- Research Article
- Open access
- Published:
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 input-output 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 half-line in terms of input-output methods. This paper was the starting point for an entire collection of studies dedicated to the input-output 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 half-line (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 half-line, 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 half-line 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 input-output 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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ1_HTML.gif)
belong to .
Example 2.5.
Let be the linear space of all
with the property that
. With respect to the norm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ2_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ3_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ4_HTML.gif)
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 so-called input-output techniques. The input-output 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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ5_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ6_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ7_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ8_HTML.gif)
Proof.
According to Theorem 3.7 and Definition 3.2(iii) we have that there is such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ9_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ10_HTML.gif)
Since there is
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ11_HTML.gif)
Let be a continuous function with
and
, for
. Then, the function
is continuous and from (3.5) and (3.6) we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ12_HTML.gif)
Let and let
. We consider the functions
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ13_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ14_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ15_HTML.gif)
According to relation (3.4) we observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ16_HTML.gif)
and using the invariance under translations of the space we deduce that
. From
, for all
, we have that
. Thus we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ17_HTML.gif)
From , for all
, we have that
, for all
, which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ18_HTML.gif)
Setting , from relations (3.10)–(3.13) it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ19_HTML.gif)
Using relations (3.7) and (3.14) we deduce that Taking into account that
does not depend on
or
, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ20_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ21_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ22_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ23_HTML.gif)
Using similar arguments with those in the proof of Theorem 3.9 we obtain that there is a continuous function with
,
, for all
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ24_HTML.gif)
Let and
. Then, there is
such that
and
, for all
. We consider the functions
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ25_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ26_HTML.gif)
where . We have that
, so
. Using relation (3.17) we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ27_HTML.gif)
From this inequality, since we deduce that
. An easy computation shows that the pair
satisfies (), so
. Then, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ28_HTML.gif)
Using relation (3.17) we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ29_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ30_HTML.gif)
Since , for all
, we deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ31_HTML.gif)
This shows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ32_HTML.gif)
From relations (3.19)–(3.27) it follows that . Taking into account that
does not depend on
or
, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ33_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ34_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ35_HTML.gif)
Then is a decreasing function.
Let endowed with the norm
, for all
. For every
we consider the operator
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ36_HTML.gif)
Then is an evolution family on
.
We prove that the pair is admissible for
. Let
. Then
. We consider the function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ37_HTML.gif)
We have that is correctly defined and an easy computation shows that the pair
satisfies (). We set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ38_HTML.gif)
We prove that . Since
, from
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ39_HTML.gif)
we have that . Let
. Then, there is
such that
. It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ40_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ41_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ42_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ43_HTML.gif)
Let . Using Lemma 2.6 and integrating in (3.37) we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ44_HTML.gif)
For in (3.39) we obtain that
.
From relation (3.38) we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ45_HTML.gif)
Integrating in relation (3.40) on and using Lemma 2.6, we deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ46_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ47_HTML.gif)
or equivalently
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ48_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F347670/MediaObjects/13662_2010_Article_1280_Equ49_HTML.gif)
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 input-output techniques with respect to the associated integral equation.
References
Daleckii JL, Krein MG: Stability of Solutions of Differential Equations in Banach Space, Translations of Mathematical Monographs. Volume 4. American Mathematical Society, Providence, RI, USA; 1974:vi+386.
Liu JH, N'Guérékata GM, Van Minh N: Topics on Stability and Periodicity in Abstract Differential Equations, Series on Concrete and Applicable Mathematics. Volume 6. World Scientific, Hackensack, NJ, USA; 2008:x+208.
Liu Q, Van Minh N, Nguerekata G, Yuan R: Massera type theorems for abstract functional differential equations. Funkcialaj Ekvacioj 2008,51(3):329-350. 10.1619/fesi.51.329
Massera JL, Schäffer JJ: Linear Differential Equations and Function Spaces, Pure and Applied Mathematics. Volume 21. Academic Press, New York, NY, USA; 1966:xx+404.
Van Minh N, Räbiger F, Schnaubelt R: Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line. Integral Equations and Operator Theory 1998,32(3):332-353. 10.1007/BF01203774
Van Minh N, N'Guérékata GM, Yuan R: Lectures on the Asymptotic Behavior of Solutions of Differential Equations. Nova Science, New York, NY, USA; 2008:viii+65.
Pazy A: Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences. Volume 44. Springer, New York, NY, USA; 1983:viii+279.
Perron O: Die Stabilitätsfrage bei Differentialgleichungen. Mathematische Zeitschrift 1930,32(1):703-728. 10.1007/BF01194662
Sasu B, Sasu AL:Exponential dichotomy and
-admissibility on the half-line. Journal of Mathematical Analysis and Applications 2006,316(2):397-408. 10.1016/j.jmaa.2005.04.047
Sasu B: Uniform dichotomy and exponential dichotomy of evolution families on the half-line. Journal of Mathematical Analysis and Applications 2006,323(2):1465-1478. 10.1016/j.jmaa.2005.12.002
Sasu AL, Sasu B: Exponential dichotomy on the real line and admissibility of function spaces. Integral Equations and Operator Theory 2006,54(1):113-130. 10.1007/s00020-004-1347-z
Sasu AL: Exponential dichotomy for evolution families on the real line. Abstract and Applied Analysis 2006, 2006:-6.
Sasu AL, Sasu B: Exponential dichotomy and admissibility for evolution families on the real line. Dynamics of Continuous, Discrete & Impulsive Systems. Series A 2006,13(1):1-26.
Sasu AL, Sasu B:Discrete admissibility,
-spaces and exponential dichotomy on the real line. Dynamics of Continuous, Discrete & Impulsive Systems. Series A 2006,13(5):551-561.
Sasu B, Sasu AL:Exponential trichotomy and
-admissibility for evolution families on the real line. Mathematische Zeitschrift 2006,253(3):515-536. 10.1007/s00209-005-0920-8
Sasu AL: Integral equations on function spaces and dichotomy on the real line. Integral Equations and Operator Theory 2007,58(1):133-152. 10.1007/s00020-006-1478-5
Acknowledgment
This work is supported by the Exploratory Research Project PN 2 ID 1081 code 550/2009.
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Sasu, A.L. Pairs of Function Spaces and Exponential Dichotomy on the Real Line. Adv Differ Equ 2010, 347670 (2010). https://doi.org/10.1155/2010/347670
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DOI: https://doi.org/10.1155/2010/347670