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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 539026, 36 pages
http://dx.doi.org/10.1155/2011/539026
Research Article

Translation Invariant Spaces and Asymptotic Properties of Variational Equations

Department of Mathematics, Faculty of Mathematics and Computer Science, West University of Timişoara, V. Pârvan Boulelvard 4, 300223 Timişoara, Romania

Received 11 November 2010; Accepted 12 February 2011

Academic Editor: Josef Diblík

Copyright © 2011 Adina Luminiţa Sasu and Bogdan Sasu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We present a new perspective concerning the study of the asymptotic behavior of variational equations by employing function spaces techniques. We give a complete description of the dichotomous behaviors of the most general case of skew-product flows, without any assumption concerning the flow, the cocycle or the splitting of the state space, our study being based only on the solvability of some associated control systems between certain function spaces. The main results do not only point out new necessary and sufficient conditions for the existence of uniform and exponential dichotomy of skew-product flows, but also provide a clear chart of the connections between the classes of translation invariant function spaces that play the role of the input or output classes with respect to certain control systems. Finally, we emphasize the significance of each underlying hypothesis by illustrative examples and present several interesting applications.

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 [118] 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 [35, 1419]). 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 [19, 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 [2022]) 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, 1417, 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 + = { 𝑡 𝑡 0 } , and let = { 𝑡 𝑡 0 } . For every 𝐴 , 𝜒 𝐴 denotes the characteristic function of the set 𝐴 . Let ( , ) 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 ( , ) is called normed function space if there is a mapping | | 𝐵 𝐵 + such that the following properties hold:(i) | 𝑢 | 𝐵 = 0 if and only if 𝑢 = 0 a.e.;(ii) | 𝛼 𝑢 | 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 𝑢 𝑡 , 𝑢 𝑡 ( 𝑠 ) = 𝑢 ( 𝑠 𝑡 ) belongs to 𝐵 and | 𝑢 𝑡 | 𝐵 = | 𝑢 | 𝐵 .

Let 𝒞 𝑐 ( , ) 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, 𝒞 𝑐 ( , ) 𝐵 and(i)for every 𝑡 > 0 there is 𝑐 ( 𝑡 ) > 0 such that 𝑡 0 | 𝑢 ( 𝜏 ) | 𝑑 𝜏 𝑐 ( 𝑡 ) | 𝑢 | 𝐵 , for all 𝑢 𝐵 ;(ii)if 𝐵 𝐿 1 ( , ) , then there is a continuous function 𝛾 𝐵 𝐿 1 ( , ) .

Remark 2.4. Let 𝐵 𝒯 ( ) . Then, the following properties hold:
(i) if 𝐽 is a bounded interval, then 𝜒 𝐽 𝐵 .
(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 ( ) . If 𝜈 > 0 and 𝑒 𝜈 is defined by 𝑒 𝜈 𝑒 ( 𝑡 ) = 𝜈 𝑡 , 𝑡 0 , 0 , 𝑡 < 0 , ( 2 . 1 ) then it is easy to see that 𝑒 𝜈 ( 𝑡 ) = 𝑛 = 0 𝑒 𝜈 𝑡 𝜒 [ 𝑛 , 𝑛 + 1 ) ( 𝑡 ) 𝑛 = 0 𝑒 𝜈 𝑛 𝜒 [ 𝑛 , 𝑛 + 1 ) ( 𝑡 ) , 𝑡 . ( 2 . 2 ) It follows that 𝑒 𝜈 𝐵 and | 𝑒 𝜈 | 𝐵 | 𝜒 [ 0 , 1 ) | 𝐵 / ( 1 𝑒 𝜈 ) .

Example 2.6. (i) If 𝑝 [ 1 , ) , then 𝐿 𝑝 ( , ) = { 𝑢 ( , ) | 𝑢 ( 𝑡 ) | 𝑝 𝑑 𝑡 < } , with respect to the norm 𝑢 𝑝 = ( | 𝑢 ( 𝑡 ) | 𝑝 𝑑 𝑡 ) 1 / 𝑝 , is a Banach function space which belongs to 𝒯 ( ) .
(ii) The linear space 𝐿 ( , ) of all measurable essentially bounded functions 𝑢 with respect to the norm 𝑢 = e s s s u p 𝑡 | 𝑢 ( 𝑡 ) | is a Banach function space which belongs to 𝒯 ( ) .

Example 2.7 (Orlicz spaces). Let 𝜑 + + be a nondecreasing left continuous function which is not identically 0 or on ( 0 , ) , and let 𝑌 𝜑 ( 𝑡 ) = 𝑡 0 𝜑 ( 𝑠 ) 𝑑 𝑠 . If 𝑢 ( , ) let 𝑀 𝜑 ( 𝑢 ) = 𝑌 𝜑 | | | | 𝑢 ( 𝑠 ) 𝑑 𝑠 . ( 2 . 3 ) The linear space 𝑂 𝜑 ( , ) = { 𝑢 ( , ) 𝑘 > 0 such that 𝑀 𝜑 ( 𝑘 𝑢 ) < } , with respect to the norm | 𝑢 | 𝜑 = i n f 𝑘 > 0 𝑀 𝜑 𝑢 𝑘 , 1 ( 2 . 4 ) is a Banach function space called the Orlicz space associated to 𝜑 . It is easy to see that 𝑂 𝜑 ( , ) is invariant under translations.

Remark 2.8. A remarkable example of Orlicz space is represented by 𝐿 𝑝 ( , ) , for every 𝑝 [ 1 , ] . This can be obtained for 𝜑 ( 𝑡 ) = 𝑝 𝑡 𝑝 1 , if 𝑝 [ 1 , ) and for [ ] , 𝜑 ( 𝑡 ) = 0 , 𝑡 0 , 1 , 𝑡 > 1 , i f 𝑝 = . ( 2 . 5 )

Lemma 2.9. If 𝜑 ( 1 ) < , then 𝑂 𝜑 ( , ) 𝒯 ( ) .

Proof. Let 𝑣 𝒞 𝑐 ( , ) . Then, there are 𝑎 , 𝑏 , 𝑎 < 𝑏 such that 𝑣 ( 𝑡 ) = 0 , for all 𝑡 ( 𝑎 , 𝑏 ) . 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 | | | | 𝑣 ( 𝑡 ) 𝑀 𝜒 [ a , 𝑏 ] ( 𝑡 ) , 𝑡 . ( 2 . 6 ) We observe that 𝑀 𝜑 𝜒 [ 𝑎 , 𝑏 ] = 𝑌 𝜑 𝜒 [ 𝑎 , 𝑏 ] ( 𝜏 ) 𝑑 𝜏 = ( 𝑏 𝑎 ) 𝑌 𝜑 ( 1 ) ( 𝑏 𝑎 ) 𝜑 ( 1 ) < . ( 2 . 7 ) This implies that 𝜒 [ 𝑎 , 𝑏 ] 𝑂 𝜑 ( , ) . Using (2.6), we deduce that 𝑣 𝑂 𝜑 ( , ) . So, 𝒞 𝑐 ( , ) 𝑂 𝜑 ( , ) .
Since 𝑌 𝜑 is nondecreasing with l i m 𝑡 𝑌 𝜑 ( 𝑡 ) = , there is 𝑞 > 0 such that 𝑌 𝜑 ( 𝑡 ) > 1 , for all 𝑡 𝑞 .
Let 𝑡 1 and let 𝑢 𝑂 𝜑 ( , ) { 0 } . Taking into account that 𝑌 𝜑 is a convex function and using Jensen's inequality (see, e.g., [26]), we deduce that 𝑌 𝜑 1 𝑡 𝑡 0 | | | | 𝑢 ( 𝜏 ) | 𝑢 | 𝜑 1 𝑑 𝜏 𝑡 𝑡 0 𝑌 𝜑 | | | | 𝑢 ( 𝜏 ) | 𝑢 | 𝜑 𝑑 𝜏 𝑀 𝜑 𝑢 | 𝑢 | 𝜑 1 . ( 2 . 8 ) This implies that 1 𝑡 𝑡 0 | | | | 𝑢 ( 𝜏 ) | 𝑢 | 𝜑 𝑑 𝜏 𝑞 , 𝑡 1 . ( 2 . 9 ) In addition, using (2.9), we have that 𝑡 0 | | | | 𝑢 ( 𝜏 ) 𝑑 𝜏 1 0 | | | | 𝑢 ( 𝜏 ) 𝑑 𝜏 𝑞 | 𝑢 | 𝜑 [ , 𝑡 0 , 1 ) . ( 2 . 1 0 ) Taking 𝑐 ( 0 , ) ( 0 , ) , 𝑐 ( 𝑡 ) = m a x { 𝑞 𝑡 , 𝑞 } , from relations (2.9) and (2.10), it follows that 𝑡 0 | | | | 𝑢 ( 𝜏 ) 𝑑 𝜏 𝑐 ( 𝑡 ) | 𝑢 | 𝜑 , 𝑡 0 . ( 2 . 1 1 )
Since the function 𝑐 does not depend on 𝑢 , we obtain that 𝑂 𝜑 ( , ) 𝒯 ( ) .

