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
where A is the generator of a -semigroup , the space of all bounded linear operators from a Banach space X into itself. Here, is a time scale which is an additive semigroup with the property that for any such that . 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 . For a real scalar dynamic equation, stability and instability results were obtained by Gard and Hoffacker . Pötzche  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  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 , DaCunha , Du and Tien , Hoffacker and Tisdell , Martynyuk , and Peterson and Raffoul .
The theory of dynamic equations on time scales was introduced by Stefan Hilger in 1988 , in order to unify continuous and discrete calculus [4,15]. A time scale is a nonempty closed subset of . The forward jump operator is defined by (supplemented by ) and the backward jump operator is defined by (supplemented by ). The graininess function is given by . A point is said to be right-dense if , right-scattered if , left-dense if , left-scattered if , isolated if , and dense if . A time scale is said to be discrete if t is left-scattered and right-scattered for all , and it is called continuous if t is right-dense and left-dense at the same time for all . Suppose that has the topology inherited from the standard topology on . We define the time scale interval . Open intervals and open neighborhoods are defined similarly. A set we need to consider is which is defined as if has a left-scattered maximum M, and otherwise. A function is called right dense continuous, or just rd-continuous, if
where F is an antiderivative of f. Every rd-continuous function has an antiderivative and is an antiderivative of f, i.e., , . Equations which include Δ-derivatives are called dynamic equations. We refer the reader to the very interesting monographs of Bohner and Peterson [5,6].
is regressive if A is regressive. We say that a real valued function on is regressive (resp. positively regressive) if (resp. ), . The family of all regressive functions (resp. positively regressive functions) is denoted by (resp. ).
has the unique solution
Here, is the exponential operator function. For more details, see . When and is a real valued function, Eq. (1.1) yields
whose solution has the closed form
This implies that the claim is true.
In the sequel, we denote by for a time scale which is an additive semigroup with the property that for any such that . In this case, is called a semigroup time scale. We assume X is a Banach space. Finally, we assume that is a -semigroup on , that is, it satisfies
If in addition , then T is called a uniformly continuous semigroup. A linear operator A is called the generator  of a -semigroup T if
where the domain of A is the set of all for which the above limit exists uniformly in t. Clearly, when , the concept of the generator defined by relation (1.10) coincides with the classical definition by Hille. See .
In Section 2 of this paper we present some results from  that we need in our study. One of them is that an abstract Cauchy problem
has the unique solution
when A is the generator of the -semigroup T. When , we get the classical existence and uniqueness theorem of the abstract Cauchy problem (1.11); see . The other results include some properties of T and its generator A, which we use in the subsequent sections. The solution is a function of the variables t, τ and the initial value . Generally, we consider τ and as parameters. Therefore, when we investigate the asymptotic behavior of with respect to , we must investigate whether or not the asymptotic behavior uniformly depends on τ or . Accordingly, there are many types of stability which we give in Section 3.
S. K. Choi, D. M. Im, and N. Koo in , Theorem 3.5] proved that the stability of the time variant abstract Cauchy problem
where , and is the family of all real matrices is equivalent to the boundedness of all its solutions. DaCunha in  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
has the unique solution
At first, we establish some properties of T and its generator A which we use to arrive at our aim.
Also, we have
On the other hand,
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). □
Theorem 2.1 implies that
Theorem 2.4Equation (2.1) has the unique solution
On the other hand, we have
3 Types of stability
In this section, the definitions of the various types of stability for dynamic equations of the form
Definition 3.2 Equation (3.1) is said to be uniformly stable if, for each , there exists a independent on any initial point such that, for any two solutions and of Eq. (3.1), the inequality implies , for all , .
Definition 3.7 Equation (3.1) is said to be uniformly exponentially stable if there exists with and there is independent on any initial point such that, for any two solutions and of Eq. (3.1), we have , for all , .
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
Lemma 4.1The following statements are equivalent:
Lemma 4.2The following statements are equivalent:
S. K. Choi, D. M. Im, and N. Koo in , Theorem 3.5] proved that the stability of (1.12) is equivalent to the boundedness of all its solutions when , where is the family of all real matrices. Also, DaCunha in  proved that the uniform stability of (1.12) is equivalent to the uniform boundedness of all its solutions with respect to the initial point , when .
Theorem 4.3The following statements are equivalent:
Thus, for every , is bounded. By the uniform boundedness theorem , is bounded.
5 A characterization of global asymptotic stability
Theorem 5.1The following statement are equivalent:
Consequently, we obtain
(ii) ⟹ (iii) Condition (ii) implies that is bounded for every . The uniform boundedness theorem insures the boundedness of . Consequently, is stable, and by our assumption, is globally asymptotically stable.
(iii) ⟹ (iv) Condition (iii) implies that is bounded for every . Again the uniform boundedness theorem guarantees the boundedness of . Consequently, is uniformly stable by Theorem 4.3, and by our assumption, is uniformly asymptotically stable. □
6 A characterization of exponential stability and uniform exponential stability
In the following two theorems, we extend the results of DaCunha , Theorem 2.2] when to the case where A is the generator of T.
Theorem 6.3The following statements are equivalent:
This implies that
By same way as in the proof of Theorem 6.3, we can obtain the following result.
Theorem 6.4The following statements are equivalent:
From Theorem 5.1 (Theorem 6.4), Lemma 6.1 (Lemma 6.2), and relation (1.7), we get the following result.
Choi in  gave an example to illustrate many types of stability. He considered the linear dynamic system
where is a time scale and and investigated some types of stability of Eq. (7.1) when A is regressive, i.e., for all . In this case the equation has the unique solution , where is the matrix exponential function. It is given by
The following stability results  for (7.1) were obtained in different cases of .
The authors declare that they have no competing interests.
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
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