scholarly journals Global asymptotic behavior of the difference equation yn+1=α⋅e−(nyn+(n−k)yn−k)β+nyn+(n−k)yn−k

2009 ◽  
Vol 22 (4) ◽  
pp. 595-599 ◽  
Author(s):  
I. Ozturk ◽  
F. Bozkurt ◽  
S. Ozen
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
The Difference

scholarly journals On the Asymptotic Behavior of a Difference Equation with Maximum

2008 ◽  
Vol 2008 ◽  
pp. 1-6 ◽  
Author(s):  
Fangkuan Sun
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Positive Solution ◽  
Positive Solutions ◽  
The Difference

We study the asymptotic behavior of positive solutions to the difference equationxn=max{A/xn-1α,B/xn−2β},n=0,1,…,where0<α, β<1, A,B>0. We prove that every positive solution to this equation converges tox∗=max{A1/(α+1),B1/(β+1)}.

scholarly journals Global Asymptotic Stability of a Rational System

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Lin-Xia Hu ◽  
Xiu-Mei Jia
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Stability ◽  
Difference Equation ◽  
Positive Equilibrium ◽  
Asymptotically Stable ◽  
Major Conclusion ◽  
Initial Value ◽  
Globally Asymptotically Stable ◽  
Rational System ◽  
The Difference

The main goal of this paper is to investigate the global asymptotic behavior of the difference equationxn+1=β1xn/A1+yn,yn+1=β2xn+γ2yn/xn+yn,n=0,1,2,…withβ1,β2,γ2,A1∈(0,∞)and the initial value(x0,y0)∈[0,∞)×[0,∞)such thatx0+y0≠0. The major conclusion shows that, in the case whereγ2<β2, if the unique positive equilibrium(x-,y-)exists, then it is globally asymptotically stable.

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 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 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 Asymptotic properties of solutions of third order difference equations

2020 ◽  
Vol 14 (1) ◽  
pp. 1-19
Author(s):  
Janusz Migda ◽  
Małgorzata Migda ◽  
Magdalena Nockowska-Rosiak
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Asymptotic Properties ◽  
Harmonic Approximation ◽  
Sufficient Conditions ◽  
Degree Of Approximation ◽  
Third Order ◽  
Order Difference ◽  
Geometric Approximation ◽  
The Difference

We consider the difference equation of the form ?(rn?(pn?xn)) = anf (x?(n)) + bn. We present sufficient conditions under which, for a given solution y of the equation ?(rn?(pn?yn)) = 0, there exists a solution x of the nonlinear equation with the asymptotic behavior xn = yn + zn, where z is a sequence convergent to zero. Our approach allows us to control the degree of approximation, i.e., the rate of convergence of the sequence We examine two types of approximation: harmonic approximation when zn = o(ns), s ? 0, and geometric approximation when zn = o(?n), ? ? (0, 1).

scholarly journals On the Global Attractivity of a Max-Type Difference Equation

2009 ◽  
Vol 2009 ◽  
pp. 1-5 ◽  
Author(s):  
Ali Gelişken ◽  
Cengiz Çinar
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Positive Solution ◽  
Positive Solutions ◽  
Global Attractivity ◽  
Period 2 ◽  
The Difference

We investigate asymptotic behavior and periodic nature of positive solutions of the difference equation , where and . We prove that every positive solution of this difference equation approaches or is eventually periodic with period 2.

scholarly journals On Boundedness of Solutions of the Difference Equation xn+1=(pxn+qxn−1)/(1+xn) for q>1+p>1

2009 ◽  
Vol 2009 ◽  
pp. 1-11 ◽  
Author(s):  
Hongjian Xi ◽  
Taixiang Sun ◽  
Weiyong Yu ◽  
Jinfeng Zhao
Keyword(s):  
Difference Equation ◽  
Boundedness Of Solutions ◽  
The Difference
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