Example 2.10. If 𝜑 + + defined by 𝜑 ( 0 ) = 0 , 𝜑 ( 𝑡 ) = 1 , for 𝑡 ( 0 , 1 ] and 𝜑 ( 𝑡 ) = 𝑒 𝑡 1 , for 𝑡 > 1 , then according to Lemma 2.9 we have that the Orlicz space 𝑂 𝜑 ( , ) 𝒯 ( ) . Moreover, it is easy to see that 𝑂 𝜑 ( , ) is a proper subspace of 𝐿 1 ( , ) .

Example 2.11. Let 𝑝 [ 1 , ) and let 𝑀 𝑝 ( , ) be the linear space of all 𝑢 ( , ) with s u p 𝑡 𝑡 𝑡 + 1 | 𝑢 ( 𝑠 ) | 𝑝 𝑑 𝑠 < . With respect to the norm 𝑢 𝑀 𝑝 = s u p 𝑡 𝑡 𝑡 + 1 | | | | 𝑢 ( 𝑠 ) 𝑝 𝑑 𝑠 1 / 𝑝 , ( 2 . 1 2 ) this is a Banach function space which belongs to 𝒯 ( ) .

Remark 2.12. If 𝐵 𝒯 ( ) , then 𝐵 𝑀 1 ( , ) .
Indeed, let 𝑐 ( 1 ) > 0 be such that 1 0 | 𝑢 ( 𝜏 ) | 𝑑 𝜏 𝑐 ( 1 ) | 𝑢 | 𝐵 , for all 𝑢 𝐵 . If 𝑢 𝐵 we observe that 𝑡 𝑡 + 1 | | | | 𝑢 ( 𝜏 ) 𝑑 𝜏 = 1 0 | | 𝑢 𝑡 | | | | 𝑢 ( 𝜉 ) 𝑑 𝜉 𝑐 ( 1 ) 𝑡 | | 𝐵 = 𝑐 ( 1 ) | 𝑢 | 𝐵 , 𝑡 , ( 2 . 1 3 ) so 𝑢 𝑀 1 ( , ) .

In what follows, we will introduce three remarkable subclasses of 𝒯 ( ) , 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 𝐵 𝒯 ( ) . The mapping 𝐹 𝐵 ( 0 , ) + , 𝐹 𝐵 ( 𝑡 ) = | 𝜒 [ 0 , 𝑡 ) | 𝐵 is called the fundamental function of the space 𝐵 .

Remark 2.14. If 𝐵 𝒯 ( ) , then the fundamental function 𝐹 𝐵 is nondecreasing.

Notation 1. We denote by 𝒬 ( ) the class of all Banach function spaces 𝐵 𝒯 ( ) with the property that s u p 𝑡 > 0 𝐹 B ( 𝑡 ) = .

Lemma 2.15. If 𝜑 ( 𝑡 ) ( 0 , ) , for all 𝑡 > 0 , then 𝑂 𝜑 ( , ) 𝒬 ( ) .

Proof. It is easy to see that 𝑌 𝜑 is strictly increasing, continuous with 𝑌 𝜑 ( 0 ) = 0 and 𝑌 𝜑 ( 𝑡 ) ( 𝑡 1 ) 𝜑 ( 1 ) , for all 𝑡 > 1 , so l i m 𝑡 𝑌 𝜑 ( 𝑡 ) = . Hence, 𝑌 𝜑 is bijective.
Let 𝑡 > 0 . Since 𝑀 𝜑 1 𝑘 𝜒 [ 0 , 𝑡 ) = 𝑡 𝑌 𝜑 1 𝑘 , 𝑘 > 0 , ( 2 . 1 4 ) it follows that 𝑀 𝜑 ( ( 1 / 𝑘 ) 𝜒 [ 0 , 𝑡 ) ) 1 if and only if 1 / 𝑌 𝜑 1 ( 1 / 𝑡 ) 𝑘 . This implies that 𝐹 𝑂 𝜑 ( , ) 1 ( 𝑡 ) = 𝑌 𝜑 1 ( 1 / 𝑡 ) , 𝑡 > 0 . ( 2 . 1 5 ) Since 𝑌 𝜑 1 ( 0 ) = 0 , from (2.15), we obtain that 𝑂 𝜑 ( , ) 𝒬 ( ) .

Another distinctive subclass of 𝒯 ( ) is introduced in the following.

Notation 2. Let ( ) denote the class of all Banach function spaces 𝐵 𝒯 ( ) with the property that 𝐵 𝐿 1 ( , ) .

Remark 2.16. According to Remark 2.2, we have that if 𝐵 ( ) , then there is a continuous function 𝛾 + such that 𝛾 𝐵 𝐿 1 ( , ) .

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 𝒯 ( ) —the rearrangement invariant spaces, see the definitions below.

Definition 2.17. Let 𝑢 , 𝑣 ( , ) . We say that 𝑢 and 𝑣 are equimeasurable if for every 𝑡 > 0 the sets { 𝑠 | 𝑢 ( 𝑠 ) | > 𝑡 } and { 𝑠 | 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 .

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

Remark 2.19. If 𝐵 ( ) , then 𝐵 is an interpolation space between 𝐿 1 ( , ) and 𝐿 ( , ) (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 𝜑 ( 1 ) < , then 𝑂 𝜑 ( , ) ( ) .

Lemma 2.21. Let 𝐵 ( ) and let 𝜈 > 0 . Then for every 𝑢 𝐵 , the functions 𝜑 𝑢 , 𝜓 𝑢 defined by 𝜑 𝑢 ( 𝑡 ) = 𝑡 𝑒 𝜈 ( 𝑡 𝜏 ) 𝑢 ( 𝜏 ) 𝑑 𝜏 , 𝜓 𝑢 ( 𝑡 ) = 𝑡 𝑒 𝜈 ( 𝜏 𝑡 ) 𝑢 ( 𝜏 ) 𝑑 𝜏 ( 2 . 1 6 ) belong to 𝐵 . Moreover, there is 𝛾 𝐵 , 𝜈 > 0 which depends only on 𝐵 and 𝜈 such that | | 𝜑 𝑢 | | 𝐵 𝛾 𝐵 , 𝜈 | 𝑢 | 𝐵 , | | 𝜓 𝑢 | | 𝐵 𝛾 𝐵 , 𝜈 | 𝑢 | 𝐵 , 𝑢 𝐵 . ( 2 . 1 7 )

Proof. We consider the operators 𝑍 𝐿 ( , ) 𝐿 ( , ) , ( 𝑍 ( 𝑢 ) ) ( 𝑡 ) = 𝑡 𝑒 𝜈 ( 𝑡 𝜏 ) 𝑢 ( 𝜏 ) 𝑑 𝜏 , 𝑊 𝐿 ( , ) 𝐿 ( , ) , ( 𝑊 ( 𝑢 ) ) ( 𝑡 ) = 𝑡 𝑒 𝜈 ( 𝜏 𝑡 ) 𝑢 ( 𝜏 ) 𝑑 𝜏 . ( 2 . 1 8 ) We have that 𝑍 and 𝑊 are correctly defined bounded linear operators. Moreover, the restrictions 𝑍 | 𝐿 1 ( , ) 𝐿 1 ( , ) and 𝑊 | 𝐿 1 ( , ) 𝐿 1 ( , ) are correctly defined and bounded linear operators. Since 𝐵 ( ) , then, from Remark 2.19, we have that 𝐵 is an interpolation space between 𝐿 1 ( , ) and 𝐿 ( , ) . This implies that the restrictions 𝑍 | 𝐵 𝐵 𝐵 and 𝑊 | 𝐵 𝐵 𝐵 are correctly defined and bounded linear operators. Setting 𝛾 𝐵 , 𝜈 = m a x { 𝑍 | 𝐵 , 𝑊 | 𝐵 } , the proof is complete.

