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Integral Equations and Exponential Trichotomy of SkewProduct Flows
Advances in Difference Equations volume 2011, Article number: 918274 (2011)
Abstract
We are interested in an open problem concerning the integral characterizations of the uniform exponential trichotomy of skewproduct flows. We introduce a new admissibility concept which relies on a double solvability of an associated integral equation and prove that this provides several interesting asymptotic properties. The main results will establish the connections between this new admissibility concept and the existence of the most general case of exponential trichotomy. We obtain for the first time necessary and sufficient characterizations for the uniform exponential trichotomy of skewproduct flows in infinitedimensional spaces, using integral equations. Our techniques also provide a nice link between the asymptotic methods in the theory of difference equations, the qualitative theory of dynamical systems in continuous time, and certain related control problems.
1. Introduction
Exponential trichotomy is the most complex asymptotic property of evolution equations, being firmly rooted in bifurcation theory of dynamical systems. The concept proceeds from the central manifold theorem and mainly relies on the decomposition of the state space into a direct sum of three invariant closed subspaces: the stable subspace, the unstable subspace, and the neutral subspace such that the behavior of the solution on the stable and unstable subspaces is described by exponential decay backward and forward in time and, respectively, the solution is bounded on the neutral subspace. The concept of exponential trichotomy for differential equations has the origin in the remarkable works of Elaydi and Hájek (see [1, 2]). Elaydi and Hájek introduced the concept of exponential trichotomy for linear and nonlinear differential systems and proved a number of notable properties in these cases (see [1, 2]). These works were the starting points for the development of this subject in various directions (see [3–8], and the references therein). In [5] the author gave necessary and sufficient conditions for exponential trichotomy of difference equations by examining the existence of a bounded solution of the corresponding inhomogeneous system. Paper [4] brings a valuable contribution to the study of the exponential trichotomy. In this paper Elaydi and Janglajew obtained the first inputoutput characterization for exponential trichotomy (see Theorem 4, page 423). More precisely, the authors proved that a system of difference equations with a invertible matrix on , has an (EH)trichotomy if and only if the associated inhomogeneous system has at least one bounded solution on for every bounded input . In [4] the applicability area of exponential trichotomy was extended, by introducing new concepts of exponential dichotomy and exponential trichotomy. The authors proposed two different methods: in the first approach the authors used the tracking method and in the second approach they introduced a discrete analogue of dichotomy and trichotomy in variation.
A new step in the study of the exponential trichotomy of difference equations was made in [3], where Cuevas and Vidal obtained the structure of the range of each trichotomy projection associated with a system of difference equations which has weighted exponential trichotomy. This approach allows them to deduce the connections between weighted exponential trichotomy and the trichotomy on and as well as to present some applications to the case of nonhomogeneous linear systems. In [8] the authors deduce the explicit formula in terms of the trichotomy projections for the solution of the nonlinear system associated with a system of difference equations which has weighted exponential trichotomy. The first study for exponential trichotomy of variational difference equations was presented in [6], the methods being provided directly for the infinitedimensional case. There we obtained necessary and sufficient conditions for uniform exponential trichotomy of variational difference equations in terms of the solvability of an associated discretetime control system.
Starting with the ideas delineated by the pioneering work of Perron (see [9]) and developed later in remarkable works by Coppel (see [10]), Daleckii and Krein (see [11]), Massera and Schäffer (see [12]) one of the most operational tool in the study of the asymptotic behavior of an evolution equation is represented by the inputoutput conditions. These methods arise from control theory and often provide characterizations of the asymptotic properties of dynamical systems in terms of the solvability of some associated control systems (see [4, 6, 13–21]). According to our knowledge, in the existent literature, there are no inputoutput integral characterizations for uniform exponential trichotomy of skewproduct flows. Moreover, the territory of integral admissibility for exponential trichotomy of skewproduct flows was not explored yet. These facts led to a collection of open questions concerning this topic and, respectively, concerning the operational connotations and consequences in the framework of general variational systems.
The aim of the present paper is to present for the first time a study of exponential trichotomy of skewproduct flows from the new perspective of the integral admissibility. We treat the most general case of exponential trichotomy of skewproduct flows (see Definition 2.4) which is a direct generalization of the exponential dichotomy (see [13, 14, 19–22]) and is tightly related to the behavior described by the central manifold theorem. Our methods will be based on the connections between the asymptotic properties of variational difference equations, the qualitative behavior of skewproduct flows, and control type techniques, providing an interesting interference between the discretetime and the continuoustime behavior of variational systems. We also emphasize that our central purpose is to deduce a characterization for uniform exponential trichotomy without assuming a priori the existence of the projection families, without supposing the invariance with respect to the projection families or the invertibility on the unstable subspace or on the bounded subspace.
We will introduce a new concept of admissibility which relies on a double solvability of an associated integral equation and on the uniform boundedness of the norm of solution relative to the norm of the input function. Using detailed and constructive methods we will prove that this assures the existence of the uniform exponential trichotomy (with all its properties), without any additional hypothesis on the skewproduct flow. Moreover, we will show that the admissibility is also a necessary condition for uniform exponential trichotomy. Thus, we deduce the premiere characterization of the uniform exponential trichotomy of skewproduct flows in terms of the solvability of an associated integral equation. The results are obtained in the most general case, being applicable to any class of variational equations described by skewproduct flows.
2. Basic Definitions and Preliminaries
In this section, for the sake of clarity, we will give some basic definitions and notations and we will present some auxiliary results.
Let be a real or a complex Banach space. The norm on and on , the Banach algebra of all bounded linear operators on , will be denoted by . The identity operator on will be denoted by .
Throughout the paper denotes the set of real numbers and denotes the set of real integers. If then we denote and .
Notations

