A Generalization for Theorems of Datko and Barbashin Type

Abstract

The goal of the paper is to give some characterizations for the uniform exponential stability of evolution families by unifying the discrete-time versions of the Barbashin-type theorem and the Datko-type theorem.

1. Introduction

In operator theory, a bounded linear operator family () is called an evolution family if(i), the identity;(ii), the evolution property;(iii)for each element , the orbit is continuous. This notion occupies a particularly important role in representing solutions of the nonautonomous linear differential equation on Banach spaces when the operators , , are linear and unbounded. It is difficult to prove the existence of evolution families when studying on infinite dimensional Banach spaces. In fact, the conditions are obtained in several special cases of (1). We will not continue, in this present paper, the existence problem. Instead, we will assume that the evolution family exists and then study its stability.

We say that admits a uniform exponential growth if there are two positive numbers and such that, for every , . When , is called uniformly exponentially stable (u.e.s.). During the past decade, an increasing attention is devoted to the stability of evolution families. For example, in 1970, Datko showed that the evolution family with the uniform exponential growth is u.e.s. if, for each element , In his result, the integral is taken according to the first variable of the evolution family. A similar characterization is named by Barbashin in 1967 when integrating with respect to the second variable: Two results become the starting point for the works [1–3], where the discrete versions were established: (i)(2) .(ii)(3) .Particularly, we can find in [4] the initial studies on unifying the discrete-time versions of the Datko-type theorem and the Barbashin-type theorem. Let us restate the following: the uniform exponential stability is equivalent to the condition where the nondecreasing sequences and belong to class ; that is, if . Following this idea, [5] provides another characterization: It is worth mentioning here that conditions (4) and (5) become the Datko-type theorem when . When , we get the Barbashin-type theorem. Naturally, we raise a question to consider the following assumption: The goal of the paper is to show that condition (6) is equivalent to the uniform exponential stability of the evolution family. The obtained result is an extension of classical theorems due to Barbashin and Datko.

2. Preliminaries

We start the paper with some notations. As usual, we denote by and the set of positive numbers and positive integers, respectively. We write for the set of integers with . Let be a Banach space. The norm on the space of bounded linear operators on is also denoted by . Let denote the set of all nondecreasing positive sequences with . Let us denote by the set of functionals defined on the set of positive sequences with the following conditions:(i) if .(ii)there is so that for every and every .(iii) for every .Notation means the characteristic function of a set . The following lemma is derived from [6, 7].

Lemma 1. If and then , where .

By , we denote the set of all functions with the following properties:(i) for all .(ii)for each , there exists such that . For simplicity, we will use the symbol “sup” instead of . In the whole paper, we always assume that the evolution family admits the uniform exponential growth with the constants and . The following lemmas play an important part in the proof.

Lemma 2. Let . Assume that there exist and satisfying Then is bounded on .

Proof. In the first step, we demonstrate that . Indeed, for each , we have two cases. The first case is . In this case, . The second case is . By we can fix such that . Thus, . Hence, . Next, we evaluate that . For each , there are also two cases.
If then .
If then . We estimate This implies the desired result.

Lemma 3. Let . Assume that there are two constants and and such that . Then there exist such that

Proof.
Step 1. By induction, we now prove that For and , (10) holds. Assume that (10) is true when . Consider . We have that Hence, (10) is true.
Step 2. We prove that there exists such that Indeed, for each , there exist and such that . We have that where and .
Step 3. We prove that there exist such that If then, from the second step, we obtain the desired result. Now we consider the case as . For each , there are two situations as follows. The first situation is . With this situation, we estimate The second situation is . We estimate We rewrite From (15) and (17), we can choose and .
Step 4. For each , there are two situations. The first situation is . Then . The second situation is . Then . We have that This implies the desired result.

As a consequence, we have the following.

Lemma 4. If there exist , and satisfying the hypotheses then is uniformly bounded.

Lemma 5. If there are two constants and , and such that for all then is uniformly exponentially stable.

3. Main Results

Given an evolution family , we define the mapping by

The first characterization is given by the following.

Theorem 6. is u.e.s. if and only if there exist , , , , , and such that (1) (2).

Proof. Let us prove the necessity. Suppose that is uniformly exponentially stable; that is, . We have that Thus we only take , , , and ; .
Now let us prove the sufficiency. Fix . Let . Denote . From the evolution property we estimate It follows that Acting on two both sides, we have that Hence, By Lemma 1, we can fix such that . Now we derive that . From the evolution property we estimate Taking into account that does not depend on . Using Lemma 5, is uniformly exponentially stable. The proof completes.

Our main result is as follows.

Theorem 7. is u.e.s. if and only if there exist , , , , , and such that

Proof. The necessity is clear. Let us prove the sufficiency. Fix . Let . It is obvious that Acting on two both sides of the inequality above, we obtain Therefore, On the other hand, from the evolution property we estimate Now we rewrite the last inequality We consider three cases as follows.
Case 1 (). In (36), let . We get Using Lemma 5 for the case of the sequence , there exists such that Applying Theorem 6, is uniformly exponentially stable.
Case 2 ( and ). With these conditions, it follows from (36) that Let . Using the continuity of the mapping , we obtain Hence, by Lemma 5 for the case of the sequence , there exists such that Using Theorem 6, is uniformly exponentially stable.
Case 3 (, ). In (36), let . We obtain Let . Using the continuity of the mapping , we obtain Using Lemma 5 for the case of the sequence , there exists such that Using Theorem 6, is uniformly exponentially stable.
Case 4 (, ). In this case, we see that Now we again apply Theorem 6 to obtain the desired result.

From Theorem 6, we get the following.

Corollary 8. is u.e.s. if and only if there exist , , and such that

Proof. Using Theorem 7 for and , the proof completes.

Remark 9. Note that if we choose in Corollary 8   () then we obtain Barbashin’s theorem (Datko’s theorem).

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author gratefully acknowledge a postgraduate research scholarship of Nanyang Technological University. The author is deeply grateful to the referees for helpful suggestions.