Notations
If 𝑋 is a Banach space, for every Banach function space 𝐵 𝒯 ( ) , we denote by 𝐵 ( , 𝑋 ) the space of all Bochner measurable functions 𝑣 𝑋 with the property that the mapping 𝑁 𝑣 + , 𝑁 𝑣 ( 𝑡 ) = 𝑣 ( 𝑡 ) belongs to 𝐵 . With respect to the norm 𝑣 𝐵 ( , 𝑋 ) | | 𝑁 = 𝑣 | | 𝐵 , ( 2 . 1 9 ) 𝐵 ( , 𝑋 ) is a Banach space. We also denote by 𝒞 0 , 𝑐 ( , 𝑋 ) the linear space of all continuous functions 𝑣 𝑋 with compact support contained in ( 0 , ) . It is easy to see that 𝒞 0 , 𝑐 ( , 𝑋 ) 𝐵 ( , 𝑋 ) , for all 𝐵 𝒯 ( ) .

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 𝒯 ( ) 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 ( 𝑋 ) —the Banach algebra of all bounded linear operators on 𝑋 , will be denoted by . Let ( Θ , 𝑑 ) be a metric space.

Definition 3.1. A continuous mapping 𝜎 Θ × Θ is called a flow on Θ if 𝜎 ( 𝜃 , 0 ) = 𝜃 and 𝜎 ( 𝜃 , 𝑠 + 𝑡 ) = 𝜎 ( 𝜎 ( 𝜃 , 𝑠 ) , 𝑡 ) , for all ( 𝜃 , 𝑠 , 𝑡 ) Θ × 2 .

Definition 3.2. A pair 𝜋 = ( Φ , 𝜎 ) is called a skew-product flow on 𝑋 × Θ if 𝜎 is a flow on Θ and the mapping Φ Θ × + ( 𝑋 ) called cocycle, satisfies the following conditions:(i) Φ ( 𝜃 , 0 ) = 𝐼 𝑑 and Φ ( 𝜃 , 𝑡 + 𝑠 ) = Φ ( 𝜎 ( 𝜃 , 𝑠 ) , 𝑡 ) Φ ( 𝜃 , 𝑠 ) , for all ( 𝜃 , 𝑡 , 𝑠 ) Θ × 2 + ;(ii)there are 𝑀 1 and 𝜔 > 0 such that Φ ( 𝜃 , 𝑡 ) 𝑀 𝑒 𝜔 𝑡 , for all ( 𝜃 , 𝑡 ) Θ × + ;(iii)for every ( 𝑥 , 𝜃 ) 𝑋 × Θ , the mapping 𝑡 Φ ( 𝜃 , 𝑡 ) 𝑥 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 Θ = , then let 𝜎 ( 𝜃 , 𝑡 ) = 𝜃 + 𝑡 and let { 𝑈 ( 𝑡 , 𝑠 ) } 𝑡 𝑠 be an evolution family on the Banach space 𝑋 . Setting Φ 𝑈 ( 𝜃 , 𝑡 ) = 𝑈 ( 𝜃 + 𝑡 , 𝜃 ) , we observe that 𝜋 𝑈 = ( Φ 𝑈 , 𝜎 ) is a skew-product flow.(ii)Let { 𝑇 ( 𝑡 ) } 𝑡 0 be a 𝒞 0 -semigroup on the Banach space 𝑋 and let Θ be a metric space. ( i i ) 1 If 𝜎 is an arbitrary flow on Θ and Φ 𝑇 ( 𝜃 , 𝑡 ) = 𝑇 ( 𝑡 ) , then 𝜋 𝑇 = ( Φ 𝑇 , 𝜎 ) is a skew-product flow. ( i i ) 2 Let 𝜎 Θ × Θ , 𝜎 ( 𝜃 , 𝑡 ) = 𝜃 be the projection flow on Θ and let { 𝑃 ( 𝜃 ) } 𝜃 Θ ( 𝑋 ) be a uniformly bounded family of projections such that 𝑃 ( 𝜃 ) 𝑇 ( 𝑡 ) = 𝑇 ( 𝑡 ) 𝑃 ( 𝜃 ) , for all ( 𝜃 , 𝑡 ) Θ × + . If Φ 𝑃 ( 𝜃 , 𝑡 ) = 𝑃 ( 𝜃 ) 𝑇 ( 𝑡 ) , then 𝜋 𝑃 = ( Φ 𝑃 , 𝜎 ) 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 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 ̇ 𝑥 ( 𝑡 ) = 𝐴 ( 𝜎 ( 𝜃 , 𝑡 ) ) 𝑥 ( 𝑡 ) , ( 𝜃 , 𝑡 ) Θ × + . ( A ) A cocycle Φ Θ × + ( 𝑋 ) is said to be a solution of A if for every 𝜃 Θ , there is a dense subset 𝐷 𝜃 𝐷 ( 𝐴 ( 𝜃 ) ) such that for every initial condition 𝑥 𝜃 𝐷 𝜃 the mapping 𝑡 𝑥 ( 𝑡 ) = Φ ( 𝜃 , 𝑡 ) 𝑥 𝜃 is differentiable on + , for every 𝑡 + 𝑥 ( 𝑡 ) 𝐷 ( 𝐴 ( 𝜎 ( 𝜃 , 𝑡 ) ) ) and the mapping 𝑡 𝑥 ( 𝑡 ) satisfies A .

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 ( 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 × + ;(ii) Φ ( 𝜃 , 𝑡 ) 𝑥 𝐾 𝑥 , for all 𝑡 0 , all 𝑥 R a n g e 𝑃 ( 𝜃 ) and all 𝜃 Θ ;(iii)the restriction Φ ( 𝜃 , 𝑡 ) | K e r 𝑃 ( 𝜃 ) K e r 𝑃 ( 𝜎 ( 𝜃 , 𝑡 ) ) is an isomorphism, for all ( 𝜃 , 𝑡 ) Θ × + ;(iv) Φ ( 𝜃 , 𝑡 ) 𝑦 ( 1 / 𝐾 ) 𝑦 , for all 𝑡 0 , all 𝑦 K e r 𝑃 ( 𝜃 ) and all 𝜃 Θ ;(v) s u p 𝜃 Θ 𝑃 ( 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 ( ) . We will emphasize that the class 𝒯 ( ) 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 ( ) . Let 𝜋 = ( Φ , 𝜎 ) be a skew-product flow on 𝑋 × Θ . We associate with 𝜋 the input-output control system 𝐸 𝜋 = ( 𝐸 𝜃 ) 𝜃 Θ , where for every 𝜃 Θ 𝑓 ( 𝑡 ) = Φ ( 𝜎 ( 𝜃 , 𝑠 ) , 𝑡 𝑠 ) 𝑓 ( 𝑠 ) + 𝑡 𝑠 Φ ( 𝜎 ( 𝜃 , 𝜏 ) , 𝑡 𝜏 ) 𝑣 ( 𝜏 ) 𝑑 𝜏 , 𝑡 𝑠 , ( 𝐸 𝜃 ) such that the input function 𝑣 𝒞 0 , 𝑐 ( , 𝑋 ) and the output function 𝑓 𝑂 ( , 𝑋 ) .

