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Pairs of Function Spaces and Exponential Dichotomy on the Real Line
Advances in Difference Equations volume 2010, Article number: 347670 (2010)
Abstract
We provide a complete diagram of the relation between the admissibility of pairs of Banach function spaces and the exponential dichotomy of evolution families on the real line. We prove that if and are two Banach function spaces with the property that either or , then the admissibility of the pair implies the existence of the exponential dichotomy. We study when the converse implication holds and show that the hypotheses on the underlying function spaces cannot be dropped and that the obtained results are the most general in this topic. Finally, our results are applied to the study of exponential dichotomy of semigroups.
1. Introduction
In the study of the asymptotic behavior of evolution equations the inputoutput conditions are very efficient tools, with wide applicability area, and give a nice connection between control theory and the qualitative theory of differential equations (see [1–16] and the reference therein). Starting with the pioneering work of Perron (see [8]) these methods were developed and improved in remarkable books (see [1, 4, 6]). A new and interesting perspective on this framework was proposed in [5], where the authors presented a complete study of stability, expansiveness, and dichotomy of evolution families on the halfline in terms of inputoutput methods. This paper was the starting point for an entire collection of studies dedicated to the inputoutput techniques and their applications to the qualitative theory of differential and difference equations.
If one analyzes the dichotomous properties of differential equations, then it is easily seen that there are some main technical differences between the case of evolution families on the halfline (see [5, 9, 10]) and the case of evolution families on the real line (see [11–16]), which require a distinct analysis for each case. For instance, when one determines sufficient conditions for the existence of exponential dichotomy on the halfline, an important hypothesis is that the initial stable subspace is closed and complemented (see, e.g., [5, Theorem 4.3] or [9, Theorem 3.3]). This assumption may be dropped when we study the exponential dichotomy on the real line (see, e.g., [11, Theorem 5.1] or [16, Theorem 5.3]). These facts implicitly generate the differences between the admissibility concepts used on the real line compared with those used on the halfline and also interesting technical approaches in each case.
The aim of the present paper is to provide new and very general conditions for the existence of exponential dichotomy on the real line. We consider the problem of finding connections between the solvability of an integral equation and the existence of exponential dichotomy of evolution families on the real line. The main purpose is to obtain a complete diagram and a classification of the classes of function spaces that may be used in the study of exponential dichotomy via admissibility.
For the beginning we will present the previous results in this topic and the main objectives will be clearly specified in the context of the actual state of knowledge. We denote by the class of all Banach sequence spaces which are invariant under translations, contain the continuous functions with compact support, satisfy an integral property and if , then there is a continuous function . We consider the subclass of satisfying the ideal property. We associate two subclasses of : —the class of all Banach function spaces with unbounded fundamental function and —the class of all Banach function spaces which contain at least a nonintegrable function. A pair of function spaces is called admissible for an evolution family on the Banach space if for every test function in the input space there exists a unique solution function in the output space for the associated integral equation given by the variation of constants formula (see Definition 3.5 below).
For the first time, we have proposed in [11] a sufficient condition for exponential dichotomy, using certain Banach function spaces which are invariant under translations and we obtained the following theorem.
Theorem 1.1.
If and the pair is admissible for an evolution family , then is exponentially dichotomic.
Our study has been continued and extended in [16], both for uniform dichotomy and exponential dichotomy. According to the proof of Theorem 4.8 in [16] we may give the following sufficient condition for uniform dichotomy.
Theorem 1.2.
If , , and the pair is admissible for an evolution family , then is uniformly dichotomic.
From the proof of Theorem 5.3(i) in [16] we deduce the following sufficient condition for exponential dichotomy.
Theorem 1.3.
If , , and the pair is admissible for an evolution family , then is exponentially dichotomic.
Taking into account the above results and their consequences, the natural question arises whether, in the general case, the output space may belong to the class and if so, which is the most general class where the input space should belong to. The aim of the present paper is to answer this question and to provide a complete study of the exponential dichotomy on the real line via integral admissibility. The answer to the above question will establish clearly how should one modify the hypotheses of Theorem 1.2 such that the admissibility of the pair implies the existence of the exponential dichotomy.
We will prove that if and , then the admissibility of the pair is a sufficient condition for exponential dichotomy. Consequently, we will deduce a complete diagram of the study of exponential dichotomy on the real line in terms of the admissibility of function spaces (see Theorem 3.11). Specifically, if and are two Banach function spaces with the property that either or , then the admissibility of the pair implies the existence of the exponential dichotomy. Also, in certain conditions, we deduce that the exponential dichotomy of an evolution family is equivalent with the admissibility of the pair .
By an example we motivate our techniques and show that the hypotheses from our main results cannot be removed. Precisely, if and are such that and , then we prove that the admissibility of the pair does not imply the exponential dichotomy. Moreover, we show that the obtained results and their consequences are the most general in this topic.
Finally, our results are applied at the study of the exponential dichotomy of semigroups. Using function spaces which are invariant under translations, we obtain a classification of the classes of input and output spaces which may be used in the study of exponential dichotomy of semigroups in terms of inputoutput techniques with respect to associated integral equations.
2. Preliminaries: Banach Function Spaces
In this section, for the sake of clarity, we present some definitions and notations and we introduce the main classes of function spaces that will be used in our study. Let be the linear space of all Lebesgue measurable functions , identifying the functions equal almost everywhere.
Definition 2.1.
A linear subspace of is called normed function space if there is a mapping such that

