scholarly journals Stability of Hyperbolic Equilibrium Solution of Second Order Nonlinear Rational Difference Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-21 ◽  
Author(s):  
S. Atawna ◽  
R. Abu-Saris ◽  
E. S. Ismail ◽  
I. Hashim
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Global Asymptotic Stability ◽  
Equilibrium Solution ◽  
Initial Conditions ◽  
Second Order ◽  
Real Numbers ◽  
Positive Real ◽  
Rational Difference Equation ◽  
Hyperbolic Equilibrium

Our goal in this paper is to investigate the global asymptotic stability of the hyperbolic equilibrium solution of the second order rational difference equation xn+1=α+βxn+γxn-1/A+Bxn+Cxn-1, n=0,1,2,…, where the parameters A≥0 and B, C, α, β, γ are positive real numbers and the initial conditions x-1, x0 are nonnegative real numbers such that A+Bx0+Cx-1>0. In particular, we solve Conjecture 5.201.1 proposed by Camouzis and Ladas in their book (2008) which appeared previously in Conjecture 11.4.2 in Kulenović and Ladas monograph (2002).

scholarly journals Stability of Nonhyperbolic Equilibrium Solution of Second Order Nonlinear Rational Difference Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-12
Author(s):  
S. Atawna ◽  
R. Abu-Saris ◽  
E. S. Ismail ◽  
I. Hashim
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Global Asymptotic Stability ◽  
Equilibrium Solution ◽  
Initial Conditions ◽  
Second Order ◽  
Real Numbers ◽  
Positive Real ◽  
Rational Difference Equation ◽  
Hyperbolic Equilibrium

This is a continuation part of our investigation in which the second order nonlinear rational difference equation xn+1=(α+βxn+γxn-1)/(A+Bxn+Cxn-1), n=0,1,2,…, where the parameters A≥0 and B, C, α, β, γ are positive real numbers and the initial conditions x-1, x0 are nonnegative real numbers such that A+Bx0+Cx-1>0, is considered. The first part handled the global asymptotic stability of the hyperbolic equilibrium solution of the equation. Our concentration in this part is on the global asymptotic stability of the nonhyperbolic equilibrium solution of the equation.

scholarly journals Global Asymptotic Stability and Naimark-Sacker Bifurcation of Certain Mix Monotone Difference Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-22
Author(s):  
M. R. S. Kulenović ◽  
S. Moranjkić ◽  
M. Nurkanović ◽  
Z. Nurkanović
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Global Asymptotic Stability ◽  
Initial Conditions ◽  
Real Numbers ◽  
Positive Real ◽  
Local Asymptotic Stability ◽  
Stable Periodic ◽  
Rational Difference Equation ◽  
Unknown Period

We investigate the global asymptotic stability of the following second order rational difference equation of the form xn+1=Bxnxn-1+F/bxnxn-1+cxn-12,  n=0,1,…, where the parameters B, F, b, and c and initial conditions x-1 and x0 are positive real numbers. The map associated with this equation is always decreasing in the second variable and can be either increasing or decreasing in the first variable depending on the parametric space. In some cases, we prove that local asymptotic stability of the unique equilibrium point implies global asymptotic stability. Also, we show that considered equation exhibits the Naimark-Sacker bifurcation resulting in the existence of the locally stable periodic solution of unknown period.

scholarly journals Boundedness of solutions and stability of certain second-order difference equation with quadratic term

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Emin Bešo ◽  
Senada Kalabušić ◽  
Naida Mujić ◽  
Esmir Pilav
Keyword(s):  
Difference Equation ◽  
Initial Conditions ◽  
Second Order ◽  
Quadratic Term ◽  
Real Numbers ◽  
Positive Real ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
Rational Difference Equation

AbstractWe consider the second-order rational difference equation $$ {x_{n+1}=\gamma +\delta \frac{x_{n}}{x^{2}_{n-1}}}, $$xn+1=γ+δxnxn−12, where γ, δ are positive real numbers and the initial conditions $x_{-1}$x−1 and $x_{0}$x0 are positive real numbers. Boundedness along with global attractivity and Neimark–Sacker bifurcation results are established. Furthermore, we give an asymptotic approximation of the invariant curve near the equilibrium point.

scholarly journals Asymptotic Stability for a Class of Nonlinear Difference Equations

2010 ◽  
Vol 2010 ◽  
pp. 1-10 ◽  
Author(s):  
Chang-you Wang ◽  
Shu Wang ◽  
Zhi-wei Wang ◽  
Fei Gong ◽  
Rui-fang Wang
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Difference Equations ◽  
Global Asymptotic Stability ◽  
Initial Conditions ◽  
Real Numbers ◽  
Nonlinear Difference Equations ◽  
Positive Real ◽  
Fractional Difference ◽  
Fractional Difference Equation

We study the global asymptotic stability of the equilibrium point for the fractional difference equationxn+1=(axn-lxn-k)/(α+bxn-s+cxn-t),n=0,1,…, where the initial conditionsx-r,x-r+1,…,x1,x0are arbitrary positive real numbers of the interval(0,α/2a),l,k,s,tare nonnegative integers,r=max⁡⁡{l,k,s,t}andα,a,b,care positive constants. Moreover, some numerical simulations are given to illustrate our results.

scholarly journals Global Stability of a Rational Difference Equation

2010 ◽  
Vol 2010 ◽  
pp. 1-17 ◽  
Author(s):  
Guo-Mei Tang ◽  
Lin-Xia Hu ◽  
Gang Ma
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Global Stability ◽  
Positive Solutions ◽  
Open Problem ◽  
Global Asymptotic Stability ◽  
Initial Conditions ◽  
Nonlinear Difference Equation ◽  
Real Numbers ◽  
Rational Difference Equation

We consider the higher-order nonlinear difference equation with the parameters, and the initial conditions are nonnegative real numbers. We investigate the periodic character, invariant intervals, and the global asymptotic stability of all positive solutions of the above-mentioned equation. In particular, our results solve the open problem introduced by Kulenović and Ladas in their monograph (see Kulenović and Ladas, 2002).

scholarly journals On the Dynamics of a Higher-Order Rational Difference Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-8 ◽  
Author(s):  
A. M. Ahmed
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Global Asymptotic Stability ◽  
Higher Order ◽  
Real Numbers ◽  
Rational Difference Equation

The aim of this paper is to investigate the global asymptotic stability and the periodic character for the rational difference equationxn+1=αxn-1/(β+γΠi=lkxn-2ipi),  n=0,1,2,…, where the parametersα,β,γ,pl,pl+1,…,pkare nonnegative real numbers, andl,kare nonnegative integers such thatl≤k.

scholarly journals Qualitative Behavior of Rational Difference Equation of Big Order

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
M. M. El-Dessoky
Keyword(s):  
Global Convergence ◽  
Difference Equation ◽  
Initial Conditions ◽  
Qualitative Behavior ◽  
Real Numbers ◽  
Positive Real ◽  
Recursive Sequence ◽  
Rational Difference Equation

We investigate the global convergence, boundedness, and periodicity of solutions of the recursive sequencexn+1=axn-l+bxn-x/c+dxn-lxn-k,n=0,1,…,where the parametersa,  b,  c,anddare positive real numbers, and the initial conditionsx-t,x-t+1,…,x-1andx0are positive real numbers wheret=maxk,l.

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 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.

Sign in / Sign up

Export Citation Format

Share Document