Definition 3.6. The pair ( 𝑂 ( , 𝑋 ) , 𝐼 ( , 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 ( , 𝑋 ) there exists 𝑓 𝑂 ( , 𝑋 ) such that the pair ( 𝑓 , 𝑣 ) satisfies 𝐸 𝜃 ;(ii)if 𝑣 𝒞 0 , 𝑐 ( , 𝑋 ) and 𝑓 𝑂 ( , 𝑋 ) are such that the pair ( 𝑓 , 𝑣 ) satisfies 𝐸 𝜃 , then 𝑓 𝑂 ( , 𝑋 ) 𝐿 𝑣 𝐼 ( , 𝑋 ) .

Remark 3.7. (i) According to this admissibility concept, it is sufficient to choose all the input functions from the space 𝒞 0 , 𝑐 ( , 𝑋 ) , and, thus, we point out that 𝒞 0 , 𝑐 ( , 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 ( , 𝑋 ) . Actually, condition (ii) shows that the norm of each output function in the space 𝑂 ( , 𝑋 ) is bounded by the norm of the input function in the space 𝐼 ( , 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 ( , 𝑋 ) , 𝐼 ( , 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 ( , 𝑋 ) there exists a unique 𝑓 𝑂 ( , 𝑋 ) such that the pair ( 𝑓 , 𝑣 ) satisfies 𝐸 𝜃 .

In what follows we will analyze the implications of the uniform admissibility of the pair ( 𝑂 ( , 𝑋 ) , 𝐼 ( , 𝑋 ) ) with 𝐼 , 𝑂 𝒯 ( ) 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 𝜆 𝑥 , 𝜃 𝑋 , 𝜆 𝑥 , 𝜃 ( 𝑡 ) = Φ ( 𝜃 , 𝑡 ) 𝑥 , 𝑡 0 , 0 , 𝑡 < 0 , ( 3 . 1 ) called the trajectory determined by the vector 𝑥 and the point 𝜃 Θ .

For every 𝜃 Θ , we denote by ( 𝜃 ) the linear space of all functions 𝜑 𝑋 with the property that 𝜑 ( 𝑡 ) = Φ ( 𝜎 ( 𝜃 , 𝑠 ) , 𝑡 𝑠 ) 𝜑 ( 𝑠 ) , 𝑠 𝑡 0 . ( 3 . 2 ) For every 𝜃 Θ , we consider the stable subset 𝒮 ( 𝜃 ) = 𝑥 𝑋 𝜆 𝑥 , 𝜃 𝑂 ( , 𝑋 ) ( 3 . 3 ) and, respectively, the unstable subset 𝒰 ( 𝜃 ) = { 𝑥 𝑋 𝜑 𝑂 ( , 𝑋 ) ( 𝜃 ) w i t h 𝜑 ( 0 ) = 𝑥 } . ( 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 × + , the following assertions hold: (i) Φ ( 𝜃 , 𝑡 ) 𝒮 ( 𝜃 ) 𝒮 ( 𝜎 ( 𝜃 , 𝑡 ) ) ;(ii) Φ ( 𝜃 , 𝑡 ) 𝒰 ( 𝜃 ) = 𝒰 ( 𝜎 ( 𝜃 , 𝑡 ) ) .

Proof. The property (i) is immediate. To prove the assertion (ii) let 𝑀 , 𝜔 > 0 be given by Definition 3.2(ii). Let ( 𝜃 , 𝑡 ) Θ × ( 0 , ) . Let 𝑥 𝒰 ( 𝜃 ) . Then, there is 𝜑 𝑂 ( , 𝑋 ) ( 𝜃 ) with 𝜑 ( 0 ) = 𝑥 . We set 𝑦 = Φ ( 𝜃 , 𝑡 ) 𝑥 , and we consider [ ] , 𝜓 𝑋 , 𝜓 ( 𝑠 ) = 0 , 𝑠 > 𝑡 , Φ ( 𝜃 , 𝑠 ) 𝑥 , 𝑠 0 , 𝑡 𝜑 ( 𝑠 ) , 𝑠 < 0 . ( 3 . 5 ) We observe that 𝜓 ( 𝑠 ) 𝜑 ( 𝑠 ) + 𝑀 𝑒 𝜔 𝑡 𝜒 [ 0 , 𝑡 ] ( 𝑠 ) 𝑥 , for all 𝑠 , and since 𝜑 𝑂 ( , 𝑋 ) , we deduce that 𝜓 𝑂 ( , 𝑋 ) . Using the fact that 𝜑 ( 𝜃 ) , we obtain that 𝜓 ( 𝑠 ) = Φ ( 𝜎 ( 𝜃 , 𝜏 ) , 𝑠 𝜏 ) 𝜓 ( 𝜏 ) , 𝜏 𝑠 𝑡 . ( 3 . 6 ) Then, we define the function 𝛿 𝑋 , 𝛿 ( 𝑠 ) = 𝜓 ( 𝑠 + 𝑡 ) and since 𝑂 ( , 𝑋 ) is invariant under translations, we deduce that 𝛿 𝑂 ( , 𝑋 ) . Moreover, from (3.6), it follows that 𝛿 ( 𝑟 ) = Φ ( 𝜎 ( 𝜃 , 𝜉 + 𝑡 ) , 𝑟 𝜉 ) 𝛿 ( 𝜉 ) = Φ ( 𝜎 ( 𝜎 ( 𝜃 , 𝑡 ) , 𝜉 ) , 𝑟 𝜉 ) 𝛿 ( 𝜉 ) , 𝜉 𝑟 0 . ( 3 . 7 ) The relation (3.7) implies that 𝛿 ( 𝜎 ( 𝜃 , 𝑡 ) ) , so 𝑦 = 𝛿 ( 0 ) 𝒰 ( 𝜎 ( 𝜃 , 𝑡 ) ) .
Conversely, let 𝑧 𝒰 ( 𝜎 ( 𝜃 , 𝑡 ) ) . Then, there is ( 𝜎 ( 𝜃 , 𝑡 ) ) 𝑂 ( , 𝑋 ) with ( 0 ) = 𝑧 . Taking 𝑞 𝑋 , 𝑞 ( 𝑠 ) = ( 𝑠 𝑡 ) , we have that 𝑞 𝑂 ( , 𝑋 ) and 𝑞 ( 𝑠 ) = Φ ( 𝜎 ( 𝜃 , 𝜏 ) , 𝑠 𝜏 ) 𝑞 ( 𝜏 ) , 𝜏 𝑠 𝑡 . ( 3 . 8 ) In particular, for 𝜏 𝑠 0 , from (3.8), we deduce that 𝑞 ( 𝜃 ) . This implies that 𝑞 ( 0 ) 𝒰 ( 𝜃 ) . Then, 𝑧 = ( 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 × + the restriction Φ ( 𝜃 , 𝑡 ) | 𝒰 ( 𝜃 ) 𝒰 ( 𝜎 ( 𝜃 , 𝑡 ) ) 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 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 ( , 𝑋 ) , 𝐼 ( , 𝑋 ) ) of spaces from the class 𝒯 ( ) 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 ( , 𝑋 ) , 𝐼 ( , 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 .

