#### Abstract

Let be a strongly continuous and -periodic evolution family acting on a complex Banach space . We prove that if and then the growth bound of the family is less than or equal to .

#### 1. Introduction

Knowing the rate of convergence of an iterative process, we can control the speed of its convergence. This helps us to obtain the limit of the process, with the desired accuracy, in a determined time. On the other hand, in the study of certain dynamical systems or differential equations, we meet many times -valued functions , defined on , having the property that the map is bounded on , for a suitable real number . Here, and in the following, stands for a real or complex Banach space. Clearly, every has the same property. The infimum of all real numbers which verify the given boundedness condition is called exponential growth of the function . In this note, we obtain an estimate of the exponential growth of a periodic evolution family that satisfies an integral condition, originally given by Datko. The theoretical result allows us to estimate the -norm of the solution of a periodic Cauchy problem with time-varying coefficients, which is naturally led by the one-dimensional heat equation.

The classical theorem of Datko [1] states that a strongly continuous evolution family , acting on a real or complex Banach space , is uniformly exponentially stable (i.e., there are two positive constants and ) such that if, for some (and then for all) , one has As usual, the norm of and of is denoted by . Here, denotes the Banach algebra of all the bounded linear operators acting on . By , we denote the spectrum of the linear operator , and when it is bounded, its spectral radius is defined by We use classical notations for the set of real numbers, complex numbers, and integer numbers. The set of all nonnegative integer numbers will be denoted by .

It is known that if is a strongly continuous semigroup on a complex Banach space and if and are such that then . See [2], where the case of semigroups acting on Hilbert spaces is analyzed, and see [3, pages 81-82] for the general case.

In this note, we extend this result to periodic strongly continuous evolution families acting on real or complex Banach spaces. Moreover, we prove the result with instead of .

As an application of our theoretical results, we provide a simple example, which apparently cannot be treated using the results described above for semigroups.

A family is called a *strongly continuous and* -*periodic* (for some ) *evolution family* if it satisfies the following.(1) for all .(2) for all .(3) for all .(4)The map is continuous for every and is called *exponentially bounded* if there exist and , such that
The growth bound of an exponentially bounded evolution family is the infimum of all for which there exists such that (5) is fulfilled. It is known [4] that
The family is uniformly exponentially stable if its growth bound is negative. A -periodic and strongly continuous evolution family is uniformly exponentially stable if and only if is negative or, equivalently, if the spectral radius of the monodromy operator is less than one. The next result from the abstract theory of operators, originally given by Müller [5], will be useful in what follows.

Lemma 1. *Let be a complex Banach space, and let . If the spectral radius of is greater than or equal to 1, then, for all and any sequence with (as and , there exists a unit vector , such that
*

Moreover, this result remains valid for real Banach spaces provided that the operator is power bounded; that is, is finite. See [6] for further details.

#### 2. Preliminary Results

The following two lemmas will be useful in the proof of Theorem 5 below. Its proof is essentially contained in [6, Theorem 1.2]. Because it has an interest in itself, we state it as a separate statement and infer its proof for the sake of completeness.

Lemma 2. *Let be a nonincreasing function such that . For each , there exists a nonincreasing function , having the properties *(1)*, for all ,*(2)*,
*(3)*, for all .*

*Proof. *Let such that , when . Next, choose the integers in such a way that for all and for all . For example, we can take .

Define the function by
Clearly, is nonincreasing, , and .

We claim that for all . Indeed, choose the integers and such that , . Then, and . Now,
yields

Lemma 3. *Let be a strongly continuous and -periodic evolution family on a complex Banach space such that . Then, for any and every nonincreasing function , with , there exist and a unit vector , such that
*

*Proof. * *Step 1.* Define by
The map is nonincreasing, and .

Since , there exists a norm one vector , such that
This follows by Lemma 1 with , , and . For , let be an integer number, such that . We infer
where . Thus,
*Step 2.* Let us choose such that . By Lemma 2, we can find a nonincreasing function , such that , , and , for all .

In view of the first step, applied to the function , we may choose a unit vector , such that
Denote
Then, , and for all , we have that . Thus,
Now, let us choose , such that
and let
We infer

which completes the proof.

