Translation Invariant Spaces and Asymptotic Properties of Variational Equations

and Applied Analysis 3 We consider the general setting of variational equations described by skew-product flows, and we associate a control system on the real line. Beside obtaining new conditions for the existence of uniform or exponential dichotomy of skew-product flows, the main aim is to clarify the chart of the connections between the classes of translation invariant function spaces that play the role of the input class or of the output class with respect to the associated control system, proposing a merger between the functional methods proceeding from interpolation theory and the qualitative techniques from the asymptotic theory of dynamical systems in infinite dimensional spaces. We consider the most general case of skew-product flows, without any assumption concerning the flow or the cocycle, without any invertibility property, and we work without assuming any initial splitting of the state space and without imposing any invariance property. Our central aim is to establish the existence of the dichotomous behaviors with all their properties see Definitions 3.5 and 4.1 based only on the minimal solvability of an associated control system described at every point of the base space by an integral equation on the real line. First, we deduce conditions for the existence of uniform dichotomy of skewproduct flows and we discuss the technical consequences implied by the solvability of the associated control system between two appropriate translation invariant spaces. We point out, for the first time, that an adequate solvability on the real line of the associated integral control system see Definition 3.6 implies both the existence of the uniform dichotomy projections as well as their uniform boundedness. Next, the attention focuses on the exponential behavior on the stable and unstable manifold, preserving the solvability concept from the previous section and modifying the properties of the input and the output spaces. Thus, we deduce a clear overview on the representative classes of function spaces which should be considered in the detection of the exponential dichotomy of skew-product flows in terms of the solvability of associated control systems on the real line. The obtained results provide not only new necessary and sufficient conditions for exponential dichotomy, but also a complete diagram of the specific delimitations between the classes of function spaces which may be considered in the study of the exponential dichotomy compared with those from the uniform dichotomy case. Moreover, we point out which are the specific properties of the underlying spaces which make a difference between the sufficient hypotheses and the necessary conditions for the existence of exponential dichotomy of skew-product flows. Finally, we motivate our techniques by illustrative examples and present several interesting applications of the main theorems which generalize the input-output type results of previous research in this topic, among, we mention the well-known theorems due to Perron 11 , Daleckii and Krein 21 , Massera and Schäffer 22 , Van Minh et al. 7 , and so forth. 2. Banach Function Spaces: Basic Notations and Preliminaries In this section, for the sake of clarity, we recall several definitions and properties of Banach function spaces, and, also, we establish the notations that will be used throughout the paper. Let denote the set of real numbers, let {t ∈ : t ≥ 0}, and let − {t ∈ : t ≤ 0}. For everyA ⊂ , χA denotes the characteristic function of the setA. LetM , be the linear space of all Lebesgue measurable functions u : → identifying the functions which are equal almost everywhere. Definition 2.1. A linear subspace B ⊂ M , is called normed function space if there is a mapping | · |B : B → such that the following properties hold: 4 Abstract and Applied Analysis i |u|B 0 if and only if u 0 a.e.; ii |αu|B |α||u|B, for all α, u ∈ × B; iii |u v|B ≤ |u|B |v|B , for all u, v ∈ B; iv if |u t | ≤ |v t | a.e. t ∈ and v ∈ B, then u ∈ B and |u|B ≤ |v|B . If B, | · |B is complete, then B is called a Banach function space. Remark 2.2. If B, | · |B is a Banach function space and u ∈ B, then also |u · | ∈ B. Definition 2.3. A Banach function space B, | · |B is said to be invariant under translations if for every u, t ∈ B × the function ut : → , ut s u s − t belongs to B and |ut|B |u|B. Let Cc , be the linear space of all continuous functions v : → with compact support. We denote by T the class of all Banach function spaces B which are invariant under translations, Cc , ⊂ B and i for every t > 0 there is c t > 0 such that ∫ t 0 |u τ |dτ ≤ c t |u|B, for all u ∈ B; ii if B \ L1 , / ∅, then there is a continuous function γ ∈ B \ L1 , . Remark 2.4. Let B ∈ T . Then, the following properties hold: i if J ⊂ is a bounded interval, then χJ ∈ B. ii if un → u in B, then there is a subsequence ukn ⊂ un which converges to u a.e. see, e.g., 25 . Remark 2.5. Let B ∈ T . If ν > 0 and eν : → is defined by eν t ⎧ ⎨ ⎩ e−νt, t ≥ 0, 0, t < 0, 2.1 then it is easy to see that