(i)
if and only if a.e.;

(ii)
, for all ;

(iii)
, for all ;

(iv)
if and a.e. then ;

(v)
if , then .
If is complete, then is called Banach function space.
Definition 2.2.
A Banach function space is said to be invariant under translations if for every , the function belongs to and .
Notations 1.
Let denote the linear space of all continuous functions with compact support. Throughout this paper, we denote by the class of all Banach function spaces , which are invariant under translations, , and satisfy the following conditions:

(i)
for every there is such that , for all ;

(ii)
if then there is a continuous function .
For examples of Banach function spaces from the class we refer to [11].
Let be the class of all Banach function spaces with the property that if a.e. and , then .
For every we denote by the characteristic function of the set . Then, if , we have that , for every with .
Definition 2.3.
Let . The mapping is called the fundamental function of the space .
For the proof of the next proposition we refer to [16, Proposition 2.8].
Proposition 2.4.
Let and . If is a function, which belongs to and with the property that belongs to , then the functions
belong to .
Example 2.5.
Let be the linear space of all with the property that . With respect to the norm
this is a Banach function space which belongs to .
Lemma 2.6.
If , then .
Proof.
Let be such that , for all . Then, we have that
Notations 2.
In what follows we denote by

(i)
the class of all Banach function spaces with ;

(ii)
the class of all Banach function spaces with the property that ;

(iii)
the class of all Banach function spaces with the property that for every in , the function belongs to .
Remark 2.7.

(i)
For examples of Banach function spaces from the class we refer to [16, Proposition 2.9].

(ii)
If then there is a continuous function with .
Notation 1.
Let be the space of all continuous functions with , which is Banach space with respect to the norm .
Lemma 2.8.
Let be a Banach function space with . Then .
Proof.
Let . Let . Then there is an unbounded increasing sequence such that , for all and all . Setting we have that
From the above inequality we deduce that is fundamental in the Banach space , so there is such that in . According to [16, Lemma 2.4] there is a subsequence such that a.e. This implies that a.e., so in . Thus and the proof is complete.
Notation 2.
Let be a real or complex Banach space. For every we denote by the linear space of all Bochner measurable functions with the property that the mapping lies in . With respect to the norm , is a Banach space.
3. Exponential Dichotomy for Evolution Families on the Real Line
Let be a real or complex Banach space. The norm on and on , the Banach algebra of all bounded linear operators on , will be denoted by . Denote by the identity operator on . First, we remind some basic definitions.
Definition 3.1.
A family of bounded linear operators on is called an evolution family if the following properties hold:

(i)
and , for all ;

(ii)
for every and every the mapping is continuous on and the mapping is continuous on ;

(iii)
there are and such that , for all .
Definition 3.2.
An evolution family is said to be uniformly dichotomic if there are a family of projections and a constant such that

(i)
, for all ;

(ii)
the restriction is an isomorphism, for all ;

(iii)
, for all and all ;

(iv)
, for all and all .
Definition 3.3.
An evolution family is said to be exponentially dichotomic if there exist a family of projections and two constants and such that