Proof. Let 𝐿 > 0 be given by Definition 3.6 and let 𝑀 , 𝜔 > 0 be given by Definition 3.2. Let 𝛼 [ 0 , 2 ] be a continuous function with s u p p 𝛼 ( 0 , 1 ) and 1 0 𝛼 ( 𝜏 ) 𝑑 𝜏 = 1 .
(i) Let 𝜃 Θ and let 𝑥 𝒮 ( 𝜃 ) . We consider the functions 𝑣 𝑋 , 𝑣 ( 𝑡 ) = 𝛼 ( 𝑡 ) Φ ( 𝜃 , 𝑡 ) 𝑥 , 𝑓 𝑋 , 𝑓 ( 𝑡 ) = Φ ( 𝜃 , 𝑡 ) 𝑥 , 𝑡 1 , 𝑡 0 𝛼 [ ( 𝜏 ) 𝑑 𝜏 Φ ( 𝜃 , 𝑡 ) 𝑥 , 𝑡 0 , 1 ) , 0 , 𝑡 < 0 . ( 3 . 9 ) Then, 𝑣 𝒞 0 𝑐 ( , 𝑋 ) and 𝑓 ( 𝑡 ) 𝜆 𝑥 , 𝜃 ( 𝑡 ) , 𝑡 . ( 3 . 1 0 ) Since 𝑥 𝒮 ( 𝜃 ) , we have that 𝜆 𝑥 , 𝜃 𝑂 ( , 𝑋 ) . Then, from (3.10), we obtain that 𝑓 𝑂 ( , 𝑋 ) . An easy computation shows that the pair ( 𝑓 , 𝑣 ) satisfies 𝐸 𝜃 . Then, 𝑓 𝑂 ( , 𝑋 ) 𝐿 𝑣 𝐼 ( , 𝑋 ) . ( 3 . 1 1 ) From 𝑣 ( 𝑡 ) 𝛼 ( 𝑡 ) 𝑀 𝑒 𝜔 𝑥 , for all 𝑡 , we obtain that 𝑣 𝐼 ( , 𝑋 ) 𝑀 𝑒 𝜔 | 𝛼 | 𝐼 𝑥 .
Let 𝑡 2 . From Φ ( 𝜃 , 𝑡 ) 𝑥 𝑀 𝑒 𝜔 [ Φ ( 𝜃 , 𝑠 ) 𝑥 , 𝑠 𝑡 1 , 𝑡 ) , ( 3 . 1 2 ) it follows that Φ ( 𝜃 , 𝑡 ) 𝑥 𝜒 [ 𝑡 1 , 𝑡 ) ( 𝑠 ) 𝑀 𝑒 𝜔 𝑓 ( 𝑠 ) , 𝑠 . ( 3 . 1 3 ) Since 𝑂 is invariant under translations, we deduce that Φ ( 𝜃 , 𝑡 ) 𝑥 𝐹 𝑂 ( 1 ) 𝑀 𝑒 𝜔 𝑓 𝑂 ( , 𝑋 ) . ( 3 . 1 4 ) Using relations (3.11) and (3.14), we have that Φ ( 𝜃 , 𝑡 ) 𝑥 𝑀 2 𝑒 2 𝜔 𝐿 | 𝛼 | 𝐼 𝐹 𝑂 ( 1 ) 𝑥 , 𝑡 2 . ( 3 . 1 5 ) Since Φ ( 𝜃 , 𝑡 ) 𝑥 𝑀 𝑒 2 𝜔 𝑥 , for all 𝑡 [ 0 , 2 ) , setting 𝐾 = m a x { ( 𝑀 2 𝑒 2 𝜔 𝐿 | 𝛼 | 𝐼 ) / 𝐹 𝑂 ( 1 ) , 𝑀 𝑒 2 𝜔 } we deduce that Φ ( 𝜃 , 𝑡 ) 𝑥 𝐾 𝑥 , for all 𝑡 0 . Taking into account that 𝐾 does not depend on 𝜃 or 𝑥 , it follows that Φ ( 𝜃 , 𝑡 ) 𝑥 𝐾 𝑥 , 𝑡 0 , 𝑥 𝒮 ( 𝜃 ) , 𝜃 Θ . ( 3 . 1 6 )
(ii) Let 𝜃 Θ and let ( 𝑥 𝑛 ) 𝒮 ( 𝜃 ) with 𝑥 𝑛 𝑛 𝑥 . For every 𝑛 , we consider the sequence 𝑣 𝑛 𝑋 , 𝑣 𝑛 ( 𝑡 ) = 𝛼 ( 𝑡 ) Φ ( 𝜃 , 𝑡 ) 𝑥 𝑛 , 𝑓 𝑛 𝑋 , 𝑓 𝑛 ( 𝑡 ) = Φ ( 𝜃 , 𝑡 ) 𝑥 𝑛 , 𝑡 1 , 𝑡 0 𝛼 ( 𝜏 ) 𝑑 𝜏 Φ ( 𝜃 , 𝑡 ) 𝑥 𝑛 [ , 𝑡 0 , 1 ) , 0 , 𝑡 < 0 . ( 3 . 1 7 ) We have that 𝑣 𝑛 𝒞 0 𝑐 ( , 𝑋 ) , for all 𝑛 and using similar arguments with those used in relation (3.10), we obtain that 𝑓 𝑛 𝑂 ( , 𝑋 ) , for all 𝑛 . An easy computation shows that the pair ( 𝑓 𝑛 , 𝑣 𝑛 ) satisfies 𝐸 𝜃 . Let 𝑣 𝑋 , 𝑣 ( 𝑡 ) = 𝛼 ( 𝑡 ) Φ ( 𝜃 , 𝑡 ) 𝑥 . Then, 𝑣 𝒞 0 𝑐 ( , 𝑋 ) . According to our hypothesis there is, 𝑓 𝑂 ( , 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 0 𝑐 ( , 𝑋 ) , 𝑔 𝑛 𝑂 ( , 𝑋 ) , and the pair ( 𝑔 𝑛 , 𝑢 𝑛 ) satisfies 𝐸 𝜃 . This implies that 𝑓 𝑛 𝑓 𝑂 ( , 𝑋 ) 𝑣 𝐿 𝑛 𝑣 𝐼 ( , 𝑋 ) , 𝑛 . ( 3 . 1 8 ) From 𝑣 𝑛 ( 𝑡 ) 𝑣 ( 𝑡 ) 𝛼 ( 𝑡 ) 𝑀 𝑒 𝜔 𝑥 𝑛 𝑥 , for all 𝑡 and all 𝑛 , we deduce that 𝑣 𝑛 𝑣 𝐼 ( , 𝑋 ) 𝑀 𝑒 𝜔 | 𝛼 | 𝐼 𝑥 𝑛 𝑥 , 𝑛 . ( 3 . 1 9 ) From (3.18) and (3.19), it follows that 𝑓 𝑛 𝑛 𝑓 in 𝑂 ( , 𝑋 ) . From Remark 2.4(ii), we have that there is a subsequence ( 𝑓 𝑘 𝑛 ) and a negligible set 𝐴 such that 𝑓 𝑘 𝑛 ( 𝑡 ) 𝑛 𝑓 ( 𝑡 ) , for all 𝑡 𝐴 . In particular, it follows that there is 𝑟 > 1 such that 𝑓 ( 𝑟 ) = l i m 𝑛 𝑓 𝑘 𝑛 ( 𝑟 ) = l i m 𝑛 Φ ( 𝜃 , 𝑟 ) 𝑥 𝑘 𝑛 = Φ ( 𝜃 , 𝑟 ) 𝑥 . ( 3 . 2 0 ) Because the pair ( 𝑓 , 𝑣 ) satisfies 𝐸 𝜃 , we obtain that 𝑓 ( 𝑡 ) = Φ ( 𝜎 ( 𝜃 , 𝑟 ) , 𝑡 𝑟 ) 𝑓 ( 𝑟 ) = Φ ( 𝜃 , 𝑡 ) 𝑥 , 𝑡 𝑟 . ( 3 . 2 1 ) This shows that 𝑓 ( 𝑡 ) = 𝜆 𝑥 , 𝜃 ( 𝑡 ) , for all 𝑡 𝑟 . Then, from 𝜆 𝑥 , 𝜃 ( 𝑡 ) 𝑓 ( 𝑡 ) + 𝑀 𝑒 𝜔 𝑟 𝑥 𝜒 [ 0 , 𝑟 ) ( 𝑡 ) , 𝑡 , ( 3 . 2 2 ) using the fact that 𝑓 𝑂 ( , 𝑋 ) and Remark 2.4(i), we obtain that 𝜆 𝑥 , 𝜃 𝑂 ( , 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 ( , 𝑋 ) , 𝐼 ( , 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 .

