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Open Access Research

Stability of abstract dynamic equations on time scales

Alaa E Hamza1* and Karima M Oraby2

Author Affiliations

1 Department of Mathematics, Faculty of Science, Cairo University, Cairo, Egypt

2 Department of Mathematics, Faculty of Science, Suez Canal University, Suez, Egypt

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Advances in Difference Equations 2012, 2012:143  doi:10.1186/1687-1847-2012-143

The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2012/1/143


Received:28 February 2012
Accepted:25 July 2012
Published:12 August 2012

© 2012 Hamza and Oraby; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we investigate many types of stability, like uniform stability, asymptotic stability, uniform asymptotic stability, global stability, global asymptotic stability, exponential stability, uniform exponential stability, of the homogeneous first-order linear dynamic equations of the form

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M1">View MathML</a>

where A is the generator of a <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M2">View MathML</a>-semigroup <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M3">View MathML</a>, the space of all bounded linear operators from a Banach space X into itself. Here, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M4">View MathML</a> is a time scale which is an additive semigroup with the property that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M5">View MathML</a> for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M6">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M7">View MathML</a>. Finally, we give an illustrative example for a nonregressive homogeneous first-order linear dynamic equation and we investigate its stability.

1 Introduction and preliminaries

The history of asymptotic stability of dynamic equations on a time scale goes back to Aulbach and Hilger [3]. For a real scalar dynamic equation, stability and instability results were obtained by Gard and Hoffacker [12]. Pötzche [20] provides sufficient conditions for the uniform exponential stability in Banach spaces, as well as spectral stability conditions for time-varying systems on time scales. Doan, Kalauch, and Siegmund [10] established a necessary and sufficient condition for the existence of uniform exponential stability and characterized the uniform exponential stability of a system by the spectrum of its matrix. Properties of exponential stability of a time varying dynamic equation on a time scale have been also investigated recently by Bohner and Martynyuk [7], DaCunha [9], Du and Tien [11], Hoffacker and Tisdell [16], Martynyuk [17], and Peterson and Raffoul [19].

The theory of dynamic equations on time scales was introduced by Stefan Hilger in 1988 [14], in order to unify continuous and discrete calculus [4,15]. A time scale <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M8">View MathML</a> is a nonempty closed subset of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M9">View MathML</a>. The forward jump operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M10">View MathML</a> is defined by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M11">View MathML</a> (supplemented by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M12">View MathML</a>) and the backward jump operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M13">View MathML</a> is defined by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M14">View MathML</a> (supplemented by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M15">View MathML</a>). The graininess function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M16">View MathML</a> is given by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M17">View MathML</a>. A point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M18">View MathML</a> is said to be right-dense if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M19">View MathML</a>, right-scattered if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M20">View MathML</a>, left-dense if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M21">View MathML</a>, left-scattered if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M22">View MathML</a>, isolated if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M23">View MathML</a>, and dense if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M24">View MathML</a>. A time scale <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M8">View MathML</a> is said to be discrete if t is left-scattered and right-scattered for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M18">View MathML</a>, and it is called continuous if t is right-dense and left-dense at the same time for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M18">View MathML</a>. Suppose that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M8">View MathML</a> has the topology inherited from the standard topology on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M9">View MathML</a>. We define the time scale interval <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M30">View MathML</a>. Open intervals and open neighborhoods are defined similarly. A set we need to consider is <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M31">View MathML</a> which is defined as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M32">View MathML</a> if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M8">View MathML</a> has a left-scattered maximum M, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M34">View MathML</a> otherwise. A function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M35">View MathML</a> is called right dense continuous, or just rd-continuous, if

(i) f is continuous at every right-dense point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M18">View MathML</a>;

(ii) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M37">View MathML</a> exists (finite) for every left-dense point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M18">View MathML</a>.

The set of rd-continuous functions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M35">View MathML</a> will be denoted by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M40">View MathML</a>.