Theorem 4. *Let be a strongly continuous and -periodic evolution family on a complex Banach space , and let . The following three statements are equivalent.*(i)*, for all unit vectors . *(ii)*For every , one has
*(iii)*The growth bound, , is negative.*

The equivalence of (i) and (iii) is given, for example, in [7, Theorem 3.3], and (ii) implying (i) is obvious. We have only to prove that (i) implies (ii). To do this, let , with and . Let and such that Then,

#### 3. The Result and Its Consequences

Theorem 5. *Let be a strongly continuous and -periodic evolution family acting on a complex Banach space . Suppose that the integral condition (22) is fulfilled for any and some (and then for all) . Then, there exists such that .*

*Proof. *Let . The family
is a -periodic evolution, family and . Let with , and let be fixed. By applying Lemma 3 to the -periodic evolution family , there exist and a norm one vector , such that
or equivalently
Hence, via Theorem 4,
That is, . Since and were arbitrary, the last inequality yields that is, .

Corollary 6. *Let be a strongly continuous and -periodic evolution family acting on a complex Banach space . If and are such that
**
then .*

Corollary 7. *Let and be a -semigroup on a Banach space X. If
**
then
*

Theorem 8. *Let be a strongly continuous and -periodic evolution family acting on a real or complex Banach space . Assume that is negative, and, in addition, the family
**
is uniformly bounded, and let . Then, there exists such that , where is defined in (22). *

Actually, under the addition boundedness assumption, the monodromy operator associated with the family , that is, , is power bounded, and we can use [6, Theorem 2.1].

#### 4. Some Examples and Remarks

Now we present two examples which illustrate the theoretical result.

*Example 1. *Let be a continuous and -periodic function, and let us define . Clearly, the family is a norm continuous -periodic evolution family on . If
Then, for every and every , one has

*Example 2. *Let us choose to be the state space. Endowed with the usual inner product and norm, it becomes a complex Hilbert space, and, in addition, the one parameter family , given by
where , is a strongly continuous semigroup on . The semigroup is generated by the linear operator given by , and the maximal domain of is the set of all such that and are absolutely continuous, , and . Moreover, is a self-adjoint operator for every [8, Example 1.3, pages 178, 198]. Consider the nonautonomous Cauchy problem
where is a given function in , and is a function having the following properties.(1) is -periodic for some .(2)There exist and such that

Let . As is well known, the solution of the above Cauchy problem satisfies the evolution condition
where . See [9, Example ].

We can estimate a positive constant having the property that the map is bounded on .

Indeed,
On the other hand

Hence,

Now by Theorem 8, we get , with being given in (24).

*Remark 9. *Even in the case of Hilbert spaces, the inequality
could be sharp.

Indeed, let be the Hilbert space of all -valued sequences verifying

Let be fixed, and set
Obviously, the family is a strongly continuous semigroup on , and

*Remark 10. *It seems that the theoretical result of this paper cannot be applied to wave equations. We try to justify this statement in the following.

Let be a closed operator on a complex Banach space , and let . Consider the Cauchy Problem

Recall that a classical solution of is a function such that for all , and holds. A function is called a mild solution for if
for all . Any -valued mild solution of is a classical solution. The mild solution of leads to the notion of cosine function. See [10] or [11, pages 206–221], for further details.

Recall that a strongly continuous function is called cosine function if(1), where is the identity operator on ,(2) for all and .It is well known that any cosine function is exponentially bounded; that is, there are and such that for all . Moreover, the uniform growth bound of any cosine function is nonnegative [11, Proposition ]. Any cosine function has a generator. More exactly, there exist a positive and a linear operator such that and
See [11, Proposition ]. On the other hand, if is a generator of a cosine function, the mild solution of the Cauchy Problem is given by , . Finally, we remind the reader that the result of this paper refers to exponentially stable evolution families, and then it cannot be applied to cosine functions.

#### Acknowledgment

The authors thank the anonymous referees for their useful suggestions on the preliminary version of this paper. In particular, the authors added the final remarks at the suggestion of the referees.