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The Scientific World Journal
Volume 2013 (2013), Article ID 284630, 6 pages
http://dx.doi.org/10.1155/2013/284630
Research Article

On Weak Exponential Expansiveness of Evolution Families in Banach Spaces

College of Science, China University of Mining and Technology, Xuzhou, Jiangsu 221008, China

Received 5 June 2013; Accepted 9 July 2013

Academic Editors: G. Bonanno, F. J. Garcia-Pacheco, and F. Minhós

Copyright © 2013 Tian Yue et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The aim of this paper is to give several characterizations for the property of weak exponential expansiveness for evolution families in Banach spaces. Variants for weak exponential expansiveness of some well-known results in stability theory (Datko (1973), Rolewicz (1986), Ichikawa (1984), and Megan et al. (2003)) are obtained.

1. Introduction

In recent years, the exponential stability theory of one parameter semigroups of operators and evolution families has witnessed significant development. A number of long-standing open problems have been solved, and the theory seems to have obtained a certain degree of maturity. One of the most important results of the stability theory is due to Datko, who proved in 1970 in [1] that a strongly continuous semigroup of operators is uniformly exponentially stable if and only if for each vector from the Banach space the function lies in . Later, Pazy generalizes the result in [2] for , . In 1973, Dakto [3] generalized the results above and proved that an evolutionary process with uniform exponential growth is uniformly exponentially stable if and only if there exists an exponent such that , for each . This result was improved by Rolewicz in 1986 (see [4]). In [5, 6], the authors generalized the results above in the case of -semigroups and evolutionary process, respectively, and presented a unified treatment in terms of Banach function spaces.

In the last few years, new concepts of exponential expansiveness and in particular, of exponential instability, have been introduced and characterized (see[714]). The cases of uniform exponential instability have been considered in [8] for evolution families and in [10] for linear skew-product flows.

In the present paper, we introduce the concept of weak exponential expansiveness for evolution families which is an extension of classical concept of exponential expansiveness. Our main objective is to give some characterizations for weak exponential expansiveness properties of evolution families in Banach spaces, and variants for weak exponential expansiveness of some well-known results in stability theory (Datko [3], Rolewicz [4], Ichikawa [15], and Megan et al. [8]) are obtained.

2. Preliminaries

Let be a real or complex Banach space. The norm on and on the space of all bounded linear operators on will be denoted by .

Definition 1. A family of bounded linear operators is called an evolution family if the following conditions are satisfied:(i), the identity operator on , for all ;(ii) for all ;(iii)there exist and such that for all and ;(iv)for every and every , the mapping is continuous on ;(v)for every , the operator is injective.

Definition 2. An evolution family is called uniformly expansive if there exists a constant such that for all and .

Definition 3. An evolution family is said to be uniformly exponentially expansive if there are such that for all and .

Remark 4. It is obvious that an evolution family is uniformly exponentially expansive if and only if there are such that for all and .

Remark 5. If the evolution family is uniformly exponentially expansive, then it is uniformly expansive. The converse is not necessarily valid. To show this, we consider the following example.

Example 6. Let be a monotone increasing and bounded continuous function, . The evolution family defined by for all and .

Proof. As a first step, we prove that is uniformly expansive. The evolution family satisfies the inequality for all and , where . Hence, is uniformly expansive.
As a second step, we prove that is not uniformly exponentially expansive. If we suppose that is uniformly exponentially expansive, then, by Remark 4, there exist some constants such that for all .
In particular, for , we obtain , which is absurd for . Hence, is not uniformly exponentially expansive.

Definition 7. An evolution family is called weakly exponentially expansive if there are such that for all there exists with for all .

Remark 8. If the evolution family is uniformly exponentially expansive, then it is weakly exponentially expansive.

The following example shows that the converse is not valid.