A function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M35">View MathML</a> is called delta differentiable <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M42">View MathML</a> at <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M43">View MathML</a> provided there exists an α such that for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M44">View MathML</a> there is a neighborhood U of t with

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M45">View MathML</a>

In this case, we denote the α by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M46">View MathML</a>; and if f is differentiable for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M43">View MathML</a>, then f is said to be differentiable on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M8">View MathML</a>. If f is differentiable at <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M43">View MathML</a>, then it is easy to see that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M50">View MathML</a>

A function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M51">View MathML</a> is called an antiderivative of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M35">View MathML</a> if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M53">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M43">View MathML</a>. The Cauchy integral is defined by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M55">View MathML</a>

where F is an antiderivative of f. Every rd-continuous function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M35">View MathML</a> has an antiderivative and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M57">View MathML</a> is an antiderivative of f, i.e., <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M53">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M43">View MathML</a>. Equations which include Δ-derivatives are called dynamic equations. We refer the reader to the very interesting monographs of Bohner and Peterson [5,6].

Definition 1.1 A mapping <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M60">View MathML</a> is called regressive if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M61">View MathML</a> is invertible for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M18">View MathML</a>, and we say that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M63">View MathML</a>

is regressive if A is regressive. We say that a real valued function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M64">View MathML</a> on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M8">View MathML</a> is regressive (resp. positively regressive) if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M66">View MathML</a> (resp. <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M67">View MathML</a>), <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M68">View MathML</a>. The family of all regressive functions (resp. positively regressive functions) is denoted by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M69">View MathML</a> (resp. <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M70">View MathML</a>).

It is well known that if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M71">View MathML</a>, the space of all right dense continuous and regressive bounded functions from <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M8">View MathML</a> to <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M73">View MathML</a>, then the initial value problem (IVP)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M74">View MathML</a>

(1.1)

has the unique solution

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M75">View MathML</a>

Here, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M76">View MathML</a> is the exponential operator function. For more details, see [2]. When <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M77">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M78">View MathML</a> is a real valued function, Eq. (1.1) yields

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M79">View MathML</a>

(1.2)

whose solution has the closed form

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M80">View MathML</a>

(1.3)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M81">View MathML</a>

(1.4)

and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M82">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M83">View MathML</a> is the principal logarithm function. It is evident that when <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M84">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M85">View MathML</a>, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M86">View MathML</a>

(1.5)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M87">View MathML</a>

(1.6)

It can be seen that for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M88">View MathML</a> with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M89">View MathML</a>, the following claim is true

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M90">View MathML</a>

(1.7)

Indeed, by taking <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M91">View MathML</a> in Eqs. (1.5) and (1.6), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M92">View MathML</a>

(1.8)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M93">View MathML</a>

(1.9)

This implies that the claim is true.

In the sequel, we denote by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M4">View MathML</a> for a time scale which is an additive semigroup with the property that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M5">View MathML</a> for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M6">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M7">View MathML</a>. In this case, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M8">View MathML</a> is called a semigroup time scale. We assume X is a Banach space. Finally, we assume that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M99">View MathML</a> is a <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M2">View MathML</a>-semigroup on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M8">View MathML</a>, that is, it satisfies

(i) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M102">View MathML</a> for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M103">View MathML</a> (the semigroup property).

(ii) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M104">View MathML</a> (I is the identity operator on X).

(iii) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M105">View MathML</a> (i.e., <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M106">View MathML</a> is continuous at 0) for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M107">View MathML</a>.

If in addition <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M108">View MathML</a>, then T is called a uniformly continuous semigroup. A linear operator A is called the generator [1] of a <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M2">View MathML</a>-semigroup T if

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M110">View MathML</a>

(1.10)

where the domain <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M111">View MathML</a> of A is the set of all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M107">View MathML</a> for which the above limit exists uniformly in t. Clearly, when <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M113">View MathML</a>, the concept of the generator defined by relation (1.10) coincides with the classical definition by Hille. See [13].

In Section 2 of this paper we present some results from [1] that we need in our study. One of them is that an abstract Cauchy problem

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M114">View MathML</a>

(1.11)

has the unique solution

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M115">View MathML</a>

when A is the generator of the <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M2">View MathML</a>-semigroup T. When <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M117">View MathML</a>, we get the classical existence and uniqueness theorem of the abstract Cauchy problem (1.11); see [21]. The other results include some properties of T and its generator A, which we use in the subsequent sections. The solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M118">View MathML</a> is a function of the variables t, τ and the initial value <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M119">View MathML</a>. Generally, we consider τ and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M119">View MathML</a> as parameters. Therefore, when we investigate the asymptotic behavior of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M121">View MathML</a> with respect to <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M8">View MathML</a>, we must investigate whether or not the asymptotic behavior uniformly depends on τ or <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M119">View MathML</a>. Accordingly, there are many types of stability which we give in Section 3.

