1. 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 input-output techniques. Their paper carried out a beautiful connection between the classical techniques originating in the pioneering works of Perron [11] and Maĭzel [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 [20–22]) 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 skew-product 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 , and let . For every , denotes the characteristic function of the set . Let be the linear space of all Lebesgue measurable functions identifying the functions which are equal almost everywhere.

Definition 2.1. A linear subspace is called normed function space if there is a mapping such that the following properties hold:(i) if and only if a.e.;(ii), for all ;(iii), for all ;(iv)if a.e. and , then and . If is complete, then is called a Banach function space.

Remark 2.2. If is a Banach function space and , then also .

Definition 2.3. A Banach function space is said to be invariant under translations if for every the function belongs to and .

Let be the linear space of all continuous functions with compact support. We denote by the class of all Banach function spaces which are invariant under translations, and(i)for every there is such that , for all ;(ii)if , then there is a continuous function .

Remark 2.4. Let . Then, the following properties hold:
(i) if is a bounded interval, then .
(ii) if in , then there is a subsequence which converges to a.e. (see, e.g., [25]).

Remark 2.5. Let . If and is defined by then it is easy to see that It follows that and .

Example 2.6. (i) If , then , with respect to the norm , is a Banach function space which belongs to .
(ii) The linear space of all measurable essentially bounded functions with respect to the norm is a Banach function space which belongs to .

Example 2.7 (Orlicz spaces). Let be a nondecreasing left continuous function which is not identically 0 or on , and let . If let The linear space such that , with respect to the norm is a Banach function space called the Orlicz space associated to . It is easy to see that is invariant under translations.

Remark 2.8. A remarkable example of Orlicz space is represented by , for every . This can be obtained for , if and for

Lemma 2.9. If , then .

Proof. Let . Then, there are such that , for all . Since is continuous on , there is such that , for all . Then, we have that We observe that This implies that . Using (2.6), we deduce that . So, .
Since is nondecreasing with , there is such that , for all .
Let and let . Taking into account that 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 Taking , from relations (2.9) and (2.10), it follows that
Since the function does not depend on , we obtain that .

Example 2.10. If defined by , for and , for , then according to Lemma 2.9 we have that the Orlicz space . Moreover, it is easy to see that is a proper subspace of .

Example 2.11. Let and let be the linear space of all with . With respect to the norm this is a Banach function space which belongs to .

Remark 2.12. If , then .
Indeed, let be such that , for all . If we observe that so .

In what follows, we will introduce three remarkable subclasses of , 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.

Definition 2.13. Let . The mapping is called the fundamental function of the space .

Remark 2.14. If , then the fundamental function is nondecreasing.

Notation 1. We denote by the class of all Banach function spaces with the property that .

Lemma 2.15. If , for all , then .

Proof. It is easy to see that is strictly increasing, continuous with and , for all , so . Hence, is bijective.
Let . Since it follows that if and only if . This implies that Since , from (2.15), we obtain that .

Another distinctive subclass of is introduced in the following.

Notation 2. Let denote the class of all Banach function spaces with the property that .

Remark 2.16. According to Remark 2.2, we have that if , then there is a continuous function such that .

We will also see, in this paper, that the necessary conditions for the existence of exponential dichotomy should be expressed using another remarkable subclass of —the rearrangement invariant spaces, see the definitions below.

Definition 2.17. Let . We say that and are equimeasurable if for every the sets and have the same measure.

Definition 2.18. A Banach function space is rearrangement invariant if for every equimeasurable functions with , we have that and .

Notation 3. We denote by the class of all Banach function spaces which are rearrangement invariant.

Remark 2.19. If , then is an interpolation space between , and (see [27, Theorem  2.2, page 106]).

Remark 2.20. The Orlicz spaces are rearrangement invariant (see [27]). Using Lemma 2.9, we deduce that if , then .

Lemma 2.21. Let and let . Then for every , the functions defined by belong to . Moreover, there is which depends only on and such that

Proof. We consider the operators We have that and are correctly defined bounded linear operators. Moreover, the restrictions and are correctly defined and bounded linear operators. Since , then, from Remark 2.19, we have that is an interpolation space between , and . This implies that the restrictions and are correctly defined and bounded linear operators. Setting , the proof is complete.

Notations
If is a Banach space, for every Banach function space , we denote by the space of all Bochner measurable functions with the property that the mapping belongs to . With respect to the norm is a Banach space. We also denote by the linear space of all continuous functions with compact support contained in . It is easy to see that , for all .

3. Uniform Dichotomy for Skew-Product Flows

In this section, we start our investigation by studying the existence of a more flexible asymptotic behavior, called uniform dichotomy. This is described 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 introduced in the previous section, provides several interesting conclusions concerning the qualitative behavior of the solutions of variational equations.

