On the uniform exponential stability of linear skew-product semiflows

The problem of uniform exponential stability of linear skew-product semiflows on locally compact metric space with Banach fibers, is discussed. It is established a connection between the uniform exponential stability of linear skewproduct semiflows and some admissibility-type condition. This approach is based on the method of “test functions”, using a very large class of function spaces, the so-called Orlicz spaces.


Introduction
It is known that many equations from physics can be cast as an abstract Cauchy problem (A) x (t) = A(t)x(t), x(s) = x s ∈ D(A(s)), t ≥ s, t, s ∈ J, on a Banach space X, where J = R or J = R + .Some systems can be described by a Cauchy problem, as for instance the heat equation, the Schrödinger equation, or certain population equations.Also, even the Maxwell's equations can be transformed into a Cauchy problem with slightly more effort.But also seemingly unrelated problems such as delay equations, Markov processes or Boltzmann's equations can be interpreted as Cauchy problems by choosing the "right" state space.Suppose now that the Cauchy problem (A) is well-posed, which means that there exists an evolution family of operators {U (t, s)} t≥s , which solves the problem above (x(t) = U (t, s)x s ).If {A(t)} t≥0 is a bounded linear operators family which belongs to M 1 i.e. sup t≥0 t+1 t A(τ ) dτ < ∞ , then the problem (A) is well-posed and generate a reversible evolution family.
If f is a X -valued locally integrable function on I we will consider the inhomogeneous equation: In this context we refer to a function u(•) as a mild solution of (A, f) if it is the solution of the inhomogeneous mild equation ( * ) u(t) = U (t, s)u(s) + t s U (t, τ )f (τ )dτ , t ≥ s, t, s ∈ J.
The connection between the uniform exponential stability of (A) and the solution of ( * ) is originating in the famous paper of O. Perron [18] from 1930.This idea was extensively analyzed also in the monographs of Massera-Schäffer [16] and Daleckij-Krein [7].
An important extension of these above facts can be found when we are dealing with the theory of linear skew-product (semi)flows instead of the particular case of evolution families.As it is known the linear skew-product semiflows arise as solution operators for variational equation where σ is a semiflow on a locally compact space and A(θ) an unbounded linear operator on a Banach space X , for each θ ∈ Θ.It is also relatively known that the qualitative behavior of (semi-)flows on (locally) compact spaces or (σ -)finite measure spaces is pointed out by concepts as stability or exponential dichotomy of the associated linear skewproduct (semi-)flow.The theory of linear skew-product flows with finite dimensional fibers has come in our days into widespread usage in the area in asymptotics theory of differential equations (see e.g.[9,10,22,23,26]).Also, it is well known the approach of J. Hale (10) which stressed that this theory should be extended in the more general framework of the infinite dimensional setting.In fact, in the last decade many contributions were done in the study of linear skew-product flows with Banach fibers over a compact metric space [1,2,3].Also, many applications were done in [2,9,10,11,25].
In the finite dimensional context the Sacker-Sell spectrum offered an interesting way to describe all of these properties (see [9,23,24,25]).This kind of approach was extended in the last decade to the case of normcontinuous cocycles on infinite dimensional Banach spaces by Latushkin and Stepin.Also, it is interesting to see that almost all interesting infinite dimensional situations, as for instance flows originating from partial differential equations and functional differential equations, only yield strongly continuous cocycles.In this context, there has been studied the dichotomy of linear skew-product semiflows defined on compact spaces (see [2,3,4,5]), and on a locally compact spaces, respectively (see [15]).The idea of associating an evolution semigroup in the expanded case of exponential stability or dichotomy of linear skew-product flow on locally compact metric space Θ , has its origins in the works of Latushkin and Stepin [13], respectively Latushkin, Montgomery-Smith and Schnaubelt [14].
We can notice in this context the idea of "autonomization" of nonautonomous problems by passing from the case of evolution families or more general of linear skew-product semiflows to associated evolution semigroups.
In contrast to this "philosophy", the present paper shows that we can characterize also the exponential stability in terms of the admissibility of some suitable function spaces in a direct way, without so-called evolution semigroups.But of course, this is not the first aim of our paper.
We are setting an connection between the uniform exponential stability of a linear skew-product semiflow on a locally compact metric space and the admissibility of a function space from a large class of function spaces, the so-called Orlicz spaces.Therefore, it is considered a concept of exponential stability for linear skew-product semiflows, which is an extension of the classical concept of exponential stability for time-dependent linear differential equations in Banach spaces (see e.g [6,7,8,16,17]).
Until now, the most common function spaces used for the connection between admissibility and stability were L p -spaces, which are in particular Orlicz spaces, so with this approach it can be obtained also an unified treatment, generalizing the above results.

Preliminaries
In the next we list the principal notations and symbols.For X a Banach space we denote by M(R + , X) the space of all measurable functions (in the sense of Bochner) from R + to X and by: Also for Θ a locally compact metric space we denote by C b (Θ, X) the space of all bounded continuous functions from Θ to X .
It is known that are Banach spaces endowed with the respectively norms: In order to simplify the notations we put for more convenience, we recall the definition of Orlicz spaces.
Let ψ : R + → R + be a non-decreasing function such that, ψ(t) > 0, for all t > 0. Define A function Ψ of this form is called a Young function.For a measurable function f : R + → R and the Young function Ψ , we define The set L Ψ of all f for which there exists k > 0 that M Ψ (kf ) < ∞ is easily checked to be a linear space and endowed with the Luxemburg norm The connection between Orlicz spaces and the L p spaces is given by Remark 2.2.L Ψ = L p if and only if Ψ(t) = t p , for all t ≥ 0. The "only if" part is obvious.Conversely if L Ψ = L p then we have that χ [0,t] Ψ = χ [0,t] p , for all t > 0, and so Ψ −1 (s) = s 1 p , for all s > 0, which implies that Ψ(t) = t p , for all t ≥ 0.
As it is known there are many other interesting examples of Orlicz spaces, but when it is involved an admissibility condition to prove the exponential stability of some differential system it is very convenient to use a small input space, in order to verify the admissibility condition for not so many "test" functions.
In order to find a "tiny" function space which can be used for instance as "input space", in our approach, we present bellow a function space which is contained in all L p , for p ∈ [1, ∞), and which is not a L p -type space.In what follows, if L Ψ is a Orlicz space we denote by