S. K. Choi, D. M. Im, and N. Koo in [8], Theorem 3.5] proved that the stability of the time variant abstract Cauchy problem

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M124">View MathML</a>

(1.12)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M125">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M126">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M127">View MathML</a> is the family of all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M128">View MathML</a> real matrices is equivalent to the boundedness of all its solutions. DaCunha in [9] defined the concepts of uniform stability and uniform exponential stability. These two concepts involve the boundedness of the solutions of the regressive time varying linear dynamic Eq. (1.12). He established a characterization of uniform stability and uniform exponential stability in terms of the transition matrix for system (1.12). Also, he illustrated the relationship between the uniform asymptotic stability and the uniform exponential stability.

In Section 4, we extend these results for the case where A is the generator of T and we prove that the concepts of stability and uniform stability are same.

Sections 5 and 6 are devoted to establishing characterizations for many other types of stability, like asymptotic stability, uniform asymptotic stability, global asymptotic stability, exponential stability, and uniform exponential stability for the abstract Cauchy problem (1.11).

We end this paper with a new illustrative example including non-regressive dynamic equation and we investigate its stability.

2 The existence and uniqueness of solutions of dynamic equations

Our aim in this section is to prove that the first order initial value problem

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M129">View MathML</a>

(2.1)

has the unique solution

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M130">View MathML</a>

when A is the generator of a <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M2">View MathML</a>-semigroup <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M132">View MathML</a>.

At first, we establish some properties of T and its generator A which we use to arrive at our aim.

Theorem 2.1For<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M107">View MathML</a>, the following statements are true:

1. For<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M18">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M135">View MathML</a>

(2.2)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M136">View MathML</a>

(2.3)

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M137">View MathML</a>

(2.4)

2. For<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M18">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M139">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M140">View MathML</a>

(2.5)

Proof 1. Set <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M141">View MathML</a>. Then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M142">View MathML</a>

Also, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M143">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M144">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M145">View MathML</a>.

2. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M146">View MathML</a> be a number in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M8">View MathML</a>. We have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M148">View MathML</a>

 □

Theorem 2.2For<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M149">View MathML</a>, the following statements are true:

1. For<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M18">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M151">View MathML</a>and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M152">View MathML</a>

(2.6)

2. For<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M153">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M154">View MathML</a>

(2.7)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M155">View MathML</a>

(2.8)

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M156">View MathML</a>

(2.9)

Proof 1. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M149">View MathML</a>. It is evident that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M158">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M159">View MathML</a>.

Now, we show that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M160">View MathML</a> solves the initial value problem

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M161">View MathML</a>

We have either <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M162">View MathML</a> or <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M163">View MathML</a>. The case <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M162">View MathML</a> implies

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M165">View MathML</a>

On the other hand,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M166">View MathML</a>

When <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M163">View MathML</a>, we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M168">View MathML</a>

2. Relations (2.7) and (2.8) can be obtained by integrating both sides of Eq. (2.6) from s to t. Relation (2.9) follows from Eqs. (2.5) and (2.7). □

Corollary 2.3IfAis the generator of a<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M2">View MathML</a>-semigroupTon<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M8">View MathML</a>, then<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M111">View MathML</a>is dense in X andAis a closed linear operator.

Proof For every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M107">View MathML</a> and fixed <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M18">View MathML</a>, set

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M174">View MathML</a>

Theorem 2.1 implies that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M175">View MathML</a>

By the same theorem, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M176">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M177">View MathML</a>. So <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M178">View MathML</a>, the closure of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M111">View MathML</a>, is equal to X. The linearity of A is evident.

