scholarly journals On Global Attractivity of a Class of Nonautonomous Difference Equations

2010 ◽  
Vol 2010 ◽  
pp. 1-13 ◽  
Author(s):  
Wanping Liu ◽  
Xiaofan Yang ◽  
Jianqiu Cao
Keyword(s):  
Positive Solution ◽  
Difference Equations ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Higher Order ◽  
Positive Equilibrium ◽  
Global Behavior ◽  
Open Problems ◽  
The Family ◽  
Recursive Equations

We mainly investigate the global behavior to the family of higher-order nonautonomous recursive equations given byyn=(p+ryn-s)/(q+ϕn(yn-1,yn-2,…,yn-m)+yn-s),n∈ℕ0, withp≥0,r,q>0,s,m∈ℕand positive initial values, and present some sufficient conditions for the parameters and mapsϕn:(ℝ+)m→ℝ+,n∈ℕ0, under which every positive solution to the equation converges to zero or a unique positive equilibrium. Our main result in the paper extends some related results from the work of Gibbons et al. (2000), Iričanin (2007), and Stević (vol. 33, no. 12, pages 1767–1774, 2002; vol. 6, no. 3, pages 405–414, 2002; vol. 9, no. 4, pages 583–593, 2005). Besides, several examples and open problems are presented in the end.

scholarly journals Global attractivity for a nonautonomous Nicholson’s equation with mixed monotonicities

Nonlinearity ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 589-607
Author(s):  
Teresa Faria ◽  
Henrique C Prates
Keyword(s):  
Local Stability ◽  
Schwarzian Derivative ◽  
Global Attractivity ◽  
Recent Literature ◽  
Sufficient Conditions ◽  
Time Dependent ◽  
Positive Equilibrium ◽  
Time Varying ◽  
Monotone Functions ◽  
Open Problems

Abstract We consider a Nicholson’s equation with multiple pairs of time-varying delays and nonlinear terms given by mixed monotone functions. Sufficient conditions for the permanence, local stability and global attractivity of its positive equilibrium K are established. The main novelty here is the construction of a suitable auxiliary difference equation x n+1 = h(x n ) with h having negative Schwarzian derivative, and its application to derive the attractivity of K for a model with one or more pairs of time-dependent delays. Our criteria depend on the size of some delays, improve results in recent literature and provide answers to open problems.

scholarly journals Oscillatory behavior of nonlinear Hilfer fractional difference equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Tuğba Yalçın Uzun
Keyword(s):  
Difference Equations ◽  
Sufficient Conditions ◽  
Higher Order ◽  
Oscillatory Behavior ◽  
Fractional Difference ◽  
Fractional Difference Equations ◽  
All Solutions ◽  
Alpha Beta

AbstractIn this paper, we study the oscillation behavior for higher order nonlinear Hilfer fractional difference equations of the type $$\begin{aligned}& \Delta _{a}^{\alpha ,\beta }y(x)+f_{1} \bigl(x,y(x+\alpha ) \bigr) =\omega (x)+f_{2} \bigl(x,y(x+ \alpha ) \bigr),\quad x\in \mathbb{N}_{a+n-\alpha }, \\& \Delta _{a}^{k-(n-\gamma )}y(x) \big|_{x=a+n-\gamma } = y_{k}, \quad k= 0,1,\ldots,n, \end{aligned}$$ Δ a α , β y ( x ) + f 1 ( x , y ( x + α ) ) = ω ( x ) + f 2 ( x , y ( x + α ) ) , x ∈ N a + n − α , Δ a k − ( n − γ ) y ( x ) | x = a + n − γ = y k , k = 0 , 1 , … , n , where $\lceil \alpha \rceil =n$ ⌈ α ⌉ = n , $n\in \mathbb{N}_{0}$ n ∈ N 0 and $0\leq \beta \leq 1$ 0 ≤ β ≤ 1 . We introduce some sufficient conditions for all solutions and give an illustrative example for our results.

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 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 Boundedness and Global Attractivity of a Higher-Order Nonlinear Difference Equation

2010 ◽  
Vol 2010 ◽  
pp. 1-17
Author(s):  
Xiu-Mei Jia ◽  
Wan-Tong Li
Keyword(s):  
Difference Equation ◽  
Global Attractor ◽  
Positive Solutions ◽  
Local Stability ◽  
Global Attractivity ◽  
Initial Conditions ◽  
Higher Order ◽  
Nonlinear Difference Equation ◽  
Positive Equilibrium ◽  
Prime Period

We investigate the local stability, prime period-two solutions, boundedness, invariant intervals, and global attractivity of all positive solutions of the following difference equation: , , where the parameters and the initial conditions . We show that the unique positive equilibrium of this equation is a global attractor under certain conditions.

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 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 On Somek-Dimensional Cyclic Systems of Difference Equations

2010 ◽  
Vol 2010 ◽  
pp. 1-11 ◽  
Author(s):  
Wanping Liu ◽  
Xiaofan Yang ◽  
Bratislav D. Iričanin
Keyword(s):  
Difference Equations ◽  
Global Attractivity ◽  
Transformation Method ◽  
Higher Order ◽  
Order Difference ◽  
Cyclic Systems ◽  
First Time

Motivated by Iričanin and Stević's paper (2006) in which for the first time were considered some cyclic systems of difference equations, here we study the global attractivity of some nonlineark-dimensional cyclic systems of higher-order difference equations. To do this, we use the transformation method from Berenhaut et al. (2007) and Berenhaut and Stević (2007). The main results in this paper also extend our recent results in the work of (Liu and Yang 2010, in press).

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