Example 2.2. If we take Ψ
Remark 2.3.L Ψ (X) is a Banach space endowed with the norm and so by that we noticed above, it follows that χ A1 ∈ L Ψ and χ A1 Ψ = 0. Now it is obvious that m(A 1 ) = 0 and so |u| < ∞ a.e.In order to prove that L Ψ (X) is a Banach space, let {f n } n∈N be a Cauchy sequence in L Ψ (X).We will show that there exists {f n k } k∈N a subsequence of {f n } and f ∈ L Ψ such that Using the fact that {f n } n∈N is Cauchy sequence, it follows that there exists {f n k } k∈N a subsequence of {f n } such that It follows that the sum −→ g, and g m (t) h(t), for all t ≥ 0.
By the other hand for all t ≥ 0 and all m ∈ N.
Knowing the fact that g ∈ L Ψ and g = h a.e., we have that Let f : R + → X the map given by, Thus we have that f n k → f a.e. and so f is measurable and for all t ∈ R + \ A 4 and all m ∈ N, therefore Thus, we have that f − f nm ∈ L Ψ for all m ∈ N * and so f ∈ L Ψ and Since {f n } n∈N is a Cauchy sequence with a convergent subsequence, then {f n } n∈N it is also convergent.

Remark 2.4. If {f
when n → ∞, then there exists {f n k } k∈N a subsequence of {f n } n∈N such that −→ f then {f n } n∈N is a Cauchy sequence and from the proof of the Remark 2.3.there exists {f n k } k∈N a subsequence of {f n } and g ∈ LΨ such that This shows that f = g a.e. and so f n k → f a.e.

Proposition 2.1. If Ψ is an Young function of the Orlicz space L
Proof.(i) First let us prove that the map b : R * In order to prove that a Ψ is nondecreasing let 0 , it results that a Ψ is a nondecreasing function.
Example 2.4.Let X be a Banach space, Θ a locally compact metric space, T = {T (t)} t≥0 a C 0 -semigroup and {U (θ)} θ∈Θ a bounded strongly continuous family of idempotent operators with the property that is a strongly continuous skew-products semi-flow.
Example 2.5.Let σ be a flow on a compact metric space Θ and T = {T (t)} t≥0 a C 0 -semigroup on X .For every strongly continuous function D : Θ → B(X) there is a linear skew-product semiflow For the proof of this fact see [2].
Example 2.7.Let Θ be a compact metric space, σ a semi-flow on Θ, X a Banach space A : Θ → B(X) a continuous mapping.If Φ(θ, t)x is the solution of the abstract Cauchy problem then the pair π = (Φ, σ) is a strongly continuous skew-product semiflow.Many of these equations are obtained when linearize the nonlinear differential equations (see for instance [25]).
Also, a large numbers of examples of strongly continuous skew-products semi-flow are provided in the recent literature (see for instance [14] ).

Definition 2.3.
A strongly continuous skew-products semi-flow π = (Φ, σ) is called uniformly exponentially stable (u.e.s) if there exists N, ν > 0 such that In what follows for f ∈ L 1 (R + , X) we define x f : Θ × R + → X the map given by From the Definition 2.2. it is easy to see that Remark 2.6.If we are in the case of the strongly continuous skewproduct semiflow provided by Example 2.4., then for all θ ∈ Θ, x f (θ, •) is the solution of the Cauchy problem

The main results
Lemma 3.1.If the Orlicz space L Ψ is admissible to π then there is Proof.We set now V : −→ g then, by Remark 2.4, there exists a subsequence −→ g a.e. for k → ∞ for all k ∈ N, θ ∈ Θ, t ≥ 0, which implies that It follows, using again the Remark 2.4, that V f = g , and hence V is closed and so, by the Closed-Graph Theorem it is also bounded.So we obtain that Proof.By the hypothesis we have that h(n + 1) ≤ ah(s) + b, for all n ∈ N and all s ∈ [n, n + 1] and from here Using again the hypothesis, we obtain that h(t) ≤ ah(n) + b ≤ ac + b, for all n ∈ N, and all t ∈ [n, n + 1].Lemma 3.3.If an Orlicz space L Ψ is admissible to π then the following statements hold: It results that The condition (ii) follows directly from (i) and Lemma 3.2.

Remark. 3.1.
It follows easily from Definition 2.3 that if π is u.e.s then for all f ∈ L ∞ (R + , X) the map u f : R + → C b (Θ, X) given by u f (t) = x f (•, t) belongs also to L ∞ (R + , C b (Θ, X)).

Example 2 . 1 .
The classical L p -spaces are examples of Orlicz spaces.
Indeed one can see that t m ≤ m!Ψ(t) for all t ≥ 0 and all m ∈ N * which implies that L Ψ ⊂ L m , for all m ∈ N * .Having in mind that L m L m+1 ⊂ L p for all p ∈ [m, m + 1], and all m ∈ N * , it follows that L Ψ ⊂ L p , for all p ∈ [m, m + 1], and all m ∈ N * .