scholarly journals Global Asymptotic Stability for a Fourth-Order Rational Difference Equation

2009 ◽  
Vol 2009 ◽  
pp. 1-7
Author(s):  
Meseret Tuba Gülpinar ◽  
Mustafa Bayram
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Fourth Order ◽  
Positive Equilibrium ◽  
Global Behavior ◽  
Asymptotically Stable ◽  
Nontrivial Solutions ◽  
Globally Asymptotically Stable ◽  
Rational Difference Equation ◽  
Positive Equilibrium Point

Our aim is to investigate the global behavior of the following fourth-order rational difference equation: , where and the initial values . To verify that the positive equilibrium point of the equation is globally asymptotically stable, we used the rule of the successive lengths of positive and negative semicycles of nontrivial solutions of the aforementioned equation.

scholarly journals Global Behavior of a Discrete Survival Model with Several Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Meirong Xu ◽  
Yuzhen Wang
Keyword(s):  
Difference Equation ◽  
Sufficient Conditions ◽  
Survival Model ◽  
Positive Equilibrium ◽  
Global Behavior ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
All Solutions ◽  
The Difference ◽  
Several Delays

The difference equationyn+1−yn=−αyn+∑j=1mβje−γjyn−kjis studied and some sufficient conditions which guarantee that all solutions of the equation are oscillatory, or that the positive equilibrium of the equation is globally asymptotically stable, are obtained.

scholarly journals Dynamics of a Family of Nonlinear Delay Difference Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Qiuli He ◽  
Taixiang Sun ◽  
Hongjian Xi
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Delay Difference Equations ◽  
Delay Difference ◽  
Nonlinear Delay

We study the global asymptotic stability of the following difference equation:xn+1=f(xn-k1,xn-k2,…,xn-ks;xn-m1,xn-m2,…,xn-mt),n=0,1,…,where0≤k1<k2<⋯<ksand0≤m1<m2<⋯<mtwith{k1,k2,…,ks}⋂‍{m1,m2,…,mt}=∅,the initial values are positive, andf∈C(Es+t,(0,+∞))withE∈{(0,+∞),[0,+∞)}. We give sufficient conditions under which the unique positive equilibriumx-of that equation is globally asymptotically stable.

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 of a Nonautonomous Difference Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Mehmet Gümüş ◽  
Özkan Öcalan
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Global Asymptotic Stability ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Initial Values ◽  
Period 2 ◽  
Prime Period

We study the following nonautonomous difference equation:xn+1=(xnxn-1+pn)/(xn+xn-1),n=0,1,…, wherepn>0is a period-2 sequence and the initial valuesx-1,x0∈(0,∞). We show that the unique prime period-2 solution of the equation above is globally asymptotically stable.

scholarly journals GLOBAL ASYMPTOTIC STABILITY FOR GENERAL SYMMETRIC RATIONAL DIFFERENCE EQUATIONS

2009 ◽  
Vol 81 (2) ◽  
pp. 251-259 ◽  
Author(s):  
CONG ZHANG ◽  
HONG-XU LI ◽  
NAN-JING HUANG
Keyword(s):  
Asymptotic Stability ◽  
Difference Equations ◽  
Global Asymptotic Stability ◽  
Global Attractivity ◽  
Positive Equilibrium ◽  
Globally Asymptotically Stable ◽  
Rational Difference Equations ◽  
Rational Difference Equation ◽  
Special Case ◽  
Appl Math

AbstractWe investigate the global asymptotic stability for positive solutions to a class of general symmetric rational difference equations and prove that the unique positive equilibrium 1 of the general symmetric rational difference equations is globally asymptotically stable. As a special case of our result, we solve the conjecture raised by Berenhaut, Foley and Stević [‘The global attractivity of the rational difference equationyn=(yn−k+yn−m)/(1+yn−kyn−m)’,Appl. Math. Lett.20(2007), 54–58].

scholarly journals Global Behavior ofxn+1=(α+βxn-k)/(γ+xn)

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Chenquan Gan ◽  
Xiaofan Yang ◽  
Wanping Liu
Keyword(s):  
Positive Integer ◽  
Difference Equation ◽  
Global Stability ◽  
Initial Conditions ◽  
Global Behavior ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
The Difference

This paper aims to investigate the global stability of negative solutions of the difference equationxn+1=(α+βxn-k)/(γ+xn),n=0,1,2,…, where the initial conditionsx-k,…,x0∈-∞,0,kis a positive integer, and the parametersβ,  γ<0,  α>0. By utilizing the invariant interval and periodic character of solutions, it is found that the unique negative equilibrium is globally asymptotically stable under some parameter conditions. Additionally, two examples are given to illustrate the main results in the end.

scholarly journals Global Behavior of a New Rational Nonlinear Higher-Order Difference Equation

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-4 ◽  
Author(s):  
Wen-Xiu Ma
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Global Asymptotic Stability ◽  
Nonnegative Integer ◽  
Equilibrium Solution ◽  
Higher Order ◽  
Positive Equilibrium ◽  
Global Behavior ◽  
Order Difference ◽  
Order Difference Equation

Let k be a nonnegative integer and c a real number greater than or equal to 1. We present qualitative global behavior of solutions to a rational nonlinear higher-order difference equation zn+1=(czn+zn-k+c-1znzn-k)/(znzn-k+c),  n≥0, with positive initial values z-k,z-k+1,⋯,z0, and show the global asymptotic stability of its positive equilibrium solution.

scholarly journals Dynamics Modeling and Analysis of SIS Epidemic Spreading in Cluster Networks

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Jingjing Tian ◽  
Shuping Li
Keyword(s):  
Equilibrium Point ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Clustering Coefficient ◽  
Positive Equilibrium ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Positive Equilibrium Point ◽  
Sis Epidemic ◽  
The Basic Reproduction Number

In this paper, we propose and study an SIS epidemic model with clustering characteristics based on networks. Using the method of the existence of positive equilibrium point, we obtain the formula of the basic reproduction number R0. Furthermore, by constructing Lyapunov function, we also prove that the disease-free equilibrium of the model is globally asymptotically stable when R0<1. When R0>1, there is only one positive equilibrium point which is globally asymptotically stable. It is also shown that the infection proportion and the basic reproduction number R0 increases as the clustering coefficient increases when the average degree of networks is fixed.

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