(i)
We consider the spaces , , and , which are Banach spaces with respect to the norm .

(ii)
If then is a Banach space with respect to the norm .

(iii)
Let be the linear space of all with the property that , for all and the set is finite.
Let be a metric space and let .
Definition 2.1.
A continuous mapping is called a flow on if and , for all .
Definition 2.2.
A pair is called (linear) skewproduct flow on if is a flow on and the mapping , called cocycle, satisfies the following conditions:

(i)
, for all ;

(ii)
, for all (the cocycle identity);

(iii)
there are and such that , for all ;

(iv)
for every the mapping is continuous.
Example 2.3.
Let be a continuous increasing function with and let . We denote by the closure of in , where denotes the space of all continuous functions and
where .
Let be a Banach space and let be a semigroup on with the infinitesimal generator . For every let . We define and we consider the system
If , then is a skewproduct flow on . For every , we note that , for all , is the strong solution of the system ( ).
For other examples which illustrate the modeling of solutions of variational equations by means of skewproduct flows as well as the existence of the perturbed skewproduct flow we refer to [21] (see Examples 2.2 and 2.4). Interesting examples of skewproduct flows which often proceed from the linearization of nonlinear equations can be found in [7, 13, 14, 22, 23], motivating the usual appellation of linear skewproduct flows.
The most complex description of the asymptotic property of a dynamical system is given by the exponential trichotomy, which provides a complete chart of the qualitative behaviors of the solutions on each fundamental manifold: the stable manifold, the central manifold, and the unstable manifold. This means that the state space is decomposed at every point of the flow's domain—the base space—into a direct sum of three invariant closed subspaces such that the solution on the first and on the third subspace exponentially decays forward and backward in time, while on the central subspace the solution had a uniform upper and lower bound (see [1–6, 8]).
Definition 2.4.
A skewproduct flow is said to be uniformly exponentially trichotomic if there are three families of projections , and two constants and such that

(i)
, for all and all ,

(ii)
, for all ,

(iii)
, for all ,

(iv)
, for all and all ,

(v)
, for all , and all ,

(vi)
, for all , and all ,

(vii)
, for all , and all ,

(viii)
the restriction is an isomorphism, for all and all .
Remark 2.5.
We note that this is a direct generalization of the classical concept of uniform exponential dichotomy (see [13, 14, 19–22, 24, 25]) and expresses the behavior described by the central manifold theorem. It is easily seen that for , for all , one obtains the uniform exponential dichotomy concept and the condition (iii) is redundant (see, e.g., [19, Lemma 2.8]).
Remark 2.6.
If a skewproduct flow is uniformly exponentially trichotomic with respect to the families of projections , , then

(i)
, for all ;

(ii)
, for all and all .
Let be a skewproduct flow on . At every point we associate with three fundamental subspaces, which will have a crucial role in the study of the uniform exponential trichotomy.
Notation
For every we denote by the linear space of all functions with
For every we consider the linear space:
called the stable subspace. We also define
called the bounded subspace and, respectively,
called the unstable subspace.
Lemma 2.7.

(i)
If for every , denotes one of the subspaces , or , then , for all .

