A New Estimation of the Growth Bound of a Periodic Evolution Family on Banach Spaces

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 timesX-valued functionsf, defined onR + := [0,∞), having the property that the map t 󳨃→ e−ωt‖f(t)‖ is bounded on


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 R + := [0, ∞), having the property that the map   →  − ‖()‖ is bounded on R + , 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  2 -norm of the solution of a periodic Cauchy problem with time-varying coefficients, which is naturally led by the onedimensional heat equation.
The classical theorem of Datko [1] states that a strongly continuous evolution family U = {(, )} ≥≥0 , acting on a real or complex Banach space , is uniformly exponentially stable (i.e., there are two positive constants  and ]) such that ‖ (, )‖ ≤  −](−) , ∀ ≥  ≥ 0, if, for some (and then for all) 1 ≤  < ∞, one has sup As usual, the norm of  and of L() is denoted by ‖⋅‖.Here, L() 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 Z + .
It is known that if T = {()} ≥0 is a strongly continuous semigroup on a complex Banach space  and if 1 ≤  < ∞ and   > 0 are such that then  0 (T) ≤ −1/(  ).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  ∈ (0, ∞) instead of 1 ≤  < ∞.
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 U := {(, )} ≥≥0 ⊂ L() is called a strongly continuous and -periodic (for some  ≥ 1) evolution family if it satisfies the following.
(4) The map  → (, ) : [, ∞) →  is continuous for every  ∈  and is called exponentially bounded if there exist  ∈ R and   ≥ 1, such that ‖ (, )‖ ≤    (−) for  ≥  ≥ 0. ( The growth bound  0 (U) of an exponentially bounded evolution family U is the infimum of all  ∈ R for which there exists   ≥ 1 such that (5) is fulfilled.It is known [4] that The family U is uniformly exponentially stable if its growth bound is negative.A -periodic and strongly continuous evolution family U = {(, ) :  ≥  ≥ 0} is uniformly exponentially stable if and only if  0 (U) is negative or, equivalently, if the spectral radius of the monodromy operator (, 0) 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  ∈ L().If the spectral radius of  is greater than or equal to 1, then, for all 0 <  < 1 and any sequence (  ) with   → 0 (as  → ∞) and ‖(  )‖ ∞ ≤ 1, there exists a unit vector  0 ∈ , such that Moreover, this result remains valid for real Banach spaces provided that the operator  is power bounded; that is, sup{‖  ‖ :  ∈ Z + } is finite.See [6] for further details.

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.
Theorem 8. Let U = {(, )} ≥≥0 be a strongly continuous and -periodic evolution family acting on a real or complex Banach space .Assume that  0 :=  0 (U) is negative, and, in addition, the family is uniformly bounded, and let 0 <  < ∞.Then, there exists Actually, under the addition boundedness assumption, the monodromy operator associated with the family U 0 , that is,  − (, 0), is power bounded, and we can use [6, Theorem 2.1].

Some Examples and Remarks
Then, for every ] > −1/(  ) and every  ≥ , one has Example 2. Let us choose  :=  2 ([0, ], C) 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 {()} ≥0 , given by where   () := ∫  0 () sin(), is a strongly continuous semigroup on .The semigroup T = {()} ≥0 is generated by the linear operator  given by  = ẍ , and the maximal domain of  is the set () of all  ∈  such that  and ẋ are absolutely continuous, ẍ ∈ , and (0) = () = 0.Moreover, () is a self-adjoint operator for every  ≥ 0 [8,Example 1.3,pages 178,198] where (⋅) is a given function in , and  : R + → [1, ∞) is a function having the following properties.
We can estimate a positive constant ] having the property that the map   →  V ‖()‖ is bounded on R + . Indeed, On the other hand Now by Theorem 8, we get  0 (U) ≤ −1/(2 2 ), with  2 being given in (24).
Let  > 0 be fixed, and set Obviously, the family T = {()} ≥0 is a strongly continuous semigroup on  2 (Z + , C), 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  2 (, 0) is a function  ∈  2 (R + , ) such that () ∈ () for all  ≥ 0, and  2 (, 0) holds.A function  ∈ (R + , ) is called a mild solution for for all  ≥ 0. Any  2 (R + , )-valued mild solution of  2 (, 0) is a classical solution.The mild solution of  2 (, 0) leads to the notion of cosine function.See [10] or [11, pages 206-221], for further details.
Recall that a strongly continuous function cos : R + → L() is called cosine function if See [11,Proposition 3.14.4].On the other hand, if  is a generator of a cosine function, the mild solution of the Cauchy Problem  2 (, 0) is given by () = cos(),  ≥ 0. 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.