Introduction
Starting from a collection of open questions related to the modeling of the equations of mathematical physics in the unified setting of dynamical systems, the study of their qualitative properties became a domain of large interest and with a wide applicability area.In this context, the interaction between the modern methods of pure mathematics and questions arising naturally from mathematical physics created a very active field of research see 1-18 and the references therein .In recent years, some interesting unsolved problems concerning the long-time behavior of dynamical systems were identified, whose potential results would be of major importance in the process of understanding, clarifying, and solving some of the essential problems belonging to a wide range of scientific domains, among, we mention: fluid mechanics, aeronautics, magnetism, ecology, population dynamics, and so forth.Generally, the asymptotic behavior of the solutions of nonlinear evolution equations arising in mathematical physics can be described in terms of attractors, which are often studied by constructing the skew-product flows of the dynamical processes.
It was natural then to independently consider and analyze the asymptotic behavior of variational systems modeled by skew-product flows see 3-5, 14-19 .In this framework, two of the most important asymptotic properties are described by uniform dichotomy and exponential dichotomy.Both properties focus on the decomposition of the state space into a direct sum of two closed invariant subspaces such that the solution on these subspaces uniformly or exponentially decays backward and forward in time, and the splitting holds at every point of the flow's domain.Precisely, these phenomena naturally lead to the study of the existence of stable and unstable invariant manifolds.It is worth mentioning that starting with the remarkable works of Coppel 20 , Daleckii and Krein 21 , and Massera and Schäffer 22 the study of the dichotomy had a notable impact on the development of the qualitative theory of dynamical systems see 1-9, 13, 14, 17, 18, 23 .A very important step in the infinite-dimensional asymptotic theory of dynamical systems was made by Van Minh et al. in 7 where the authors proposed a unified treatment of the stability, instability, and dichotomy of evolution families on the half-line via inputoutput techniques.Their paper carried out a beautiful connection between the classical techniques originating in the pioneering works of Perron 11 and Maȋzel 24 and the natural requests imposed by the development of the infinite-dimensional systems theory.They extended the applicability area of the so-called admissibility techniques developed by Massera and Schäffer in 22 , from differential equations in infinite-dimensional spaces to general evolutionary processes described by propagators.The authors pointed out that instead of characterizing the behavior of a homogeneous equation in terms of the solvability of the associated inhomogeneous equation see [20][21][22] one may detect the asymptotic properties by analyzing the existence of the solutions of the associated integral system given by the variation of constants formula.These new methods technically moved the central investigation of the qualitative properties into a different sphere, where the study strongly relied on control-type arguments.It is important to mention that the control-type techniques have been also successfully used by Palmer see 9 and by Rodrigues and Ruas-Filho see 13 in order to formulate characterizations for exponential dichotomy in terms of the Fredholm Alternative.Starting with these papers, the interaction between control theory and the asymptotic theory of dynamical systems became more profound, and the obtained results covered a large variety of open problems see, e.g., 1, 2, 12, 14-17, 23 and the references therein .
Despite the density of papers devoted to the study of the dichotomy in the past few years and in contrast with the apparent impression that the phenomenon is well understood, a large number of unsolved problems still raise in this topic, most of them concerning the variational case.In the present paper, we will provide a complete answer to such an open question.We start from a natural problem of finding suitable conditions for the existence of uniform dichotomy as well as of exponential dichotomy using control-type methods, emphasizing on the identification of the essential structures involved in such a construction, as the input-output system, the eligible spaces, the interplay between their main properties, the specific lines that make the differences between a necessary and a sufficient condition, and the proper motivation of each underlying condition.
In this paper, we propose an inedit link between the theory of function spaces and the dichotomous behavior of the solutions of infinite dimensional variational systems, which offers a deeper understanding of the subtle mechanisms that govern the control-type approaches in the study of the existence of the invariant stable and unstable manifolds.
We consider the general setting of variational equations described by skew-product flows, and we associate a control system on the real line.Beside obtaining new conditions for the existence of uniform or exponential dichotomy of skew-product flows, the main aim is to clarify the chart of the connections between the classes of translation invariant function spaces that play the role of the input class or of the output class with respect to the associated control system, proposing a merger between the functional methods proceeding from interpolation theory and the qualitative techniques from the asymptotic theory of dynamical systems in infinite dimensional spaces.
We consider the most general case of skew-product flows, without any assumption concerning the flow or the cocycle, without any invertibility property, and we work without assuming any initial splitting of the state space and without imposing any invariance property.Our central aim is to establish the existence of the dichotomous behaviors with all their properties see Definitions 3.5 and 4.1 based only on the minimal solvability of an associated control system described at every point of the base space by an integral equation on the real line.First, we deduce conditions for the existence of uniform dichotomy of skewproduct flows and we discuss the technical consequences implied by the solvability of the associated control system between two appropriate translation invariant spaces.We point out, for the first time, that an adequate solvability on the real line of the associated integral control system see Definition 3.6 implies both the existence of the uniform dichotomy projections as well as their uniform boundedness.Next, the attention focuses on the exponential behavior on the stable and unstable manifold, preserving the solvability concept from the previous section and modifying the properties of the input and the output spaces.Thus, we deduce a clear overview on the representative classes of function spaces which should be considered in the detection of the exponential dichotomy of skew-product flows in terms of the solvability of associated control systems on the real line.The obtained results provide not only new necessary and sufficient conditions for exponential dichotomy, but also a complete diagram of the specific delimitations between the classes of function spaces which may be considered in the study of the exponential dichotomy compared with those from the uniform dichotomy case.Moreover, we point out which are the specific properties of the underlying spaces which make a difference between the sufficient hypotheses and the necessary conditions for the existence of exponential dichotomy of skew-product flows.Finally, we motivate our techniques by illustrative examples and present several interesting applications of the main theorems which generalize the input-output type results of previous research in this topic, among, we mention the well-known theorems due to Perron 11

Banach Function Spaces: Basic Notations and Preliminaries
In this section, for the sake of clarity, we recall several definitions and properties of Banach function spaces, and, also, we establish the notations that will be used throughout the paper.
Let Ê denote the set of real numbers, let Ê {t ∈ Ê : t ≥ 0}, and let Ê − {t ∈ Ê : t ≤ 0}.For every A ⊂ Ê, χ A denotes the characteristic function of the set A. Let M Ê, Ê be the linear space of all Lebesgue measurable functions u : Ê → Ê identifying the functions which are equal almost everywhere.Let C c Ê, Ê be the linear space of all continuous functions v : Ê → Ê with compact support.We denote by T Ê the class of all Banach function spaces B which are invariant under translations, C c Ê, Ê ⊂ B and i for every t > 0 there is c t > 0 such that ii if u n → u in B, then there is a subsequence u k n ⊂ u n which converges to u a.e.see, e.g., 25 .
Remark 2.5.Let B ∈ T Ê .If ν > 0 and e ν : Ê → Ê is defined by then it is easy to see that

2.2
It follows that e ν ∈ B and Since v is continuous on a, b , there is M > 0 such that |v t | ≤ M, for all t ∈ a, b .Then, we have that

2.6
We observe that Let t ≥ 1 and let u ∈ O ϕ Ê, Ê \ {0}.Taking into account that Y ϕ is a convex function and using Jensen's inequality see, e.g., 26 , we deduce that This implies that In addition, using 2.9 , we have that In what follows, we will introduce three remarkable subclasses of T Ê , which will have an essential role in the study of the existence of dichotomy from the next sections.To do this, we first need the following. 2.17 Proof.We consider the operators

2.18
We have that Z and W are correctly defined bounded linear operators.Moreover, the restrictions

Notations
If X is a Banach space, for every Banach function space B ∈ T Ê , we denote by B Ê, X the space of all Bochner measurable functions v : Ê → X with the property that the mapping

2.19
B Ê, X is a Banach space.We also denote by C 0,c Ê, X the linear space of all continuous functions v : Ê → X with compact support contained in 0, ∞ .It is easy to see that C 0,c Ê, X ⊂ B Ê, X , for all B ∈ T Ê .

