-477 ◽  
Author(s):  
Stevo Stevic ◽  
Mohammed Alghamdi ◽  
Abdullah Alotaibi ◽  
Elsayed Elsayed
Keyword(s):  
Difference Equation ◽  
Closed Form ◽  
Difference Equations ◽  
Higher Order ◽  
Real Numbers ◽  
Order Difference ◽  
Initial Values ◽  
Long Term Behavior

Closed form formulas for well-defined solutions of the next difference equation xn = xn-2xn-k-2/xn-k(an + bnxn-2xn-k-2), n ? N0, where k ? N, (an)n?N0, (bn)n?N0, and initial values x-i, i = 1,k+2 are real numbers, are given. Long-term behavior of well-defined solutions of the equation when (an)n?N0 and (bn)n?N0 are constant sequences is described in detail by using the formulas. We also describe the domain of undefinable solutions of the equation. Our results explain and considerably improve some recent results in the literature.

scholarly journals Closed-Form Solution of a Rational Difference Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Tarek F. Ibrahim ◽  
Abdul Qadeer Khan ◽  
Burak Oğul ◽  
Dağistan Şimşek
Keyword(s):  
Difference Equation ◽  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Real Numbers ◽  
Positive Real ◽  
Rational Difference Equation ◽  
The Difference

In this paper, we study the solution of the difference equation Ω m + 1 = Ω m − 7 q + 6 / 1 + ∏ t = 0 5 Ω m − q + 1 t − q , where the initials are positive real numbers.

scholarly journals On a Discrete Epidemic Model

2007 ◽  
Vol 2007 ◽  
pp. 1-10 ◽  
Author(s):  
Stevo Stevic
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Positive Solutions ◽  
Epidemic Model ◽  
Asymptotic Behavior Of Solutions ◽  
Negative Numbers ◽  
Initial Values ◽  
Discrete Epidemic Model ◽  
The Difference

We investigate the global asymptotic behavior of solutions of the difference equationxn+1=(1−∑j=0k−1xn−j)(1−e−Axn),n∈ℕ0, whereA∈(0,∞),k∈{2,3,…}, and the initial valuesx−k+1,x−k+2,…,x0are arbitrary negative numbers. Asymptotics of some positive solutions of the equation are also found.

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}.

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