Proof. Let 𝐿 > 0 be given by Definition 3.6 and let 𝑀 , 𝜔 > 0 be given by Definition 3.2. Let 𝛼 [ 0 , 2 ] be a continuous function with s u p p 𝛼 ( 0 , 1 ) and 1 0 𝛼 ( 𝜏 ) 𝑑 𝜏 = 1 .
(i) Let 𝜃 Θ and let 𝑦 𝒰 ( 𝜃 ) . Then, there is 𝜑 ( 𝜃 ) 𝑂 ( , 𝑋 ) with 𝜑 ( 0 ) = 𝑦 . Let 𝑡 > 0 . We consider the functions 𝑣 𝑋 , 𝑣 ( 𝑠 ) = 𝛼 ( 𝑠 𝑡 ) Φ ( 𝜃 , 𝑠 ) 𝑦 , 𝑓 𝑋 , 𝑓 ( 𝑠 ) = 𝑠 𝛼 ( 𝜏 𝑡 ) 𝑑 𝜏 Φ ( 𝜃 , 𝑠 ) 𝑦 , 𝑠 0 , 𝜑 ( 𝑠 ) , 𝑠 < 0 . ( 3 . 2 3 ) We have that 𝑣 𝒞 0 𝑐 ( , 𝑋 ) and 𝑓 is continuous. Let 𝑚 = s u p 𝑠 [ 0 , 𝑡 + 1 ] 𝑓 ( 𝑠 ) . Then, we have that 𝑓 ( 𝑠 ) 𝜑 ( 𝑠 ) + 𝑚 𝜒 [ 0 , 𝑡 + 1 ] ( 𝑠 ) , 𝑠 . ( 3 . 2 4 ) From (3.24) and Remark 2.4(i), we deduce that 𝑓 𝑂 ( , 𝑋 ) . An easy computation shows that the pair ( 𝑓 , 𝑣 ) satisfies 𝐸 𝜃 . Then, according to our hypothesis, we have that 𝑓 𝑂 ( , 𝑋 ) 𝐿 𝑣 𝐼 ( , 𝑋 ) . ( 3 . 2 5 ) From 𝑣 ( 𝑠 ) 𝛼 ( 𝑠 𝑡 ) 𝑀 𝑒 𝜔 Φ ( 𝜃 , 𝑡 ) 𝑦 , for all 𝑠 , we obtain that 𝑣 𝐼 ( , 𝑋 ) | 𝛼 | 𝐼 𝑀 𝑒 𝜔 Φ ( 𝜃 , 𝑡 ) 𝑦 . ( 3 . 2 6 ) Since 𝑦 = 𝜑 ( 0 ) = Φ ( 𝜎 ( 𝜃 , 𝑠 ) , 𝑠 ) 𝜑 ( 𝑠 ) , for all 𝑠 [ 1 , 0 ) , we have that 𝑦 𝜒 [ 1 , 0 ) ( 𝑠 ) 𝑀 𝑒 𝜔 𝜑 ( 𝑠 ) 𝜒 [ 1 , 0 ) ( 𝑠 ) 𝑀 𝑒 𝜔 𝑓 ( 𝑠 ) , 𝑠 . ( 3 . 2 7 ) Using the invariance under translations of the space 𝑂 from relation (3.27), we obtain that 𝑦 𝐹 𝑂 ( 1 ) 𝑀 𝑒 𝜔 𝑓 𝑂 ( , 𝑋 ) . ( 3 . 2 8 ) Taking 𝐾 = ( 𝑀 2 𝑒 2 𝜔 𝐿 | 𝛼 | 𝐼 ) / 𝐹 𝑂 ( 1 ) from relations (3.25), (3.26), and (3.28), it follows that Φ ( 𝜃 , 𝑡 ) 𝑦 ( 1 / 𝐾 ) 𝑦 . Taking into account that 𝐾 does not depend on 𝑡 , 𝑦 or 𝜃 , we conclude that 1 Φ ( 𝜃 , 𝑡 ) 𝑦 𝐾 𝑦 , 𝑡 0 , 𝑦 𝒰 ( 𝜃 ) , 𝜃 Θ . ( 3 . 2 9 )
(ii) Let 𝜃 Θ and let ( 𝑦 𝑛 ) 𝒰 ( 𝜃 ) with 𝑦 𝑛 𝑦 . Then, for every 𝑛 , there is 𝜑 𝑛 𝑂 ( , 𝑋 ) ( 𝜃 ) with 𝜑 𝑛 ( 0 ) = 𝑦 𝑛 . For every 𝑛 , we consider the functions 𝑣 𝑛 𝑋 , 𝑣 𝑛 ( 𝑡 ) = 𝛼 ( 𝑡 ) Φ ( 𝜃 , 𝑡 ) 𝑦 𝑛 , 𝑓 𝑛 𝑋 , 𝑓 𝑛 ( 𝑡 ) = 𝑡 𝛼 ( 𝜏 ) 𝑑 𝜏 Φ ( 𝜃 , 𝑡 ) 𝑦 𝑛 𝜑 , 𝑡 0 , 𝑛 ( 𝑡 ) , 𝑡 < 0 . ( 3 . 3 0 ) We have that 𝑣 𝑛 𝒞 0 𝑐 ( , 𝑋 ) , and, using similar arguments with those used in relation (3.24), we deduce that 𝑓 𝑛 𝑂 ( , 𝑋 ) , for all 𝑛 . An easy computation shows that the pair ( 𝑓 𝑛 , 𝑣 𝑛 ) satisfies 𝐸 𝜃 . Let 𝑣 𝑋 , 𝑣 ( 𝑡 ) = 𝛼 ( 𝑡 ) Φ ( 𝜃 , 𝑡 ) 𝑦 . ( 3 . 3 1 ) According to our hypothesis, there is 𝑓 𝑂 ( , 𝑋 ) such that the pair ( 𝑓 , 𝑣 ) satisfies 𝐸 𝜃 . In particular, this implies that 𝑓 ( 𝜃 ) . Moreover, for every 𝑛 , the pair ( 𝑓 𝑛 𝑓 , 𝑣 𝑛 𝑣 ) satisfies 𝐸 𝜃 . According to our hypothesis, it follows that 𝑓 𝑛 𝑓 𝑂 ( , 𝑋 ) 𝑣 𝐿 𝑛 𝑣 𝐼 ( , 𝑋 ) , 𝑛 . ( 3 . 3 2 ) We have that 𝑣 𝑛 ( 𝑡 ) 𝑣 ( 𝑡 ) 𝛼 ( 𝑡 ) 𝑀 𝑒 𝜔 𝑦 𝑛 𝑦 , for all 𝑡 and all 𝑛 , so 𝑣 𝑛 𝑣 𝐼 ( , 𝑋 ) 𝑀 𝑒 𝜔 | 𝛼 | 𝐼 𝑦 𝑛 𝑦 , 𝑛 . ( 3 . 3 3 ) From (3.32) and (3.33) it follows that 𝑓 𝑛 𝑛 𝑓 in 𝑂 ( , 𝑋 ) . Then, from Remark 2.4(ii), there is a subsequence ( 𝑓 𝑘 𝑛 ) ( 𝑓 𝑛 ) and a negligible set 𝐴 such that 𝑓 𝑘 𝑛 ( 𝑡 ) 𝑛 𝑓 ( 𝑡 ) , for all 𝑡 𝐴 . In particular, there is < 0 such that 𝑓 𝑘 𝑛 ( ) 𝑛 𝑓 ( ) . Since 𝑓 , 𝑓 𝑘 𝑛 ( 𝜃 ) , we successively deduce that 𝑦 = l i m 𝑛 𝑦 𝑘 𝑛 = l i m 𝑛 𝑓 𝑘 𝑛 ( 0 ) = l i m 𝑛 Φ ( 𝜎 ( 𝜃 , ) , ) 𝑓 𝑘 𝑛 ( ) = Φ ( 𝜎 ( 𝜃 , ) , ) 𝑓 ( ) = 𝑓 ( 0 ) . ( 3 . 3 4 ) 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 ( ) 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 ( ) and let 𝜋 = ( Φ , 𝜎 ) be a skew-product flow on 𝑋 × Θ . If the pair ( 𝑂 ( , 𝑋 ) , 𝐼 ( , 𝑋 ) ) is uniformly admissible for the system ( 𝐸 𝜋 ) , then 𝜋 is uniformly dichotomic.