To prove its closeness, let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M180">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M181">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M182">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M183">View MathML</a>. In view of equality (2.7), we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M184">View MathML</a>

(2.10)

The integrand on the right-hand side of (2.10) converges to <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M185">View MathML</a> uniformly on bounded intervals. Consequently, letting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M183">View MathML</a> in (2.10), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M187">View MathML</a>

(2.11)

Dividing Eq. (2.11) by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M188">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M146">View MathML</a> and letting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M177">View MathML</a>, we see, using identity (2.4), that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M149">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M192">View MathML</a>. □

Theorem 2.4Equation (2.1) has the unique solution

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M193">View MathML</a>

Proof The existence of the solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M194">View MathML</a> follows by Theorem 2.2. To prove the uniqueness, assume that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M195">View MathML</a> is another solution. Consider the function

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M196">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M197">View MathML</a>. We have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M198">View MathML</a>

On the other hand, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M199">View MathML</a>

from which we obtain that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M200">View MathML</a> on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M201">View MathML</a>. Then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M202">View MathML</a>, i.e.<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M203">View MathML</a>. □

3 Types of stability

In this section, the definitions of the various types of stability for dynamic equations of the form

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M204">View MathML</a>

(3.1)

are presented, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M205">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M206">View MathML</a> is the delta derivative of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M207">View MathML</a> with respect to <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M43">View MathML</a>. See [8,18].

Definition 3.1 Equation (3.1) is said to be stable if, for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M209">View MathML</a> and for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M44">View MathML</a>, there exists a <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M211">View MathML</a> such that, for any two solutions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M212">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M213">View MathML</a> of Eq. (3.1), the inequality <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M214">View MathML</a> implies <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M215">View MathML</a>, for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M216">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M18">View MathML</a>.

Definition 3.2 Equation (3.1) is said to be uniformly stable if, for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M218">View MathML</a>, there exists a <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M219">View MathML</a> independent on any initial point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M220">View MathML</a> such that, for any two solutions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M212">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M213">View MathML</a> of Eq. (3.1), the inequality <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M214">View MathML</a> implies <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M215">View MathML</a>, for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M216">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M18">View MathML</a>.

Definition 3.3 Equation (3.1) is said to be asymptotically stable if it is stable and for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M209">View MathML</a>, there exists a <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M228">View MathML</a> such that, the inequality <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M229">View MathML</a> implies <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M230">View MathML</a>.

Definition 3.4 Equation (3.1) is said to be uniformly asymptotically stable if it is uniformly stable and there exists a <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M231">View MathML</a> such that for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M209">View MathML</a> the inequality <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M229">View MathML</a> implies <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M230">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M18">View MathML</a>.

Definition 3.5 Equation (3.1) is said to be globally asymptotically stable if it is stable and for any solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M212">View MathML</a> of Eq. (3.1), we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M230">View MathML</a>.

Definition 3.6 Equation (3.1) is said to be exponentially stable if there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M238">View MathML</a> with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M239">View MathML</a> such that for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M209">View MathML</a>, there is <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M241">View MathML</a> such that, for any two solutions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M212">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M243">View MathML</a> of Eq. (3.1), we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M244">View MathML</a>, for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M216">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M18">View MathML</a>.

Definition 3.7 Equation (3.1) is said to be uniformly exponentially stable if there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M238">View MathML</a> with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M239">View MathML</a> and there is <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M249">View MathML</a> independent on any initial point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M220">View MathML</a> such that, for any two solutions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M212">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M243">View MathML</a> of Eq. (3.1), we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M244">View MathML</a>, for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M216">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M18">View MathML</a>.

4 Characterization of stability and uniformly stability

In this section, we obtain some results concerning characterizations of stability and uniform stability of linear dynamic equations of the form

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M256">View MathML</a>

where A is the generator of T. The initial value problem <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257">View MathML</a> has the unique solution

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M258">View MathML</a>

(4.1)

In the following two lemmas, by linearity of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257">View MathML</a>, we get an equivalent definition of stability and uniform stability of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257">View MathML</a>.