(i)
, for all ;

(ii)
the restriction is an isomorphism, for all ;

(iii)
, for all and all ;

(iv)
, for all and all .
Remark 3.4.
It is obvious that if an evolution family is exponentially dichotomic, then it is uniformly dichotomic.
One of the most efficient tool in the study of the dichotomic behavior of an evolution family is represented by the socalled inputoutput techniques. The inputoutput method considered in this paper is the admissibility of a pair of function spaces. Indeed, let be two Banach function spaces such that and .
Definition 3.5.
The pair is said to be admissible for if for every there exists a unique such that the pair satisfies the equation
Remark 3.6.
If the pair is admissible for , then it makes sense to define the operator , where is such that the pair satisfies (). Then is a bounded linear operator (see [16, Proposition 4.4]).
Let be an evolution family on and . For every , we consider the stable subspace as the space of all with the property that the function
belongs to and we define the unstable subspace as the space of all with the property that there is a function such that and , for all .
An important information concerning the structure of the projection family associated with a uniformly dichotomic evolution family was obtained in [16, Theorem 4.8] and this is given by the following.
Theorem 3.7.
Let be an evolution family on and let be two Banach function spaces with and . If the pair is admissible for the evolution family , then is uniformly dichotomic with respect to the family of projections , where
Taking into account the results obtained in [11, 16], an interesting open question is whether in the study of exponential dichotomy, the output space may belong to the general class . To answer this question, the first purpose of this paper is to prove the following theorem.
Theorem 3.8.
Let be an evolution family on the Banach space and let be two Banach function spaces with and . If the pair is admissible for , then is uniformly exponentially dichotomic.
The proof will be constructive and therefore, we will present several intermediate results.
Theorem 3.9.
Let be an evolution family on the Banach space and let be two Banach function spaces with and . If the pair is admissible for , then there are such that
Proof.
According to Theorem 3.7 and Definition 3.2(iii) we have that there is such that
Since , from Remark 2.7(ii) we have that there is a continuous function with . Using the invariance under translations of the space , we may assume that there is such that
Since there is such that
Let be a continuous function with and , for . Then, the function is continuous and from (3.5) and (3.6) we have that
Let and let . We consider the functions
Since it follows that . Setting we observe that , for all . Since we deduce that . A simple computation shows that the pair satisfies (), so . This implies that
According to relation (3.4) we observe that
and using the invariance under translations of the space we deduce that . From , for all , we have that . Thus we obtain that
From , for all , we have that , for all , which implies that
Setting , from relations (3.10)–(3.13) it follows that
Using relations (3.7) and (3.14) we deduce that Taking into account that does not depend on or , we have that
Let and . Let and . Then, there are and such that . Using relations (3.4) and (3.15) we obtain that , which completes the proof.
Theorem.
Let be an evolution family on the Banach space and let be two Banach function spaces with and . If the pair is admissible for , then there are such that
Proof.
Let and be given by Definition 3.1. According to Theorem 3.7 and Definition 3.2(iv) we have that there is such that
Since , from Remark 2.7(ii) we have that there is a continuous function with . Using the invariance under translations of the space we may assume that there is such that
Using similar arguments with those in the proof of Theorem 3.9 we obtain that there is a continuous function with , , for all and
Let and . Then, there is such that and , for all . We consider the functions
where . We have that , so . Using relation (3.17) we have that
From this inequality, since we deduce that . An easy computation shows that the pair satisfies (), so . Then, we have that
Using relation (3.17) we have that
which implies that
Since , for all , we deduce that
This shows that
From relations (3.19)–(3.27) it follows that . Taking into account that does not depend on or , we have that
We set and . Let and . Then, there are and such that . Using relations (3.17) and (3.28) we obtain that , which completes the proof.
Now, we may give the proof of Theorem 3.8.
Proof of Theorem 3.8.
This immediately follows from Theorems 3.7, 3.9, and 3.10.
Now, we may give the main result of the paper, which establishes a complete diagram concerning the study of exponential dichotomy on the real line in terms of integral admissibility.
Theorem 3.11.
Let be an evolution family on the Banach space and let be two Banach function spaces with and . If or , then the following assertions hold:

(i)
if the pair is admissible for , then is uniformly exponentially dichotomic;

(ii)
if and one of the spaces belongs to the class , then is exponentially dichotomic if and only if the pair is admissible for .
Proof.