Uniform Dichotomy for Skew-Product Flows
In this section, we start our investigation by studying the existence of by the upper and lower uniform boundedness of the solution in a uniform way on certain complemented subspaces.
We will employ a control-type technique and we will show that the use of the function spaces, from the class T Ê introduced in the previous section, provides several interesting conclusions concerning the qualitative behavior of the solutions of variational equations.Let X be a real or complex Banach space and let I d denote the identity operator on X.The norm on X and on B X -the Banach algebra of all bounded linear operators on X, will be denoted by • .Let Θ, d be a metric space.
ii Let {T t } t≥0 be a C 0 -semigroup on the Banach space X and let Θ be a metric space.
ii 1 If σ is an arbitrary flow on Θ and Φ T θ, t : T t , then π T Φ T , σ is a skewproduct flow.
ii 2 Let σ : Θ × Ê → Θ, σ θ, t θ be the projection flow on Θ and let {P θ } θ∈Θ ⊂ B X be a uniformly bounded family of projections such that P θ T t T t P θ , for all θ, t ∈ Θ × Ê .If Φ P θ, t : P θ T t , then Starting with the remarkable work of Foias et al. see 19 , the qualitative theory of dynamical systems acquired a new perspective concerning the connections between bifurcation theory and the mathematical modeling of nonlinear equations.In 19 , the authors proved that classical equations like Navier-Stokes, Taylor-Couette, and Bubnov-Galerkin can be modeled and studied in the unified setting of skew-product flows.In this context, it was pointed out that the skew-product flows often proceed from the linearization of nonlinear equations.Thus, classical examples of skew-product flows arise as operator solutions for variational equations.
Example 3.4 The variational equation .Let Θ be a locally compact metric space and let σ be a flow on Θ.Let X be a Banach space and let {A θ : D A θ ⊆ X → X : θ ∈ Θ} be a family of densely defined closed operators.We consider the variational equation An important asymptotic behavior of skew-product flows is described by the uniform dichotomy, which relies on the splitting of the Banach space X at every point θ ∈ Θ into a direct sum of two invariant subspaces such that on the first subspace the trajectory solution is uniformly stable, on the second subspace the restriction of the cocycle is reversible and also the trajectory solution is uniformly unstable on the second subspace.This is given by the following.

Definition 3.5. A skew-product flow π
Φ, σ is said to be uniformly dichotomic if there exist a family of projections {P θ } θ∈Θ ⊂ B X and a constant K ≥ 1 such that the following properties hold: i Φ θ, t P θ P σ θ, t Φ θ, t , for all θ, t ∈ Θ × Ê ; ii Φ θ, t x ≤ K x , for all t ≥ 0, all x ∈ Range P θ and all θ ∈ Θ; iii the restriction Φ θ, t | : Ker P θ → Ker P σ θ, t is an isomorphism, for all θ, t ∈ Θ × Ê ; iv Φ θ, t y ≥ 1/K y , for all t ≥ 0, all y ∈ Ker P θ and all θ ∈ Θ; In what follows, our main attention will focus on finding suitable conditions for the existence of uniform dichotomy for skew-product flows.To do this, we will introduce an integral control system associated with a skew-product flow such that the input and the output spaces of the system belong to the general class T Ê .We will emphasize that the class T Ê has an essential role in the study of the dichotomous behavior of variational equations.
Let I, O be two Banach function spaces with I, O ∈ T Ê .Let π Φ, σ be a skewproduct flow on X × Θ.We associate with π the input-output control system E π E θ θ∈Θ , where for every θ ∈ Θ such that the input function v ∈ C 0,c Ê, X and the output function f ∈ O Ê, X .
Definition 3.6.The pair O Ê, X , I Ê, X is said to be uniformly admissible for the system E π if there is L > 0 such that for every θ ∈ Θ, the following properties hold: i for every v ∈ C 0,c Ê, X there exists f ∈ O Ê, X such that the pair f, v satisfies Remark 3.7.i According to this admissibility concept, it is sufficient to choose all the input functions from the space C 0,c Ê, X , and, thus, we point out that C 0,c Ê, X is in fact the smaller possible input space that can be used in the input-output study of the dichotomy.
ii It is also interesting to see that the norm estimation from ii reflects the presence and implicitly the structure of the space I Ê, X .Actually, condition ii shows that the norm of each output function in the space O Ê, X is bounded by the norm of the input function in the space I Ê, X uniformly with respect to θ ∈ Θ.
iii In the admissibility concept, there is no need to require the uniqueness of the output function in the property i , because this follows from condition ii .Indeed, if the pair O Ê, X , I Ê, X is uniformly admissible for the system E π , then from ii we deduce that for every θ ∈ Θ and every v ∈ C 0,c Ê, X there exists In what follows we will analyze the implications of the uniform admissibility of the pair O Ê, X , I Ê, X with I, O ∈ T Ê concerning the asymptotic behavior of skew-product flows.With this purpose we introduce two category of subspaces stable and unstable and we will point out their role in the detection of the uniform dichotomy.
For every x, θ ∈ X × Θ, we consider the function called the trajectory determined by the vector x and the point θ ∈ Θ.
For every θ ∈ Θ, we denote by F θ the linear space of all functions ϕ : Ê → X with the property that For every θ ∈ Θ, we consider the stable subset and, respectively, the unstable subset

3.4
Remark 3.8.It is easy to see that for every θ ∈ Θ, S θ , and U θ are linear subspaces.Therefore, in all what follows, we will refer S θ as the stable subspace and, respectively, U θ as the unstable subspace, for each θ ∈ Θ.
Proposition 3.9.For every θ, t ∈ Θ × Ê , the following assertions hold: Proof.The property i is immediate.To prove the assertion ii let M, ω > 0 be given by x.We set y Φ θ, t x, and we consider ϕ s , s<0.