Lemma 4.1The following statements are equivalent:

(i) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257">View MathML</a>is stable;

(ii) For every<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M209">View MathML</a>and for every<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M44">View MathML</a>, there exists<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M264">View MathML</a>such that for any solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M212">View MathML</a>of<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M267">View MathML</a>

Lemma 4.2The following statements are equivalent:

(i) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257">View MathML</a>is uniformly stable;

(ii) For every<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M44">View MathML</a>there exists<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M270">View MathML</a>such that for any solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M212">View MathML</a>of<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M273">View MathML</a>

S. K. Choi, D. M. Im, and N. Koo in [8], Theorem 3.5] proved that the stability of (1.12) is equivalent to the boundedness of all its solutions when <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M125">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M126">View MathML</a> where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M127">View MathML</a> is the family of all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M128">View MathML</a> real matrices. Also, DaCunha in [9] proved that the uniform stability of (1.12) is equivalent to the uniform boundedness of all its solutions with respect to the initial point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M220">View MathML</a>, when <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M125">View MathML</a>.

In the following theorem, we extend these results for the case where A is the generator of a <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M2">View MathML</a>-semigroup T and we prove that the concepts of stability and uniform stability are the same.

Theorem 4.3The following statements are equivalent:

(i) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257">View MathML</a>is stable.

(ii) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M282">View MathML</a>is bounded.

(iii) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257">View MathML</a>is uniformly stable.

Proof (i) ⟹ (ii) Assume <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257">View MathML</a> is stable. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M209">View MathML</a>. Fix <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M286">View MathML</a>. There exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M231">View MathML</a> such that for any solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M288">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M289">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M290">View MathML</a>

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M291">View MathML</a>. Take <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M292">View MathML</a>. Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M229">View MathML</a>, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M294">View MathML</a>

i.e.

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M295">View MathML</a>

The density of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M111">View MathML</a> in X, by Corollary 2.3, implies that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M297">View MathML</a>

Thus, for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M107">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M299">View MathML</a> is bounded. By the uniform boundedness theorem [22], <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M300">View MathML</a> is bounded.

(ii) ⟹ (iii) Assume that there is <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M301">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M302">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M18">View MathML</a>. Clearly, condition (ii) of Lemma 4.2 holds, because for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M44">View MathML</a>, choose <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M305">View MathML</a>. □

5 A characterization of global asymptotic stability

In the following result, we establish necessary and sufficient conditions for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257">View MathML</a> to be globally asymptotically stable.

Theorem 5.1The following statement are equivalent:

(i) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257">View MathML</a>is asymptotically stable;

(ii) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M308">View MathML</a>, for every<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M107">View MathML</a>;

(iii) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257">View MathML</a>is globally asymptotically stable;

(iv) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257">View MathML</a>is uniformly asymptotically stable.

Proof (i) ⟹ (ii) Suppose that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257">View MathML</a> is asymptotically stable. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M209">View MathML</a>. There exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M314">View MathML</a> such that any solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M212">View MathML</a> of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257">View MathML</a> with initial value <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M289">View MathML</a>, vanishes at ∞ whenever <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M318">View MathML</a>. Fix <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M319">View MathML</a>. Then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M320">View MathML</a>

Hence,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M321">View MathML</a>

Consequently, we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M322">View MathML</a>

By the boundedness of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M282">View MathML</a> and the density of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M111">View MathML</a> in X, we deduce that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M325">View MathML</a>

(ii) ⟹ (iii) Condition (ii) implies that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M326">View MathML</a> is bounded for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M107">View MathML</a>. The uniform boundedness theorem insures the boundedness of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M300">View MathML</a>. Consequently, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257">View MathML</a> is stable, and by our assumption, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257">View MathML</a> is globally asymptotically stable.

(iii) ⟹ (iv) Condition (iii) implies that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M331">View MathML</a> is bounded for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M107">View MathML</a>. Again the uniform boundedness theorem guarantees the boundedness of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M333">View MathML</a>. Consequently, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257">View MathML</a> is uniformly stable by Theorem 4.3, and by our assumption, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257">View MathML</a> is uniformly asymptotically stable. □

6 A characterization of exponential stability and uniform exponential stability

We need the following lemmas to establish a characterization of the exponential stability of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257">View MathML</a>. Their proofs are straightforward and will be omitted.

