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 firstorder linear dynamic equations of the form
where A is the generator of a
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 timevarying 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
(i) f is continuous at every rightdense point
(ii)
The set of rdcontinuous functions
A function
In this case, we denote the α by
A function
where F is an antiderivative of f. Every rdcontinuous function
Definition 1.1 A mapping
is regressive if A is regressive. We say that a real valued function
It is well known that if
has the unique solution
Here,
whose solution has the closed form
where
and
It can be seen that for
Indeed, by taking
This implies that the claim is true.
In the sequel, we denote by
(i)
(ii)
(iii)
If in addition
where the domain
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
has the unique solution
when A is the generator of the
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
where
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 nonregressive 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
when A is the generator of a
At first, we establish some properties of T and its generator A which we use to arrive at our aim.
Theorem 2.1For
1. For
and
2. For
and
Proof 1. Set
Also, we have
and
where
2. Let
□
Theorem 2.2For
1. For
2. For
and
Proof 1. Let
Now, we show that
We have either
On the other hand,
When
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
Proof For every
Theorem 2.1 implies that
By the same theorem,
To prove its closeness, let
The integrand on the righthand side of (2.10) converges to
Dividing Eq. (2.11) by
Theorem 2.4Equation (2.1) has the unique solution
Proof The existence of the solution
where
On the other hand, we have
from which we obtain that
3 Types of stability
In this section, the definitions of the various types of stability for dynamic equations of the form
are presented, where
Definition 3.1 Equation (3.1) is said to be stable if, for every
Definition 3.2 Equation (3.1) is said to be uniformly stable if, for each
Definition 3.3 Equation (3.1) is said to be asymptotically stable if it is stable and for every
Definition 3.4 Equation (3.1) is said to be uniformly asymptotically stable if it is uniformly stable
and there exists a
Definition 3.5 Equation (3.1) is said to be globally asymptotically stable if it is stable and for
any solution
Definition 3.6 Equation (3.1) is said to be exponentially stable if there exists
Definition 3.7 Equation (3.1) is said to be uniformly exponentially stable if there exists
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
where A is the generator of T. The initial value problem
In the following two lemmas, by linearity of
Lemma 4.1The following statements are equivalent:
(i)
(ii) For every
Lemma 4.2The following statements are equivalent:
(i)
(ii) For every
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
In the following theorem, we extend these results for the case where A is the generator of a
Theorem 4.3The following statements are equivalent:
(i)
(ii)
(iii)
Proof (i) ⟹ (ii) Assume
Let
i.e.
The density of
Thus, for every
(ii) ⟹ (iii) Assume that there is
5 A characterization of global asymptotic stability
In the following result, we establish necessary and sufficient conditions for
Theorem 5.1The following statement are equivalent:
(i)
(ii)
(iii)
(iv)
Proof (i) ⟹ (ii) Suppose that
Hence,
Consequently, we obtain
By the boundedness of
(ii) ⟹ (iii) Condition (ii) implies that
(iii) ⟹ (iv) Condition (iii) implies that
6 A characterization of exponential stability and uniform exponential stability
We need the following lemmas to establish a characterization of the exponential stability
of
Lemma 6.1
Lemma 6.2
In the following two theorems, we extend the results of DaCunha [9], Theorem 2.2] when
Theorem 6.3The following statements are equivalent:
(i)
(ii) There exists
Proof (i) ⟹ (ii) Let
Fix
Using
This implies that
(ii) ⟹ (i) Assume there exists
Let
□
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)
(ii) There exists
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
7 Example
Choi in [8] gave an example to illustrate many types of stability. He considered the linear dynamic system
where
We see that the generalized exponential function
and
The following stability results [8] for (7.1) were obtained in different cases of
(1) If
(2) If
(3) If
(4) If
Now we consider the time scale
Indeed, for
Then
Consequently,
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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