3.5
We observe that ψ s ≤ ϕ s Me ωt χ 0,t s x , for all s ∈ Ê, and since ϕ ∈ O Ê, X , we deduce that ψ ∈ O Ê, X .Using the fact that ϕ ∈ F θ , we obtain that Then, we define the function δ : Ê → X, δ s ψ s t and since O Ê, X is invariant under translations, we deduce that δ ∈ O Ê, X .Moreover, from 3.6 , it follows that The relation 3.7 implies that δ ∈ F σ θ, t , so y δ 0 ∈ U σ θ, t .
Conversely, let z ∈ U σ θ, t .Then, there is h ∈ F σ θ, t ∩ O Ê, X with h 0 z.Taking q : Ê → X, q s h s − t , we have that q ∈ O Ê, X and In particular, for τ ≤ s ≤ 0, from 3.8 , we deduce that q ∈ F θ .This implies that q 0 ∈ U θ .Then, z h 0 q t Φ θ, t q 0 ∈ Φ θ, t U θ and the proof is complete.
Remark 3.10.From Proposition 3.9 ii , we have that for every θ, t ∈ Θ × Ê the restriction We also note that according to Proposition 3.9 one may deduce that, the stable subspace and the unstable subspace are candidates for the possible splitting of the main space X required by any dichotomous behavior.
In what follows, we will study the behavior of the cocycle on the stable subspace and also on the unstable subspace and we will deduce several interesting properties of these subspaces in the hypothesis that a pair O Ê, X , I Ê, X of spaces from the class T Ê is admissible for the control system associated with the skew-product flow.

Theorem 3.11
The behavior on the stable subspace .If the pair O Ê, X , I Ê, X is uniformly admissible for the system E π , then the following assertions hold: i there is K > 0 such that Φ θ, t x ≤ K x , for all t ≥ 0, all x ∈ S θ and all θ ∈ Θ; ii S θ is a closed linear subspace, for all θ ∈ Θ.
An easy computation shows that the pair f, v satisfies E θ .Then,

3.11
From v t ≤ α t Me ω x , for all t ∈ Ê, we obtain that 12 it follows that Φ θ, t x χ t−1,t s ≤ Me ω f s , ∀s ∈ Ê.

3.13
Since O is invariant under translations, we deduce that

3.14
Using relations 3.11 and 3.14 , we have that

3.16
ii Let θ ∈ Θ and let x n ⊂ S θ with x n → n → ∞ x.For every n ∈ AE, we consider the sequence 3.17 We have that v n ∈ C 0c Ê, X , for all n ∈ AE and using similar arguments with those used in relation 3.10 , we obtain that f n ∈ O Ê, X , for all n ∈ AE.An easy computation shows that the pair According to our hypothesis there is, f ∈ O Ê, X such that the pair f, v satisfies E θ .
Taking u n v n − v and g n f n − f we observe that u n ∈ C 0c Ê, X , g n ∈ O Ê, X , and the pair g n , u n satisfies E θ .This implies that

3.18
From v n t − v t ≤ α t Me ω x n − x , for all t ∈ Ê and all n ∈ AE, we deduce that x , ∀n ∈ AE.

3.19
From 3.18 and 3.19 , it follows that 4 ii , we have that there is a subsequence f k n and a negligible set

3.20
Because the pair f, v satisfies E θ , we obtain that

3.21
This shows that f t λ x,θ t , for all t ≥ r.Then, from λ x,θ t ≤ f t Me ωr x χ 0,r t , ∀t ∈ Ê,

3.22
using the fact that f ∈ O Ê, X and Remark 2.4 i , we obtain that λ x,θ ∈ O Ê, X , so x ∈ S θ .
In conclusion, S θ is a closed linear subspace, for all θ ∈ Θ.
Theorem 3.12 The behavior on the unstable subspace .If the pair O Ê, X , I Ê, X is uniformly admissible for the system E π , then the following assertions hold: i there is K > 0 such that Φ θ, t y ≥ 1/K y , for all t ≥ 0, all y ∈ U θ and all θ ∈ Θ; ii U θ is a closed linear subspace, for all θ ∈ Θ.
i Let θ ∈ Θ and let y ∈ U θ .Then, there is ϕ ∈ F θ ∩ O Ê, X with ϕ 0 y.Let t > 0. We consider the functions

3.23
We have that v ∈ C 0c Ê, X and f is continuous.Let m sup s∈ 0,t 1 f s .Then, we have that f s ≤ ϕ s mχ 0,t 1 s , ∀s ∈ Ê.

3.24
From 3.24 and Remark 2.4 i , we deduce that f ∈ O Ê, X .An easy computation shows that the pair f, v satisfies E θ .Then, according to our hypothesis, we have that

3.27
Using the invariance under translations of the space O from relation 3.27 , we obtain that

3.29
ii Let θ ∈ Θ and let y n ⊂ U θ with y n → y.Then, for every n ∈ AE, there is ϕ n ∈ O Ê, X ∩ F θ with ϕ n 0 y n .For every n ∈ AE, we consider the functions

3.30
We have that v n ∈ C 0c Ê, X , and, using similar arguments with those used in relation 3.24 , we deduce that f n ∈ O Ê, X , for all n ∈ AE.An easy computation shows that the pair

3.31
According to our hypothesis, there is f ∈ O Ê, X such that the pair f, v satisfies E θ .In particular, this implies that f ∈ F θ .Moreover, for every n ∈ AE, the pair f n − f, v n − v satisfies E θ .According to our hypothesis, it follows that

3.32
We have that v n t − v t ≤ α t Me ω y n − y , for all t ∈ Ê and all n ∈ AE, so v n − v I Ê,X ≤ Me ω |α| I y n − y , ∀n ∈ AE.