(i)
This follows from Theorems 1.3 and 3.8.

(ii)
Necessity. Suppose that is exponentially dichotomic with respect to the family of projections and the constants . Then, we have that (see, e.g., [13]).
Let . We consider the function
where for every , denotes the inverse of the operator .
If , then using Proposition 2.4 we obtain that . Since we deduce that .
If then, since we have that . Using Proposition 2.4 it follows that .
An easy computation shows that the pair satisfies (). The uniqueness of is immediate (see, e.g., [16, the Necessity part of Theorem 5.3]). In conclusion, the pair is admissible for the evolution family .
The natural question arises whether the hypotheses from Theorem 3.11 can be dropped and also if the conditions given by this theorem are the most general in this topic. The answers are given by the following example.
Example 3.12.
Let and be such that and . Then and according to Lemma 2.8 we have that .
We consider the function
Then is a decreasing function.
Let endowed with the norm , for all . For every we consider the operator
Then is an evolution family on .
We prove that the pair is admissible for . Let . Then . We consider the function
We have that is correctly defined and an easy computation shows that the pair satisfies (). We set
We prove that . Since , from
we have that . Let . Then, there is such that . It follows that
The above inequality implies that . Since was arbitrary we obtain that there exists . Using similar arguments with those in (3.34) we deduce that .
Let . Then there is such that . It follows that
From this inequality we have that . Since was arbitrary, it follows that there exists . Thus, we deduce that , so .
To prove the uniqueness of , let be such that the pair satisfies (). Setting we have that and , for all . If , then we deduce that
Let . Using Lemma 2.6 and integrating in (3.37) we have that
For in (3.39) we obtain that .
From relation (3.38) we have that
Integrating in relation (3.40) on and using Lemma 2.6, we deduce that
For in (3.41) it follows that . Since was arbitrary we have that , so . Thus, the pair is admissible for the evolution family .
Suppose that is exponentially dichotomic with respect to the family of projections and the constants . According to [13, Proposition 3.1] we have that , which implies that , for all . Then, we obtain that
or equivalently
which is absurd. In conclusion, the pair is admissible for , but, for all that, the evolution family is not exponentially dichotomic.
4. Applications to the Case of Semigroups
In this section, by applying the central theorems from the Section 3 we deduce several consequences of the main results for the study of exponential dichotomy of semigroups. Let be a real or complex Banach space.
Definition 4.1.
A family of bounded linear operators on is said to be a semigroup if the following properties are satisfied:

(i)
, the identity operator on ;

(ii)
, for all

(iii)
, for every .
Definition 4.2.
A semigroup is said to be exponentially dichotomic if there exist a projection and two constants and such that

(i)
, for all ;

(ii)
, for all and all ;

(iii)
, for all and all ;

(iv)
the restriction is an isomorphism, for every .
Remark 4.3.

(i)
If is a semigroup, we can associate to an evolution family , by , for every .

(ii)
A semigroup is exponentially dichotomic if and only if the associated evolution family is exponentially dichotomic (see [12, Proposition 4.4]).
Let be two Banach function spaces such that and .
Definition 4.4.
The pair is said to be admissible for the semigroup if for every there is a unique such that
Theorem 4.5.
Let be a semigroup on the Banach space and let be two Banach function spaces with and . If or , then the following assertions hold:

(i)
if the pair is admissible for , then is uniformly exponentially dichotomic;

(ii)
if and one of the spaces belongs to the class , then is exponentially dichotomic if and only if the pair is admissible for .
Proof.
This follows from Theorem 3.11 and Remark 4.3(ii).
Remark 4.6.
According to the example given in the previous section we deduce that the hypothesis or cannot be removed. Moreover, Theorem 4.5 provides a complete answer concerning the study of exponential dichotomy of semigroups using inputoutput techniques with respect to the associated integral equation.
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Acknowledgment
This work is supported by the Exploratory Research Project PN 2 ID 1081 code 550/2009.
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Keywords
 Banach Space
 Real Line
 Bounded Linear Operator
 Qualitative Theory
 Evolution Family