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 Oscillation of nonlinear delay difference equations

2001 ◽  
Vol 28 (5) ◽  
pp. 301-306 ◽  
Author(s):  
Jianchu Jiang
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Delay Difference Equation ◽  
Oscillation Criteria ◽  
Delay Difference Equations ◽  
Delay Difference ◽  
Nonlinear Delay

We obtain some oscillation criteria for solutions of the nonlinear delay difference equation of the formxn+1−xn+pn∏j=1mxn−kjαj=0.

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 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 Stability and global attractivity for a class of nonlinear delay difference equations

2005 ◽  
Vol 2005 (3) ◽  
pp. 227-234 ◽  
Author(s):  
Binxiang Dai ◽  
Na Zhang
Keyword(s):  
Difference Equations ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Delay Equation ◽  
Stability Properties ◽  
The Stability ◽  
Delay Difference Equations ◽  
Delay Difference ◽  
Nonlinear Delay

A class of nonlinear delay difference equations are considered and some sufficient conditions for global attractivity of solutions of the equation are obtained. It is shown that the stability properties, both local and global, of the equilibrium of the delay equation can be derived from those of an associated nondelay equation.

scholarly journals Some New Oscillation Criteria for Fourth-Order Nonlinear Delay Difference Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Kandasamy Alagesan ◽  
Subaramaniyam Jaikumar ◽  
Govindasamy Ayyappan
Keyword(s):  
Difference Equation ◽  
Fourth Order ◽  
Difference Equations ◽  
Oscillatory Behavior ◽  
Delay Difference Equation ◽  
Oscillation Criteria ◽  
Delay Difference Equations ◽  
Delay Difference ◽  
Nonlinear Delay

In this paper, the authors studied oscillatory behavior of solutions of fourth-order delay difference equation Δc3nΔc2nΔc1nΔun+pnfun−k=0 under the conditions ∑n=n0∞cin<∞, i=1, 2, 3. New oscillation criteria have been obtained which greatly reduce the number of conditions required for the studied equation. Some examples are presented to show the strength and applicability of the main results.

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 Stability conditions for linear delay difference equations: a survey and perspectives

2015 ◽  
Vol 63 (1) ◽  
pp. 1-29 ◽  
Author(s):  
Jan Čermák
Keyword(s):  
Asymptotic Stability ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Stability Conditions ◽  
Linear Delay ◽  
Necessary And Sufficient ◽  
Delay Difference Equations ◽  
Delay Difference ◽  
Theoretical Results

Abstract The paper presents an overview of the basic results and methods for stability investigations of higher-order linear autonomous difference equations. The presented criteria formulate several types of necessary and sufficient conditions for the asymptotic stability of the zero solution of studied equations, with a special emphasize put on delay difference equations. Various comments, comparisons, examples and illustrations are given to support theoretical results.

Improved oscillation criteria for second order nonlinear delay difference equations with non-positive neutral term

2016 ◽  
Vol 56 (1) ◽  
pp. 155-165 ◽  
Author(s):  
E. Thandapani ◽  
S. Selvarangam ◽  
R. Rama ◽  
M. Madhan
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Second Order ◽  
Delay Difference Equation ◽  
Oscillation Criteria ◽  
Neutral Term ◽  
Positive Integers ◽  
Delay Difference Equations ◽  
Delay Difference ◽  
Nonlinear Delay

Abstract In this paper, we present some oscillation criteria for second order nonlinear delay difference equation with non-positive neutral term of the form $$\Delta (a_n (\Delta z_n )^\alpha ) + q_n f(x_{n - \sigma } ) = 0,\;\;\;n \ge n_0 > 0,$$ where zn = xn − pnxn−τ, and α is a ratio of odd positive integers. Examples are provided to illustrate the results. The results obtained in this paper improve and complement to some of the existing results.

scholarly journals Oscillation and nonoscillation of nonlinear neutral delay difference equations

2007 ◽  
Vol 38 (4) ◽  
pp. 323-333 ◽  
Author(s):  
E. Thandapani ◽  
P. Mohan Kumar
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Second Order ◽  
Delay Difference Equation ◽  
Neutral Delay ◽  
Delay Difference Equations ◽  
Delay Difference ◽  
Oscillation And Nonoscillation

In this paper, the authors establish some sufficient conditions for oscillation and nonoscillation of the second order nonlinear neutral delay difference equation$$ \Delta^2 (x_n-p_nx_{n-k}) + q_nf(x_{n-\ell}) = 0, ~~n \ge n_0 $$where $ \{p_n\} $ and $ \{q_n\} $ are non-negative sequences with $ 0$

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