3.33
From 3.32 and 3.33 it follows that This implies that y ∈ U θ , so U θ is a closed linear subspace.
Taking into account the above results it makes sense to study whether the uniform admissibility of a pair of function spaces from the class T Ê is a sufficient condition for the existence of the uniform dichotomy.Thus, the main result of this section is as follows.
Theorem 3.13 Sufficient condition for uniform dichotomy .Let O, I ∈ T Ê and let π Φ, σ be a skew-product flow on X × Θ.If the pair O Ê, X , I Ê, X is uniformly admissible for the system E π , then π is uniformly dichotomic.

3.35
Then, f t ≤ ϕ t λ x,θ t , for all t ∈ Ê.This implies that f ∈ O Ê, X .An easy computation shows that the pair f, 0 satisfies E θ .Then, according to our hypothesis, it follows that f O Ê,X 0, so f t 0 a.e.t ∈ Ê. Observing that f is continuous, we obtain that f t 0, for all t ∈ Ê.In particular, we have that x f 0 0.
Let θ ∈ Θ and let x ∈ X.Let v : Ê → X, v t α t Φ θ, t x.Then, v ∈ C 0c Ê, X , so there is f ∈ O Ê, X such that the pair f, v satisfies E θ .In particular, this implies that f ∈ F θ , so f 0 ∈ U θ .In addition, we observe that

3.36
Setting z x f 0 x from 3.36 , we have that λ z x ,θ t f t , for all t ≥ 1.It follows that Me ω z x χ 0,1 t , ∀t ∈ Ê.

3.37
From relation 3.37 and Remark 2.4 i we obtain that λ z x ,θ ∈ O Ê, X , so z x ∈ S θ .This shows that x z x − f 0 ∈ S θ U θ , so S θ U θ X.According to Steps 1 and 2, Theorem 3.11 ii , and Theorem 3.12 ii , we deduce that S θ ⊕ U θ X, ∀θ ∈ Θ.

3.38
For every θ ∈ Θ we denote by P θ the projection with the property that Range P θ S θ , Ker P θ U θ .
Let θ ∈ Θ and let x ∈ X.Let x θ s P θ x and let x θ u

3.41
We have that v ∈ C 0c Ê, X and f is continuous.From x θ s ∈ Range P θ S θ , we have that the function λ x θ s ,θ belongs to O Ê, X .Setting m sup t∈ 0,1 f t and observing that f t ≤ ψ t mχ 0,1 t λ x θ s ,θ t , ∀t ∈ Ê, 3.42 from 3.42 , we deduce that f ∈ O Ê, X .An easy computation shows that the pair f, v satisfies E θ .This implies that

3.45
Using the invariance under translations of the space O, from relation 3.45 we deduce that

3.47
we obtain that v I Ê,X ≤ |α| I Me ω x .

3.50
Taking into account that γ does not depend on θ or x, it follows that relation 3.50 holds, for all θ ∈ Θ and all x ∈ X, so P θ ≤ 1 γ , for all θ ∈ Θ.
Finally, from Theorem 3.11 i and Theorem 3.12 i , we conclude that π is uniformly dichotomic.
Remark 3.14.Relation 3.39 shows that the stable subspace and the instable subspace play a central role in the detection of the dichotomous behavior of a skew-product flow and gives a comprehensible motivation for their usual appellation.

Exponential Dichotomy of Skew-Product Flows
In the previous section, we have obtained sufficient conditions for the uniform dichotomy of a skew-product flow π Φ, σ on X × Θ in terms of the uniform admissibility of the pair O Ê, X , I Ê, X for the associated control system E π , where O, I ∈ T Ê .The natural question arises: which are the additional preferably minimal hypotheses under which this admissibility may provide the existence of the exponential dichotomy?In this context, the main purpose of this section is to establish which are the most general classes of Banach function spaces where O or I may belong to, such that the uniform admissibility of the pair O Ê, X , I Ê, X for the control system E π is a sufficient and also a necessary condition for the existence of exponential dichotomy.Let X be a real or complex Banach space and let Θ, d be a metric space.Let π Φ, σ be a skew-product flow on X × Θ.
Before proceeding to the next steps, we need a technical lemma.Lemma 4.2.If a skew-product flow π is exponentially dichotomic with respect to a family of projections {P θ } θ∈Θ , then sup θ∈Θ P θ < ∞.
Proof.Let K, ν > 0 be given by Definition 4.1 and let M, ω > 0 be given by Definition 3.2.For every x, θ ∈ X × Θ and every t ≥ 0, we have that which implies that
Remark 4.3.i Using Lemma 4.2, we deduce that if a skew-product flow π is exponentially dichotomic with respect to a family of projections {P θ } θ∈Θ , then π is uniformly dichotomic with respect to the same family of projections.
ii If a skew-product flow π is exponentially dichotomic with respect to a family of projections {P θ } θ∈Θ , then this family is uniquely determined see, e.g., 18 , Remark 2.5 .
Remark 4.4.In the description of any dichotomous behavior, the properties i and iii are inherent, because beside the splitting of the space ensured by the presence of the dichotomy projections, these properties reflect both the invariance with respect to the decomposition induced by each projection as well as the reversibility of the cocycle restricted to the kernel of each projection.
In this context, it is extremely important to note that if in the detection of the dichotomy one assumes from the very beginning that there exist a projection family such that the invariance property i and the reversibility condition iii hold, then the dichotomy concept is resumed to a stability property ii and to an instability condition iv , which via iii will consist only of a double stability.Thus, if in the study of the dichotomy one considers i and iii as working hypotheses, then the entire investigation is reduced to a quasitrivial case of double stability.
In conclusion, in the study of the existence of uniform or exponential dichotomy, it is essential to determine conditions which imply the existence of the projection family and also the fulfillment of all the conditions from Definition 4.1.Now let O, I be two Banach function spaces such that O, I ∈ T Ê .According to the main result in the previous section see Theorem 3.13 , if the pair O Ê, X , I Ê, X is uniformly admissible for the system E π , then π is uniformly dichotomic with respect to a family of projections {P θ } θ∈Θ with the property that Range P θ S θ , Ker P θ U θ , ∀θ ∈ Θ.