Let be a real or complex Banach space and let denote the identity operator on . The norm on and on —the Banach algebra of all bounded linear operators on , will be denoted by . Let be a metric space.

Definition 3.1. A continuous mapping is called a flow on if and , for all .

Definition 3.2. A pair is called a skew-product flow on if is a flow on and the mapping called cocycle, satisfies the following conditions:(i) and , for all ;(ii)there are and such that , for all ;(iii)for every , the mapping is continuous on .

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:(i)If , then let and let be an evolution family on the Banach space . Setting , we observe that is a skew-product flow.(ii)Let be a -semigroup on the Banach space and let be a metric space.If is an arbitrary flow on and , then is a skew-product flow.Let be the projection flow on and let be a uniformly bounded family of projections such that , for all . If , then is a skew-product flow.

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 be a locally compact metric space and let be a flow on . Let be a Banach space and let be a family of densely defined closed operators. We consider the variational equation A cocycle is said to be a solution of if for every , there is a dense subset such that for every initial condition the mapping is differentiable on , for every and the mapping satisfies .

An important asymptotic behavior of skew-product flows is described by the uniform dichotomy, which relies on the splitting of the Banach space at every point 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 is said to be uniformly dichotomic if there exist a family of projections and a constant such that the following properties hold:(i), for all ;(ii), for all , all and all ;(iii)the restriction is an isomorphism, for all ;(iv), for all , all and all ;(v).

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 . We will emphasize that the class has an essential role in the study of the dichotomous behavior of variational equations.

Let be two Banach function spaces with . Let be a skew-product flow on . We associate with the input-output control system , where for every such that the input function and the output function .

Definition 3.6. The pair is said to be uniformly admissible for the system if there is such that for every , the following properties hold:(i)for every there exists such that the pair satisfies ;(ii)if and are such that the pair satisfies , then .

Remark 3.7. (i) According to this admissibility concept, it is sufficient to choose all the input functions from the space , and, thus, we point out that 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 . Actually, condition (ii) shows that the norm of each output function in the space is bounded by the norm of the input function in the space uniformly with respect to .
(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 is uniformly admissible for the system , then from (ii) we deduce that for every and every there exists a unique such that the pair satisfies .

In what follows we will analyze the implications of the uniform admissibility of the pair with 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 , we consider the function called the trajectory determined by the vector and the point .

For every , we denote by the linear space of all functions with the property that For every , we consider the stable subset and, respectively, the unstable subset

Remark 3.8. It is easy to see that for every , and are linear subspaces. Therefore, in all what follows, we will refer as the stable subspace and, respectively, as the unstable subspace, for each .

Proposition 3.9. For every , the following assertions hold: (i);(ii).

Proof. The property (i) is immediate. To prove the assertion (ii) let be given by Definition 3.2(ii). Let . Let . Then, there is with . We set , and we consider We observe that , for all , and since , we deduce that . Using the fact that , we obtain that Then, we define the function and since is invariant under translations, we deduce that . Moreover, from (3.6), it follows that The relation (3.7) implies that , so .
Conversely, let . Then, there is with . Taking , we have that and In particular, for , from (3.8), we deduce that . This implies that . Then, and the proof is complete.

Remark 3.10. From Proposition 3.9(ii), we have that for every the restriction is surjective. 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 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 of spaces from the class is admissible for the control system associated with the skew-product flow.

Theorem 3.11 (The behavior on the stable subspace). If the pair is uniformly admissible for the system , then the following assertions hold: (i)there is such that , for all , all and all ; (ii) is a closed linear subspace, for all .

Proof. Let be given by Definition 3.6 and let be given by Definition 3.2. Let be a continuous function with and .
(i) Let and let . We consider the functions Then, and Since , we have that . Then, from (3.10), we obtain that . An easy computation shows that the pair satisfies . Then, From , for all , we obtain that .
Let . From it follows that Since is invariant under translations, we deduce that Using relations (3.11) and (3.14), we have that Since , for all , setting , we deduce that , for all . Taking into account that does not depend on or , it follows that
(ii) Let and let with . For every , we consider the sequence We have that , for all and using similar arguments with those used in relation (3.10), we obtain that , for all . An easy computation shows that the pair satisfies . Let . Then, . According to our hypothesis there is, such that the pair satisfies . Taking and we observe that , and the pair satisfies . This implies that From , for all and all , we deduce that From (3.18) and (3.19), it follows that in . From Remark 2.4(ii), we have that there is a subsequence and a negligible set such that , for all . In particular, it follows that there is such that Because the pair satisfies , we obtain that This shows that , for all . Then, from using the fact that and Remark 2.4(i), we obtain that , so .
In conclusion, is a closed linear subspace, for all .