Global stability of a linear nonautonomous delay difference equation<sup>1</sup>

Erbe, Lynn; Xia, Hao; Yu, Jing

doi:10.1080/10236199508808016

Cited by 82 publications

(31 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The initial function associated with (1.1) takes the form x(t) = ψ(t) for t ∈ [τ(t 0 ),t 0 ], where ψ is rd-continuous on [τ(t 0 ),t 0 ]. Equation (1.1) is studied extensively by Qian and Sun [13] in the case when T = R. See also related discussions on unforced delay equations by Matsunaga et al [12] in the continuous case, and by Erbe et al [6] or Zhang and Yan [14] 2 Forced delay dynamic equation in the discrete case. Other papers on delay dynamic equations include [1][2][3].…”

Section: Delay Dynamic Equation With Forcing Termmentioning

confidence: 99%

Boundedness and vanishing of solutions for a forced delay dynamic equation

Anderson¹

2006

Adv. Differ Equ.

View full text Add to dashboard Cite

We give conditions under which all solutions of a time-scale first-order nonlinear variable-delay dynamic equation with forcing term are bounded and vanish at infinity, for arbitrary time scales that are unbounded above. A nontrivial example illustrating an application of the results is provided.Copyright © 2006 Douglas R. Anderson. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Delay dynamic equation with forcing termFollowing Hilger's landmark paper [8], a rapidly expanding body of literature has sought to unify, extend, and generalize ideas from discrete calculus, quantum calculus, and continuous calculus to arbitrary time-scale calculus, where a time scale is simply any nonempty closed set of real numbers. This paper illustrates this new understanding by extending some continuous results from differential equations to dynamic equations on time scales, thus including as corollaries difference equations and q-difference equations. Throughout this work, we consider the nonlinear forced delay dynamic equationwhere T is a time scale unbounded above, f : R → R is continuous, and the functions p : T → (0,∞) and r : T → R are both right-dense continuous. Moreover, the variable delay τ : T → T is increasing with τ(t) ≤ t for all t ∈ [t 0 ,∞) T such that lim t→∞ τ(t) = ∞. The initial function associated with (1.1) takes the form Hindawi Publishing Corporation

show abstract

Section: Delay Dynamic Equation With Forcing Termmentioning

confidence: 99%

Boundedness and vanishing of solutions for a forced delay dynamic equation

Anderson¹

2006

Adv. Differ Equ.

View full text Add to dashboard Cite

show abstract

“…If n 0 ≤ k ≤ n 1 , n > n 1 , then by (4), (9), (a1) and (a2) we have h l (j) ≥ n 1 , j > n 1 + T , thus (the second sum below involves only such…”

Section: ⊓ ⊔mentioning

confidence: 99%

New stability conditions for linear difference equations using Bohl–Perron type theorems

Berezansky

Braverman

2011

Journal of Difference Equations and Applications

View full text Add to dashboard Cite

The Bohl-Perron result on exponential dichotomy for a linear difference equationstates (under some natural conditions) that if all solutions of the non-homogeneous equation with a bounded right hand side are bounded, then the relevant homogeneous equation is exponentially stable. According to its corollary, if a given equation is close to an exponentially stable comparison equation (the norm of some operator is less than one), then the considered equation is exponentially stable.For a difference equation with several variable delays and coefficients we obtain new exponential stability tests using the above results, representation of solutions and comparison equations with a positive fundamental function.

show abstract

“…This paper illustrates this new understanding by extending some discrete results from difference equations to dynamic equations on time scales. In particular, (1.1) is studied in [6] with T = Z and p ≡ 0, and in [5] in the case when T = Z with variable p. Much of the organization of and motivation for this paper arise from [5,6]. For more on delay dynamic equations, see, for…”

Section: Neutral Delay Dynamic Equationmentioning

confidence: 99%

Asymptotic behavior of solutions for neutral dynamic equations on time scales

Anderson¹

2006

Adv. Differ Equ.

View full text Add to dashboard Cite

We investigate the boundedness and asymptotic behavior of a first-order neutral delay dynamic equation on arbitrary time scales, extending some results from difference equations.Copyright © 2006 Douglas R. Anderson. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Neutral delay dynamic equationWe consider, on arbitrary time scales, the neutral delay dynamic equationwhere T is a time scale unbounded above, the variable delays k, :The coefficient functions p, q : T → R are right-dense continuous with p bounded and q ≥ 0. To clarify some notation, take −1 (t) := sup{s : (s) ≤ t}, −(n+1) (t) = −1 ( −n (t)) for t ∈ [ (t 0 ),∞) T , and n+1 (t) = ( n (t)) for t ∈ [ −3 (t 0 ),∞) T . For p and k above, let Ω be the linear set of all functions given bysolutions of (1.1) will belong to Ω. In the aftermath of Hilger's breakthrough paper [4], a rapidly diversifying body of literature has sought to unify, extend, and generalize ideas from discrete calculus, continuous calculus, and quantum calculus to arbitrary time-scale calculus, where a time scale is merely a nonempty closed set of real numbers. This paper illustrates this new understanding by extending some discrete results from difference equations to dynamic equations on time scales. In particular, (1.1) is studied in [6] with T = Z and p ≡ 0, and in [5] in the case when T = Z with variable p. Much of the organization of and motivation for this paper arise from [5,6]. For more on delay dynamic equations, see, forHindawi Publishing Corporation

show abstract

Global stability of a linear nonautonomous delay difference equation¹

Cited by 82 publications

References 16 publications

Boundedness and vanishing of solutions for a forced delay dynamic equation

Boundedness and vanishing of solutions for a forced delay dynamic equation

New stability conditions for linear difference equations using Bohl–Perron type theorems

Asymptotic behavior of solutions for neutral dynamic equations on time scales

Contact Info

Product

Resources

About

Global stability of a linear nonautonomous delay difference equation1

Cited by 82 publications

References 16 publications

Boundedness and vanishing of solutions for a forced delay dynamic equation

Boundedness and vanishing of solutions for a forced delay dynamic equation

New stability conditions for linear difference equations using Bohl–Perron type theorems

Asymptotic behavior of solutions for neutral dynamic equations on time scales

Contact Info

Product

Resources

About

Global stability of a linear nonautonomous delay difference equation¹