4.3
In what follows, we will see that by imposing some conditions either on the output space O or on the input space I, the admissibility becomes a sufficient condition for the exponential dichotomy.Proof.Let δ > 0 be such that Φ θ, t x ≤ δ x , ∀t ≥ 0, ∀x ∈ Range P θ , ∀θ ∈ Θ.

4.6
Let L > 0 be given by Definition 3.6 and let M, ω > 0 be given by Definition 3.2.

4.9
We observe that f is continuous and f t ≤ a λ x,θ t , ∀t ∈ Ê.

4.10
Since x ∈ Range P θ S θ , we have that λ x,θ ∈ O Ê, X .Then using Remark 2.4 i , we deduce that f ∈ O Ê, X .In addition, we have that v ∈ C 0c Ê, X and an easy computation shows that the pair f, v satisfies E θ .Then, according to our hypothesis, it follows that

4.12
Using relation 4.5 , we deduce that

4.14
Using the invariance under translations of the space O from relation 4.14 , we obtain that

4.18
If Φ θ, 1 x 0, then Φ θ, h x 0, so the above relation holds.Taking into account that h does not depend on θ or x, we obtain that in this case, there is h > 0 such that relation 4.6 holds.
Case 2. Suppose that I ∈ L Ê .In this situation, from Remark 2.16, we have that there is a continuous function γ : Ê → Ê such that γ ∈ I \ L 1 Ê, Ê .Since the space I is invariant under translations, we may assume that there is r > 1 such that Let β : Ê → 0, 1 be a continuous function with supp β ⊂ 0, r 1 and β t 1, for all t ∈ 1, r .Let θ ∈ Θ and let x ∈ Range P θ .We consider the functions We have that v ∈ C 0c Ê, X , f is continuous, and f t ≤ q λ x,θ t , for all t ∈ Ê.Using similar arguments with those used in relation 4.10 , we deduce that f ∈ O Ê, X .An easy computation shows that the pair f, v satisfies E θ .Then, we have that

4.22
Using relation 4.5 , we obtain that

4.25
Using the invariance under translations of the space O from relations 4.25 , 4.22 , and 4.24 we have that

4.27
Setting h r 2 and taking into account that h does not depend on θ or x, we obtain that relation 4.6 holds.

4.28
Let ν : 1/h and let K δe.Let θ ∈ Θ and let x ∈ Range P θ .Let t > 0.Then, there are k ∈ AE and τ ∈ 0, h such that t kh τ.Using relations 4.5 and 4.6 , we successively deduce that

4.29
Theorem 4. 6 The behavior on the unstable subspace .Let O, I be two Banach function spaces such that either O ∈ Q Ê or I ∈ L Ê .If the pair O Ê, X , I Ê, X is uniformly admissible for the system E π , then, there are K, ν > 0 such that Φ θ, t y ≥ 1 K e νt y , ∀t ≥ 0, ∀y ∈ Ker P θ , ∀θ ∈ Θ.

4.35
We have that v ∈ C 0c Ê, X and f is continuous.Moreover, from f t ≤ a ϕ t aMe ω r 1 y χ 0,r 1 t , ∀t ∈ Ê,

4.36
we obtain that f ∈ O Ê, X .An easy computation shows that the pair f, v satisfies E θ , so

4.37
Observing that v t α t − r , for all t ∈ Ê, the relation 4.37 becomes

4.38
From relation 4.31 , we have that

4.39
This implies that where q : r 1 0 β τ γ τ dτ.We have that v ∈ C 0c Ê, X , and, using similar arguments with those from Case 1, we obtain that f ∈ O Ê, X .An easy computation shows that the pair f, v satisfies E θ , so

4.52
Using the translation invariance of the space O from 4.52 , we obtain that

4.56
According to the previous results we may formulate now a sufficient condition for the existence of the exponential dichotomy.Moreover, for the converse implication we will show that it sufficient to chose one of the spaces in the admissible pair from the class R Ê .Thus, the main result of this section is as follows.ii Since I ⊂ O, it follows that there is α > 0 such that |u| O ≤ α|u| I , ∀u ∈ I.
Let θ ∈ Θ and let v ∈ C 0c Ê, X .We consider the function f v : Ê → X given by

4.58
We have that f v is continuous, and a direct computation shows that the pair f v , v satisfies E θ .In addition, we have that

4.59
If I ∈ R Ê , let γ I,ν > 0 be the constant given by Lemma 2.21.Then, from 4.59 and Lemma 2.21, it follows that f v ∈ I Ê, X and

4.61
If O ∈ R Ê , let γ O,ν > 0 be the constant given by Lemma 2.21.Then, from 4.59 , 4.57 and using Lemma 2.21, we successively obtain that f v ∈ O Ê, X and

4.63
Then setting L : α 1 2q Kγ from relations 4.61 and 4.62 , we have that

4.64
Now let v ∈ C 0c Ê, X and f ∈ O Ê, X be such that the pair f, v satisfies E θ .We set Since ϕ ∈ O Ê, X , from Remark 2.12 it follows that ϕ ∈ M 1 Ê, X .Then, from 4.67 , we have that