(ii)
If the skewproduct flow is uniformly exponentially trichotomic with respect to three families of projections , , then these families are uniquely determined by the conditions in Definition 2.4. Moreover one has that
(2.6)
Proof.
See [6, Lemma 5.4, Proposition 5.5, and Remark 5.3].
We will start our investigation by recalling a recent result obtained for the discretetime case. Precisely, the discrete case was treated in [6], where we formulated a first resolution concerning the characterization of the uniform exponential trichotomy in terms of the solvability of a system of variational difference equations. Indeed, we associated with the skewproduct flow the discrete inputoutput system , where for every
with and .
Definition 2.8.
The pair is said to be uniformly admissible for the skewproduct flow if there are and such that for every the following properties hold:

(i)
for every there are and such that the pairs and satisfy ( );

(ii)
if and is such that the pair satisfies ( ) then ;

(iii)
if is such that , for all , and is such that the pair satisfies ( ), then .
The first connection between an inputoutput discrete admissibility and the uniform exponential trichotomy of skewproduct flows was obtained in [6] (see Theorem 5.8) and this is given by what follows.
Theorem 2.9.
Let be a skewproduct flow on . is uniformly exponentially trichotomic if and only if the pair is uniformly admissible for .
The proof of this result relies completely on discretetime arguments and essentially uses the properties of the associated system of variational difference equations. The natural question is whether we may study the uniform exponential trichotomy property of skewproduct flows from a "continuous" point of view. On the other hand, in the spirit of the classical admissibility theory (see [10–12, 14, 15, 21]) it would be interesting to see if the uniform exponential trichotomy can be expressed in terms of the solvability of an integral equation. The aim of the next section will be to give a complete resolution to these questions. Thus, we are interested in solving for the first time the problem of characterizing the exponential trichotomy of skewproduct flows in terms of the solvability of an integral equation and also in establishing the connections between the qualitative theory of difference equations and the continuoustime behavior of dynamical systems, pointing out how the discretetime arguments provide interesting information in control problems related with the existence of the exponential trichotomy.
3. Main Results
Let be a real or a complex Banach space. In this section, we will present a complete study concerning the characterization of uniform exponential trichotomy using a special solvability of an associated integral equation. We introduce a new and natural admissibility concept and we show that the trichotomic behavior of skewproduct flows can be studied in the most general case, without any additional assumptions.
Notations
Let , which is a Banach space with respect to the norm . We consider the spaces , and let . Then and are closed linear subspaces of . Let be the space of all continuous functions with compact support and .
Let and let be the linear space of all Bochner measurable functions with the property that , which is a Banach space with respect to the norm
Let be a metric space and let be a skewproduct flow. For every we consider the integral equation
with and .
Definition 3.1.
The pair is said to beuniformly admissible for the skewproduct flow if there are and such that for every the following properties hold:

(i)
for every there are and such that the pairs and satisfy ( );

(ii)
if and are such that the pair satisfies ( ), then ;

(iii)
if is such that , for all and has the property that the pair satisfies ( ), then .
Remark 3.2.
In the above admissibility concept, the input space is a minimal one, because all the test functions belong to the space .
In what follows we will establish the connections between the admissibility and the existence of uniform exponential trichotomy.
The first main result of this paper is as follows.
Theorem 3.3.
Let be a skewproduct flow on . If the pair , is uniformly admissible for , then is uniformly exponentially trichotomic.
Proof.
We prove that the pair is uniformly admissible for .
Indeed, let and be given by Definition 3.1. We consider a continuous function with the support contained in and
Let be such that , for all . Since there is such that
Let .
Step 1.
Let . We consider the function
Then is continuous and
Since there is such that . Then, from (3.7) it follows that , so . According to our hypothesis it follows that there are and such that the pairs and satisfy ( ).
Then, for every we obtain that
Let
From (3.8) we have that
so the pair satisfies ( ). Moreover, since and , we deduce that . Since the pair satisfies ( ) we obtain that
Taking
we analogously obtain that and the pair satisfies ( ).
Step 2.
Let and let be such that the pair satisfies ( ). We consider the functions , given by
Since we have that . Observing that for every
we deduce that is continuous. Moreover, since and , from
we obtain that .
Let . We prove that
We set . If , then, taking into account the way how was defined, the relation (3.16) obviously holds. If then there is , such that . Then, we deduce that
If then and
If then
From relations (3.18) and (3.19) we have that
Then, from (3.17) and (3.20) it follows that
Since the pair satisfies ( ) we have that
From (3.21) and (3.22) we obtain that the relation (3.16) holds for all . Since was arbitrary we deduce that the pair satisfies ( ). Then according to our hypothesis we have that
In addition, we have that
and that
Taking , from relations (3.23)–(3.25) and (3.3) it follows that
Observing that , for all , from (3.26) and (3.4) we successively deduce that
Setting , from relation (3.27) we deduce that
Step 3.
Let be such that , for all and let be such that the pair satisfies ( ). We consider the functions , given by
Using analogous arguments with those used in the Step 2 we obtain that and the pair satisfies ( ). Moreover, using Lemma 2.7 we have that , for all . Then, according to our hypothesis we deduce that
Since , for all , using (3.30) and (3.5) we obtain that
Observing that
from (3.31) and (3.32) we deduce that
where .
Finally, from Steps 1–3 and relations (3.28) and (3.33) we deduce that the pair is uniformly admissible for the skewproduct flow . By applying Theorem 2.9 we conclude that is uniformly exponentially trichotomic.
The natural question arises whether the integral admissibility given by Definition 3.1 is also a necessary condition for the existence of the uniform exponential trichotomy. To answer this question, in what follows, our attention will focus on the converse implication of the result given by Theorem 3.3. Specifically, our study will motivate the admissibility concept introduced in this paper and will point out several qualitative aspects. First of all, we prove a technical result.
Proposition 3.4.
Let be a skewproduct flow which is uniformly exponentially trichotomic with respect to the families of projections . Let , let , and let be such that the pair satisfies ( ). We denote
and let denote the inverse of the operator , for all . The following assertions hold:

(i)
the functions and have the following representations
(3.35)(3.36) 
(ii)
for every there is which does not depend on or such that
(3.37)
Proof.
Since there is such that . Let be given by Definition 2.4 and let .

(i)
Since the pair satisfies ( ) we have that
(3.38)
Since from (3.38) we have that , for all . Let . Then we deduce that
For in (3.39) we obtain that , for all . This shows that relation (3.35) holds for all . For from (3.38) we have that
so relation (3.35) holds for every .
For every from (3.38) we have that . This implies that
so we deduce that
From relation (3.42) it follows that , for all . In particular, we have that relation (3.36) holds for . For from (3.38) we obtain that
which implies that
for all . Thus, we conclude that relation (3.36) holds for every .

(ii)
Let and let . Setting and using Hölder's inequality we deduce that
(3.45)
The second main result of the paper is as follows.
Theorem 3.5.
Let be a skewproduct flow. If is uniformly exponentially trichotomic, then the pair , is uniformly admissible for .
Proof.
Let be two constants and let , be the families of projections given by Definition 2.4. We set . For every and every we denote by the inverse of the operator .
For every and every we denote by
Then , for all and all and , for all .
Let and let be given by Proposition 3.4. We prove that all the properties from Definition 3.1 are fulfilled.
Let .
Step 1.
Let , . We consider the functions defined by
Since we have that and are correctly defined and continuous. Let be such that . Setting , and , we have that and , for .
We observe that , for all , which implies that
In addition, we have that , for all . This implies that
Since is continuous from relations (3.48) and (3.49) it follows that . Using similar arguments we deduce that . An easy computation shows that the pairs and satisfy ( ).
If , then we take .
Step 2.
Let and let be such that the pair satisfies ( ).
Suppose that . From Proposition 3.4 we have that
Let be such that . Since
for we deduce that . Since we obtain that
so . This implies that , for all . Moreover, using (3.51), for we have that
which implies that
Then, we obtain that
Setting from relations (3.50) and (3.55) it follows that
The case can be treated using similar arguments with those used above.
Step 3.
Let be such that , for all and let be such that the pair satisfies ( ).
Since and , for all , using Lemma 2.7 we deduce that , for all . Since the pair satisfies ( ) we obtain that
Let . Since , using relation (3.57) we have that
so , for all . This shows that, in this case, .
If is given by Step 2, we deduce that
and the proof is complete.
The central result of this paper is as follows.
Theorem 3.6.
A skewproduct flow is uniformly exponentially trichotomic if and only if the pair is uniformly admissible for .
Proof.
This follows from Theorems 3.3 and 3.5.
Remark 3.7.
The above result establishes for the first time in the literature a necessary and sufficient condition for the existence of the uniform exponential trichotomy of skewproduct flows, based on an inputoutput admissibility with respect to the associated integral equation. The chart described by our method allows a direct analysis of the asymptotic behavior of skewproduct flows, without assuming a priori the existence of a projection families, invariance properties or any reversibility properties. Moreover, the study is done in the most general case, without any additional assumptions concerning the flow or the cocycle.
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Acknowledgments
The first author is supported by CNCSISUEFISCDI, project PN IIIDEI code 1081/2008 no. 550/2009 and the second author is supported by CNCSISUEFISCDI, project PN IIIDEI code 1080/2008 no. 508/2009.
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Sasu, A.L., Sasu, B. Integral Equations and Exponential Trichotomy of SkewProduct Flows. Adv Differ Equ 2011, 918274 (2011). https://doi.org/10.1155/2011/918274
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DOI: https://doi.org/10.1155/2011/918274