Stability of abstract dynamic equations on time scales
© Hamza and Oraby; licensee Springer 2012
Received: 28 February 2012
Accepted: 25 July 2012
Published: 12 August 2012
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 .
f is continuous at every right-dense point ;
exists (finite) for every left-dense point .
The set of rd-continuous functions will be denoted by .
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. ).
This implies that the claim is true.
for every (the semigroup property).
(I is the identity operator on X).
(i.e., is continuous at 0) for each .
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 .
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.
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
when A is the generator of a -semigroup .
At first, we establish some properties of T and its generator A which we use to arrive at our aim.
- 1.For ,(2.2)
- 2.For ,
- 2.Let be a number in . We have
- 1.For , and(2.6)
- 2.For , we have(2.7)
Proof 1. Let . It is evident that , .
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 2.3 If A is the generator of a -semigroup T on , then is dense in X and A is a closed linear operator.
By the same theorem, as . So , the closure of , is equal to X. The linearity of A is evident.
Dividing Eq. (2.11) by , and letting , we see, using identity (2.4), that and . □
from which we obtain that on . Then , i.e. . □
3 Types of stability
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 3.3 Equation (3.1) is said to be asymptotically stable if it is stable and for every , there exists a such that, the inequality implies .
Definition 3.4 Equation (3.1) is said to be uniformly asymptotically stable if it is uniformly stable and there exists a such that for every the inequality implies , .
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 the following two lemmas, by linearity of , we get an equivalent definition of stability and uniform stability of .
- (ii)For every and for every , there exists such that for any solution of , we have
is uniformly stable;
- (ii)For every there exists such that for any solution of , we have
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 .
In the following theorem, we extend these results for the case where A is the generator of a -semigroup T and we prove that the concepts of stability and uniform stability are the same.
is uniformly stable.
⟹ (iii) Assume that there is such that , . Clearly, condition (ii) of Lemma 4.2 holds, because for , choose . □
5 A characterization of global asymptotic stability
In the following result, we establish necessary and sufficient conditions for to be globally asymptotically stable.
is asymptotically stable;
, for every ;
is globally asymptotically stable;
is uniformly asymptotically stable.
⟹ (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.
⟹ (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
We need the following lemmas to establish a characterization of the exponential stability of . Their proofs are straightforward and will be omitted.
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.
is exponentially stable;
- (ii)There exists with such that for any , there exists such that
By same way as in the proof of Theorem 6.3, we can obtain the following result.
is uniformly exponentially stable;
- (ii)There exists with and there exists such that for any ,
From Theorem 5.1 (Theorem 6.4), Lemma 6.1 (Lemma 6.2), and relation (1.7), we get the following result.
Corollary 6.5 If is (uniformly) exponentially stable, then is (uniformly) asymptotically stable.
If , then (7.1) is uniformly stable, exponentially stable and asymptotically stable, since as .
If , then (7.1) is uniformly stable but not asymptotically stable, since .
If with and , then (7.1) is not asymptotically stable. However, goes to zero as .
If with , then (7.1) is not asymptotically stable.
Consequently, , which implies that Eq. (7.1) is uniformly stable but is not asymptotically stable.
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