Lemma 6.1<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257">View MathML</a>is exponentially stable if and only if there exists<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M238">View MathML</a>with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M239">View MathML</a>such that for any<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M209">View MathML</a>, there exists<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M241">View MathML</a>such that for any solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M212">View MathML</a>of<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257">View MathML</a>with initial value<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M289">View MathML</a>we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M345">View MathML</a>

Lemma 6.2<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257">View MathML</a>is uniformly exponentially stable if and only if there exists<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M238">View MathML</a>with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M239">View MathML</a>and there exists<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M349">View MathML</a>such that for any<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M209">View MathML</a>, and any solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M212">View MathML</a>of<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257">View MathML</a>with initial value<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M289">View MathML</a>we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M354">View MathML</a>

In the following two theorems, we extend the results of DaCunha [9], Theorem 2.2] when <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M125">View MathML</a> to the case where A is the generator of T.

Theorem 6.3The following statements are equivalent:

(i) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257">View MathML</a>is exponentially stable;

(ii) There exists<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M238">View MathML</a>with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M239">View MathML</a>such that for any<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M209">View MathML</a>, there exists<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M360">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M361">View MathML</a>

Proof (i) ⟹ (ii) Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257">View MathML</a> be exponentially stable. Then there is <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M238">View MathML</a> with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M239">View MathML</a> such that for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M209">View MathML</a>, there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M241">View MathML</a> such that for any solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M288">View MathML</a> of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257">View MathML</a> with initial value <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M289">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M370">View MathML</a>

Fix <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M209">View MathML</a>, and let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M319">View MathML</a>. Then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M373">View MathML</a>

Using <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M111">View MathML</a> is dense in X and Corollary 2.3, we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M375">View MathML</a>

This implies that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M376">View MathML</a>

(ii) ⟹ (i) Assume there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M238">View MathML</a> with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M239">View MathML</a> such that for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M209">View MathML</a>, there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M241">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M381">View MathML</a>

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M382">View MathML</a> be any solution of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257">View MathML</a> with initial value <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M384">View MathML</a>. Then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M385">View MathML</a>

 □

By same way as in the proof of Theorem 6.3, we can obtain the following result.

Theorem 6.4The following statements are equivalent:

(i) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257">View MathML</a>is uniformly exponentially stable;

(ii) There exists<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M238">View MathML</a>with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M239">View MathML</a>and there exists<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M349">View MathML</a>such that for any<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M209">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M391">View MathML</a>

From Theorem 5.1 (Theorem 6.4), Lemma 6.1 (Lemma 6.2), and relation (1.7), we get the following result.

Corollary 6.5If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257">View MathML</a>is (uniformly) exponentially stable, then<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M257">View MathML</a>is (uniformly) asymptotically stable.

7 Example

Choi in [8] gave an example to illustrate many types of stability. He considered the linear dynamic system

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M394">View MathML</a>

(7.1)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M395">View MathML</a> is a time scale and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M396">View MathML</a> and investigated some types of stability of Eq. (7.1) when A is regressive, i.e., <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M397">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M18">View MathML</a>. In this case the equation has the unique solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M399">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M400">View MathML</a> is the matrix exponential function. It is given by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M401">View MathML</a>

(7.2)

We see that the generalized exponential function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M402">View MathML</a> is given by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M403">View MathML</a>

(7.3)

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M404">View MathML</a>

(7.4)

The following stability results [8] for (7.1) were obtained in different cases of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M8">View MathML</a>.

(1) If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M117">View MathML</a>, then (7.1) is uniformly stable, exponentially stable and asymptotically stable, since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M407">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M408">View MathML</a>.

(2) If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M409">View MathML</a>, then (7.1) is uniformly stable but not asymptotically stable, since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M410">View MathML</a>.

(3) If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M411">View MathML</a> with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M412">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M413">View MathML</a>, then (7.1) is not asymptotically stable. However, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M402">View MathML</a> goes to zero as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M408">View MathML</a>.

(4) If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M411">View MathML</a> with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M417">View MathML</a>, then (7.1) is not asymptotically stable.

Now we consider the time scale <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M418">View MathML</a> with the graininess function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M419">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M18">View MathML</a>. So A is nonregressive and the matrix exponential function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M400">View MathML</a> does not exist. On the other hand, A is the generator of the <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M2">View MathML</a>-semigroup

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M423">View MathML</a>

Indeed, for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M424">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M425">View MathML</a>

Then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M426">View MathML</a>

Consequently, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M427">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/143/mathml/M18">View MathML</a> which implies that Eq. (7.1) is uniformly stable but is not asymptotically stable.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

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