Proof. Let 𝐿 > 0 be given by Definition 3.6. Let 𝑀 , 𝜔 > 0 be given by Definition 3.2. Let 𝛼 [ 0 , 2 ] be a continuous function with s u p p 𝛼 ( 0 , 1 ) and 1 0 𝛼 ( 𝜏 ) 𝑑 𝜏 = 1 .

Step 1. We prove that 𝒮 ( 𝜃 ) 𝒰 ( 𝜃 ) = { 0 } , for all 𝜃 Θ .
Let 𝜃 Θ and let 𝑥 𝒮 ( 𝜃 ) 𝒰 ( 𝜃 ) . Then, there is 𝜑 𝑂 ( , 𝑋 ) ( 𝜃 ) with 𝜑 ( 0 ) = 𝑥 . We consider the function 𝑓 𝑋 , 𝑓 ( 𝑡 ) = Φ ( 𝜃 , 𝑡 ) 𝑥 , 𝑡 0 , 𝜑 ( 𝑡 ) , 𝑡 < 0 . ( 3 . 3 5 ) Then, 𝑓 ( 𝑡 ) 𝜑 ( 𝑡 ) + 𝜆 𝑥 , 𝜃 ( 𝑡 ) , for all 𝑡 . This implies that 𝑓 𝑂 ( , 𝑋 ) . An easy computation shows that the pair ( 𝑓 , 0 ) satisfies 𝐸 𝜃 . Then, according to our hypothesis, it follows that 𝑓 𝑂 ( , 𝑋 ) = 0 , so 𝑓 ( 𝑡 ) = 0 a.e. 𝑡 . Observing that 𝑓 is continuous, we obtain that 𝑓 ( 𝑡 ) = 0 , for all 𝑡 . In particular, we have that 𝑥 = 𝑓 ( 0 ) = 0 .

Step 2. We prove that 𝒮 ( 𝜃 ) + 𝒰 ( 𝜃 ) = 𝑋 , for all 𝜃 Θ .
Let 𝜃 Θ and let 𝑥 𝑋 . Let 𝑣 𝑋 , 𝑣 ( 𝑡 ) = 𝛼 ( 𝑡 ) Φ ( 𝜃 , 𝑡 ) 𝑥 . Then, 𝑣 𝒞 0 𝑐 ( , 𝑋 ) , so there is 𝑓 𝑂 ( , 𝑋 ) such that the pair ( 𝑓 , 𝑣 ) satisfies 𝐸 𝜃 . In particular, this implies that 𝑓 ( 𝜃 ) , so 𝑓 ( 0 ) 𝒰 ( 𝜃 ) . In addition, we observe that 𝑓 ( 𝑡 ) = Φ ( 𝜃 , 𝑡 ) 𝑓 ( 0 ) + 𝑡 0 𝛼 ( 𝜏 ) 𝑑 𝜏 Φ ( 𝜃 , 𝑡 ) 𝑥 = Φ ( 𝜃 , 𝑡 ) ( 𝑓 ( 0 ) + 𝑥 ) , 𝑡 1 . ( 3 . 3 6 ) 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 𝜆 𝑧 𝑥 , 𝜃 ( 𝑡 ) 𝑓 ( 𝑡 ) + 𝑀 𝑒 𝜔 𝑧 𝑥 𝜒 [ 0 , 1 ) ( 𝑡 ) , 𝑡 . ( 3 . 3 7 ) From relation (3.37) and Remark 2.4(i) we obtain that 𝜆 𝑧 𝑥 , 𝜃 𝑂 ( , 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 . 3 8 ) For every 𝜃 Θ we denote by 𝑃 ( 𝜃 ) the projection with the property that R a n g e 𝑃 ( 𝜃 ) = 𝒮 ( 𝜃 ) , K e r 𝑃 ( 𝜃 ) = 𝒰 ( 𝜃 ) . ( 3 . 3 9 ) Using Proposition 3.9 we obtain that Φ ( 𝜃 , 𝑡 ) 𝑃 ( 𝜃 ) = 𝑃 ( 𝜎 ( 𝜃 , 𝑡 ) ) Φ ( 𝜃 , 𝑡 ) , ( 𝜃 , 𝑡 ) Θ × + . ( 3 . 4 0 ) Let ( 𝜃 , 𝑡 ) Θ × + . From Proposition 3.9(ii), it follows that the restriction Φ ( 𝜃 , 𝑡 ) | K e r 𝑃 ( 𝜃 ) K e r 𝑃 ( 𝜎 ( 𝜃 , 𝑡 ) ) is correctly defined and surjective. According to Theorem 3.12(ii) we have that Φ ( 𝜃 , 𝑡 ) | is also injective, so this is an isomorphism, for all ( 𝜃 , 𝑡 ) Θ × + .Step 3. We prove that s u p 𝜃 Θ 𝑃 ( 𝜃 ) < .
Let 𝜃 Θ and let 𝑥 𝑋 . Let 𝑥 𝜃 𝑠 = 𝑃 ( 𝜃 ) 𝑥 and let 𝑥 𝜃 𝑢 = ( 𝐼 𝑃 ( 𝜃 ) ) 𝑥 . Since 𝑥 𝜃 𝑢 K e r 𝑃 ( 𝜃 ) = 𝒰 ( 𝜃 ) , there is 𝜓 ( 𝜃 ) 𝑂 ( , 𝑋 ) with 𝜓 ( 0 ) = 𝑥 𝜃 𝑢 . We consider the functions 𝑣 𝑋 , 𝑣 ( 𝑡 ) = 𝛼 ( 𝑡 ) Φ ( 𝜃 , 𝑡 ) 𝑥 , 𝑓 𝑋 , 𝑓 ( 𝑡 ) = Φ ( 𝜃 , 𝑡 ) 𝑥 𝜃 𝑠 , 𝑡 1 , Φ ( 𝜃 , 𝑡 ) 𝑥 𝜃 𝑢 + 𝑡 0 [ 𝛼 ( 𝜏 ) 𝑑 𝜏 Φ ( 𝜃 , 𝑡 ) 𝑥 , 𝑡 0 , 1 ) , 𝜓 ( 𝑡 ) , 𝑡 < 0 . ( 3 . 4 1 ) We have that 𝑣 𝒞 0 𝑐 ( , 𝑋 ) and 𝑓 is continuous. From 𝑥 𝜃 𝑠 R a n g e 𝑃 ( 𝜃 ) = 𝒮 ( 𝜃 ) , we have that the function 𝜆 𝑥 𝜃 𝑠 , 𝜃 belongs to 𝑂 ( , 𝑋 ) . Setting 𝑚 = s u p 𝑡 [ 0 , 1 ] 𝑓 ( 𝑡 ) and observing that 𝑓 ( 𝑡 ) 𝜓 ( 𝑡 ) + 𝑚 𝜒 [ 0 , 1 ] ( 𝑡 ) + 𝜆 𝑥 𝜃 𝑠 , 𝜃 ( 𝑡 ) , 𝑡 , ( 3 . 4 2 ) from (3.42), we deduce that 𝑓 𝑂 ( , 𝑋 ) . An easy computation shows that the pair ( 𝑓 , 𝑣 ) satisfies 𝐸 𝜃 . This implies that 𝑓 𝑂 ( , 𝑋 ) 𝐿 𝑣 𝐼 ( , 𝑋 ) . ( 3 . 4 3 ) Since 𝜓 ( 𝜃 ) , we have that 𝑥 𝜃 𝑢 = 𝜓 ( 0 ) = Φ ( 𝜎 ( 𝜃 , 𝑠 ) , 𝑠 ) 𝜓 ( 𝑠 ) , for all 𝑠 [ 1 , 0 ) . This implies that 𝑥 𝜃 𝑢 𝑀 𝑒 𝜔 𝜓 ( 𝑠 ) = 𝑀 𝑒 𝜔 [ 𝑓 ( 𝑠 ) , 𝑠 1 , 0 ) , ( 3 . 4 4 ) and we obtain that 𝑥 𝜃 𝑢 𝜒 [ 1 , 0 ) ( 𝑠 ) 𝑀 𝑒 𝜔 𝑓 ( 𝑠 ) , 𝑠 . ( 3 . 4 5 ) Using the invariance under translations of the space 𝑂 , from relation (3.45) we deduce that 𝑥 𝜃 𝑢 𝐹 𝑂 ( 1 ) 𝑀 𝑒 𝜔 𝑓 𝑂 ( , 𝑋 ) . ( 3 . 4 6 ) In addition, from 𝑣 ( 𝑡 ) 𝛼 ( 𝑡 ) 𝑀 𝑒 𝜔 𝑥 , 𝑡 , ( 3 . 4 7 ) we obtain that 𝑣 𝐼 ( , 𝑋 ) | 𝛼 | 𝐼 𝑀 𝑒 𝜔 𝑥 . ( 3 . 4 8 ) Setting 𝛾 = [ 𝐿 | 𝛼 | 𝐼 𝑀 2 𝑒 2 𝜔 / 𝐹 𝑂 ( 1 ) ] from relations (3.43), (3.46), and (3.48), we have that ( 𝐼 𝑃 ( 𝜃 ) ) 𝑥 = 𝑥 𝜃 𝑢 𝛾 𝑥 . ( 3 . 4 9 ) This implies that 𝑃 ( 𝜃 ) 𝑥 ( 1 + 𝛾 ) 𝑥 . ( 3 . 5 0 ) 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.