Example 9. Let with the Euclidean norm. Consider the evolution family generated by the matrix , where

Proof. We divide the proof into two steps.
As a first step, we prove that is weakly exponentially expansive. For every , there exist and such that .
It is easy to see that and hence for all with , which shows that is weakly exponentially expansive.
As a second step, we prove that is not uniformly exponentially expansive. If we assume that is uniformly exponentially expansive, then there exist some constants such that for all and .
In particular, for , we obtain and hence which shows that is not uniformly exponentially expansive.

3. The Main Results

Theorem 10. The following assertions are equivalent:(i) is weakly exponentially expansive;(ii)there are and such that for every there exist and with for all ;(iii) there exist and such that for each there exists with the property that for every there is with

Proof. (i)(ii) If is weakly exponentially expansive, then by Definition 7, there exist such that for all there exists with the property for all . Let satisfy that . Then, for , we have for all .
(ii)(iii) It is obvious.
(iii)(i) We define and , where and are given by (iii).
From (iii), it results that for each , there exists with the property that for every there is such that
Let , and we have that there is with By induction, we have that where
It is easy to see that () is unbounded. In fact, if () is bounded, then there exists with (). From the relation (20) and , it follows that which is a contradiction because .
So, () is unbounded, and then for , there is such that
Then, and hence

Remark 11. Theorem 10 can be considered a generalization of some results from uniform exponential instability proved in [8].

An important set in what follows is , the set of all nondecreasing functions with the properties:(f1), for all ;(f2), for every .

Theorem 12. An evolution family is weakly exponentially expansive if and only if there are and such that for every there is with for all .

Proof. Necessity. If is weakly exponentially expansive, then by Definition 7, there are such that for all there exists with for all .
Thus, the inequality (26) is satisfied for and .
Sufficiency. We assume for a contradiction that for all and there exists such that for every there is with for all .
In particular, for and , the inequality (28) implies which contradicts the inequality (26). This contradiction proves that is weakly exponentially expansive.

It makes sense to consider also the set all non-decreasing functions with the properties:(g1), for all ;(g2), for every .

Theorem 13. An evolution family is weakly exponentially expansive if and only if there are and such that for every there is with the relation (26).

Proof. Necessity. This is a simple verification for .
Sufficiency. It is similar to the proof of Theorem 12. Indeed, from (28) for and , we have which contradicts the inequality (26).

Remark 14. The preceding theorems are variants for the case of weak exponential expansiveness property of a well-known theorem due to Rolewicz [4].

Corollary 15. An evolution family is weakly exponentially expansive if and only if there are and such that for all there exists with for all .

Proof. It is immediate from Theorem 12 for .

Remark 16. Corollary 15 is the version of a well-known theorem due to Datko [3], for the case of weak exponential expansiveness of evolution families.

In the following corollary, we give a discrete version of Theorems 12 and 13.

Corollary 17. An evolution family is weakly exponentially expansive if and only if there are and such that for all there exists with for all .

Proof. Necessity. This is a simple verification for .
Sufficiency. By Definition 1, we know that , and is continuous on for all , so there exist such that
Let . We suppose that , and from (32), it results that for all .
If , in a similar way we have for all .
Applying Theorems 12 and 13, we conclude that is weakly exponentially expansive.

Corollary 18. An evolution family is weakly exponentially expansive if and only if there is such that for all there exists with for all .

Another characterization of the weak exponential expansiveness is given by the following.

Theorem 19. An evolution family is weakly exponentially expansive if and only if there are such that for every there is with for all .

Proof. Necessity. If is weakly exponentially expansive then by Definition 7, there are such that for all there is with the property that for we have for all , where .
Sufficiency. Let be such that , where and are given by (37). We suppose that is not weakly exponentially expansive. Then, by Theorem 10, for , there exists such that for all and all there is with Then, for , we have which contradicts the inequality (37), the proof is completed.

Acknowledgments

The authors would like to thank the referee for helpful suggestions and comments. This work was supported by the “Fundamental Research Funds for the Central Universities” (no. 2012LWB53).

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