4.68
For s → −∞ in 4.68 , it follows that ϕ 1 t 0 0. In addition, from 4.66 we have that

4.70
The relation 4.70 shows that
Since t 0 ∈ Ê was arbitrary, we deduce that ϕ 0, so f f v .Then, from 4.64 , we have that

4.72
Taking into account that L does not depend on θ ∈ Θ or on v ∈ C 0c Ê, X , we finally conclude that the pair O Ê, X , I Ê, X is uniformly admissible for the system E π .
Sufficiency follows from i .
Corollary 4.8.Let π Φ, σ be a skew-product flow on E X × Θ and let V be a Banach function space with V ∈ T Ê .Then, the following assertions hold: i if the pair V Ê, X , V Ê, X is uniformly admissible for the system E π , then, π is exponentially dichotomic; ii if V ∈ R Ê , then, π is exponentially dichotomic if and only if the pair V Ê, X ,V Ê, X is uniformly admissible for the system E π .
Proof.We prove that either follows that there is γ > 0 such that In particular, from v χ 0,t in relation 4.73 , we deduce that which is absurd.This shows that the assumption is false, which shows that either V ∈ Q Ê or V ∈ L Ê .By applying Theorem 4.7, we obtain the conclusion.

Applications and Conclusions
We have seen in the previous section that in the study of the exponential dichotomy of variational equations the classes Q Ê and, respectively, L Ê have a crucial role in the identification of the appropriate function spaces in the admissible pair.Moreover, it was also important to point out that it is sufficient to impose conditions either on the input space or on the output space.In this context, the natural question arises if these conditions are indeed necessary and whether our hypotheses may be dropped.The aim of this section is to answer this question.With this purpose, we will present an illustrative example of uniform admissibility, and we will discuss the concrete implications concerning the existence of the exponential dichotomy.
Let X be a Banach space.We denote by C 0 Ê, X the space of all continuous functions u : Ê → X with lim t → ∞ u t lim t → −∞ u t 0, which is a Banach space with respect to the norm |u| : sup t∈Ê u t .

5.1
We start with a technical lemma.
Proof.Let c : sup t>0 F O t .Let u ∈ C 0 Ê, Ê .Then, there is an unbounded increasing sequence In what follows, we present a concrete situation which illustrates the relevance of the hypotheses on the underlying function spaces considered in the admissible pair, for the study of the dichotomous behavior of skew-product flows.
Example 5.2.Let X Ê × Ê which is a Banach space with respect to the norm Then, there are α, β > 0 such that

5.5
Step 1.We prove that the pair O Ê, X , I Ê, X is uniformly admissible for the system E π .
Let θ ∈ Θ and let v v 1 , v 2 ∈ C 0c Ê, X and let h > 0 be such that supp v ⊂ 0, h .We consider the function f : Ê → X where f f 1 , f 2 and

5.6
We have that f is continuous and an easy computation shows that the pair f, v satisfies E θ .Since supp v ⊂ 0, h , we obtain that f 1 t 0, for all t ≤ 0 and f 2 t 0, for all t ≥ h.From we have that lim t → ∞ f 1 t 0. In addition, from Since ϕ r → ∞ as r → −∞, for s → −∞ in 5.14 , we obtain that g 1 t 0 0. In addition, for every t ≥ t 0 from relation 5.13 we have that e −t 0 g 2 t 0 t 1 t e −τ g 2 τ dτ ≤ e −t t 1 t g 2 τ dτ ≤ e −t g 2 M 1 Ê,Ê .

5.15
For t → ∞ in 5.15 we deduce that g 2 t 0 0. So, we obtain that g t 0 0. Taking into account that t 0 ∈ Ê was arbitrary it follows that g 0. This implies that f f.Then, from relation 5.11 we have that f O Ê,X ≤ αβ v I Ê,X .

5.16
We set L αβ, and, taking into account that L does not depend on θ or v, we conclude that the pair O Ê, X , I Ê, X is uniformly admissible for the system E π .
Step 2. We prove that π is not exponentially dichotomic.Suppose by contrary that π is exponentially dichotomic with respect to the family of projections {P θ } θ∈Θ and let K, ν > 0 be two constants given by Definition 4.1.In this case, according to Proposition 2.1 from 18 we have that Im P θ {x ∈ X : Φ θ, t x −→ 0 as t −→ ∞}, ∀θ ∈ Θ.

5.20
In particular, for θ 0, from 5.20 , we have that 1 t 1 ≤ Ke −νt , ∀t ≥ 0, 5.21 which is absurd.This shows that the assumption is false, so π is not exponentially dichotomic.
Remark 5.3.The above example shows that if I, O are two Banach function spaces from the class T Ê such that O / ∈ Q Ê and I / ∈ L Ê , then the uniform admissibility of the pair O Ê, X , I Ê, X for the system E π does not imply the existence of the exponential dichotomy of π.This shows that the hypotheses of the main result from the previous section are indeed necessary and emphasizes the fact that in the study of the exponential dichotomy in terms of the uniform admissibility at least one of the output space or the input space should belong to, respectively, Q Ê or L Ê .
Finally, we complete our study with several consequences of the main result, which will point out some interesting conclusions for some usual classes of spaces often used in control-type problems arising in qualitative theory of dynamical systems.We will also show that, in our approach, the input space can be successively minimized, and we will discuss several optimization directions concerning the admissibility-type techniques.
Remark 5.4.The input-output characterizations for the asymptotic properties of systems have a wider applicability area if the input space is as small as possible and the output space is very general.In our main result, given by Theorem 4.7, the input functions belong to the space C 0c Ê, X while the output space is a general Banach function space.By analyzing condition ii from Definition 3.6, we observe that the input-output characterization given by Theorem 4.7 becomes more flexible and provides a more competitive applicability spectrum when the norm on the input space is larger.
Another interesting aspect that must be noted is that the class T Ê is closed to finite intersections.Indeed, if I 1 , . . . ,I n ∈ T Ê , then we may define I : which is a Banach function space which belongs to T Ê .So, taking as input space a Banach function space which is obtained as an intersection of Banach function spaces from the class T Ê we will have a "larger" norm in our admissibility condition, and, thus the estimation will be more permissive and more general.
As a consequence of the aspects presented in the above remark we deduce the following corollaries.AE * , q 1 , . . ., q n ∈ 1, ∞ and I L q 1 Ê, Ê ∩ • • • ∩ L q n Ê, Ê ∩ L p Ê, Ê .Then, π is exponentially dichotomic if and only if the pair L p Ê, X , I Ê, X is admissible for the system E π .
Proof.This follows from Corollary 5.5.
Proof.This follows from Theorem 4.7 by observing that I ∈ L Ê .
Remark 5.8.According to Remark 2.12, the largest space from the class T Ê is M 1 Ê, Ê .Thus, considering the output space M 1 Ê, Ê , in order to obtain optimal input-output characterizations for exponential dichotomy in terms of admissibility, it is sufficient to work with smaller and smaller input spaces.