4. 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 ( , 𝑋 ) , 𝐼 ( , 𝑋 ) ) for the associated control system ( 𝐸 𝜋 ) , where 𝑂 , 𝐼 𝒯 ( ) . 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 ( , 𝑋 ) , 𝐼 ( , 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 .

Definition 4.1. A skew-product flow 𝜋 = ( Φ , 𝜎 ) is said to be exponentially dichotomic if there exist a family of projections { 𝑃 ( 𝜃 ) } 𝜃 Θ ( 𝑋 ) and two constants 𝐾 1 and 𝜈 > 0 such that the following properties hold:(i) Φ ( 𝜃 , 𝑡 ) 𝑃 ( 𝜃 ) = 𝑃 ( 𝜎 ( 𝜃 , 𝑡 ) ) Φ ( 𝜃 , 𝑡 ) , for all ( 𝜃 , 𝑡 ) Θ × + ;(ii) Φ ( 𝜃 , 𝑡 ) 𝑥 𝐾 𝑒 𝜈 𝑡 𝑥 , for all 𝑡 0 , all 𝑥 R a n g e 𝑃 ( 𝜃 ) and all 𝜃 Θ ;(iii)the restriction Φ ( 𝜃 , 𝑡 ) | K e r 𝑃 ( 𝜃 ) K e r 𝑃 ( 𝜎 ( 𝜃 , 𝑡 ) ) is an isomorphism, for all ( 𝜃 , 𝑡 ) Θ × + ;(iv) Φ ( 𝜃 , 𝑡 ) 𝑦 ( 1 / 𝐾 ) 𝑒 𝜈 𝑡 𝑦 , for all 𝑡 0 , all 𝑦 K e r 𝑃 ( 𝜃 ) and all 𝜃 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 s u p 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 1 𝐾 𝑒 𝜈 𝑡 ( 𝐼 𝑃 ( 𝜃 ) ) 𝑥 Φ ( 𝜃 , 𝑡 ) ( 𝐼 𝑃 ( 𝜃 ) ) 𝑥 𝑀 𝑒 𝜔 𝑡 𝑥 + 𝐾 𝑒 𝜈 𝑡 𝑃 ( 𝜃 ) 𝑥 𝑀 𝑒 𝜔 𝑡 + 𝐾 𝑥 + 𝐾 𝑒 𝜈 𝑡 ( 𝐼 𝑃 ( 𝜃 ) ) 𝑥 , ( 4 . 1 ) which implies that 𝑒 2 𝜈 𝑡 𝐾 2 𝑒 𝜈 𝑡 𝐾 ( 𝐼 𝑃 ( 𝜃 ) ) 𝑥 𝑀 𝑒 𝜔 𝑡 + 𝐾 𝑥 , 𝑡 0 , ( 𝑥 , 𝜃 ) 𝑋 × Θ . ( 4 . 2 ) Let > 0 be such that 𝑒 2 𝜈 𝐾 2 > 0 . Setting 𝛼 = ( 𝑒 2 𝜈 𝐾 2 ) 𝑒 𝜈 / 𝐾 and 𝛿 = ( 𝑀 𝑒 𝜔 + 𝐾 ) , it follows that ( 𝐼 𝑃 ( 𝜃 ) ) 𝑥 ( 𝛿 / 𝛼 ) 𝑥 , for all ( 𝑥 , 𝜃 ) 𝑋 × Θ . This implies that 𝐼 𝑃 ( 𝜃 ) 𝛿 / 𝛼 , for all 𝜃 Θ , so 𝑃 ( 𝜃 ) 1 + ( 𝛿 / 𝛼 ) , for all 𝜃 Θ , and the proof is complete.

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 ( ) . According to the main result in the previous section (see Theorem 3.13), if the pair ( 𝑂 ( , 𝑋 ) , 𝐼 ( , 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 R a n g e 𝑃 ( 𝜃 ) = 𝒮 ( 𝜃 ) , K e r 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.

Theorem 4.5 (The behavior on the stable subspace). Let 𝑂 , 𝐼 be two Banach function spaces such that either 𝑂 𝒬 ( ) or 𝐼 ( ) . If the pair ( 𝑂 ( , 𝑋 ) , 𝐼 ( , 𝑋 ) ) is uniformly admissible for the system ( 𝐸 𝜋 ) , then there are 𝐾 , 𝜈 > 0 such that Φ ( 𝜃 , 𝑡 ) 𝑥 𝐾 𝑒 𝜈 𝑡 𝑥 , 𝑡 0 , 𝑥 R a n g e 𝑃 ( 𝜃 ) , 𝜃 Θ . ( 4 . 4 )

Proof. Let 𝛿 > 0 be such that Φ ( 𝜃 , 𝑡 ) 𝑥 𝛿 𝑥 , 𝑡 0 , 𝑥 R a n g e 𝑃 ( 𝜃 ) , 𝜃 Θ . ( 4 . 5 ) We prove that there is > 0 such that 1 Φ ( 𝜃 , ) 𝑥 𝑒 𝑥 , 𝑥 R a n g e 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.
Case 1. Suppose that 𝑂 𝒬 ( ) . Let 𝛼 [ 0 , 2 ] be a continuous function with s u p p 𝛼 ( 0 , 1 ) such that 1 0 𝛼 ( 𝜏 ) 𝑑 𝜏 = 1 . Since s u p 𝑡 > 0 𝐹 𝑂 ( 𝑡 ) = , there is 𝑟 > 0 such that 𝐹 𝑂 ( 𝑟 ) 𝑒 𝛿 2 𝐿 | 𝛼 | 𝐼 . ( 4 . 7 )
Let 𝜃 Θ and let 𝑥 R a n g e 𝑃 ( 𝜃 ) . If Φ ( 𝜃 , 1 ) 𝑥 0 , then we consider the functions 𝑣 𝑋 , 𝑣 ( 𝑡 ) = 𝛼 ( 𝑡 ) Φ ( 𝜃 , 𝑡 ) 𝑥 , Φ ( 𝜃 , 𝑡 ) 𝑥 𝑓 𝑋 , 𝑓 ( 𝑡 ) = 𝑎 Φ ( 𝜃 , 𝑡 ) 𝑥 , 𝑡 1 , 𝑡 0 𝛼 ( 𝜏 ) [ ] , Φ ( 𝜃 , 𝜏 ) 𝑥 𝑑 𝜏 Φ ( 𝜃 , 𝑡 ) 𝑥 , 𝑡 0 , 1 0 , 𝑡 < 0 , ( 4 . 8 ) where 𝑎 = 1 0 𝛼 ( 𝜏 ) Φ ( 𝜃 , 𝜏 ) 𝑥 𝑑 𝜏 . ( 4 . 9 ) We observe that 𝑓 is continuous and