Example 3 . 3
Particular cases .The class described by skew-product flows generalizes the autonomous systems as well as the nonautonomous systems, as the following examples show:

Theorem 4 . 7
Necessary and sufficient condition for exponential dichotomy .Let π Φ, σ be a skew-product flow on E X × Θ and let O, I be two Banach function spaces with O, I ∈ T Ê such that either O ∈ Q Ê or I ∈ L Ê .The following assertions hold:(i) if the pair O Ê, X , I Ê, X is uniformly admissible for the system E π , then π is exponentially dichotomic.(ii)if I ⊂ O and one of the spaces I or O belongs to the class R Ê , then π is exponentially dichotomic if and only if the pair O Ê, X , I Ê, X is uniformly admissible for the system E π .Proof.i This follows from Theorem 3.13, Theorem 4.5, and Theorem 4.6.
Definition 2.1.A linear subspace B ⊂ M Ê, Ê is called normed function space if there is a mapping | • | B : B → Ê such that the following properties hold: i |u| B 0 if and only if u 0 a.e.; ii |αu| B |α||u| B , for all α, u ∈ Ê × B; iii |u v| B ≤ |u| B |v| B , for all u, v ∈ B; iv if |u t | ≤ |v t | a.e.t ∈ Ê and v ∈ B, then u ∈ B and |u| B ≤ |v| B .If B, | • | B is complete, then B is called a Banach function space.Remark 2.2.If B, | • | B is a Banach function space and u ∈ B, then also |u • | ∈ B. Definition 2.3.A Banach function space B, | • | B is said to be invariant under translations if for every u, t ∈ B × Ê the function u t : Ê → Ê, u t s u s − t belongs to B and |u t | B |u| B .
1/p, is a Banach function space which belongs to T Ê .ii The linear space L ∞ Ê, Ê of all measurable essentially bounded functions u : Ê → Ê with respect to the norm u ∞ ess sup t∈Ê |u t | is a Banach function space which belongs to T Ê .

Lemma 2.15. If
Definition 2.13.Let B ∈ T Ê .The mapping F B : 0, ∞ → Ê , F B t |χ 0,t | B is called the fundamental function of the space B. We denote by Q Ê the class of all Banach function spaces B ∈ T Ê with theproperty that sup t>0 F B t ∞. ϕ t ∈ 0, ∞ , for all t > 0, then O ϕ Ê, Ê ∈ Q Ê .Let L Ê denote the class of all Banach function spaces B ∈ T Ê with the property that B \ L 1 Ê, Ê / ∅.According to Remark 2.2, we have that if B ∈ L Ê , then there is a continuous function γ : Ê → Ê such that γ ∈ B \ L 1 Ê, Ê .Let u, v ∈ M Ê,Ê .We say that u and v are equimeasurable if for every t > 0 the sets {s ∈ Ê : |u s | > t} and {s ∈ Ê : |v s | > t} have the same measure.Definition 2.18.A Banach function space B, | • | B is rearrangement invariant if for every equimeasurable functions u, v with u ∈ B, we have that v ∈ B and |u| B |v| B .Moreover, there is γ B,ν > 0 which depends only on B and ν such that Setting h : r 1 and taking into account that h does not depend on y or θ we obtain that relation 4.32 holds.Suppose that I ∈ L Ê .In this situation, using Remark 2.16 and the translation invariance of the space I, we have that there is a continuous function γ : Ê → Ê with γ ∈ I \ L 1 Ê, Ê and r > 1 such that : r 1 and since h does not depend on y or θ, we have that the relation 4.32 holds.In conclusion, in both situations there is h > 0 such that Φ θ, h y ≥ e y , ∀y ∈ Ker P θ , ∀θ ∈ Θ.4.55Let ν 1/h and let K δe.Let θ ∈ Θ and let y ∈ Ker P θ .Let t > 0.Then, there are j ∈ AE and s ∈ 0, h such that t jh s.Using relations 4.31 and 4.32 , we obtain that and we have that ϕ ∈ O Ê, X and , for all t ∈ Ê and let ϕ 2 t I − P σ θ, t ϕ t , for all t ∈ Ê.
for all |t| ≥ t n and all n ∈ AE.Setting u n uχ −t n ,t n we According to Remark 2.4 ii , there exists a subsequence u k n such that u k n t → v t for a.e.t ∈ Ê.This implies that v t u t for a.e.t ∈ Ê, so v u in O.In conclusion, u ∈ O, and the proof is complete.
* .5.2From relation 5.2 , it follows that the sequence u n is fundamental in O, so this is convergent, that is, there exists v ∈ O such that u n → v in O.