Department of Mathematics, Faculty of Mathematics and Computer Science, West University of Timişoara, V. Pârvan Blvd. No. 4, 300223 Timişoara, Romania
We give new and very general characterizations for uniform exponential dichotomy of variational difference equations in terms of the admissibility of pairs of sequence spaces over with respect to an associated control system. We establish in the variational case the connections between the admissibility of certain pairs of sequence spaces over and the admissibility of the corresponding pairs of sequence spaces over . We apply our results to the study of the existence of exponential dichotomy of linear skew-product flows.
1. Introduction
In recent years, a number of papers have added important contributions to the existing literature on the relation between exponential dichotomy of systems and solvability properties of associated difference or differential equations, or the so-called admissibility properties with sequence or function spaces (see [1–10]). In the case of classical differential equations, the literature on this subject is rich and the main techniques are presented in the valuable contributions of Coffman and Schäffer [11], Coppel [12], Daleckii and Kreĭn [13], and Massera and Schäffer [14]. In the last few years, the methods have been improved and extended for general cases like those of evolution families or variational equations. The recent development in the theory of linear skew-product flows led to important generalizations of the classical results (see [4, 6, 7, 15–17]). A significant achievement was obtained in [15], where Chow and Leiva deduced the structure of the stable and unstable subspaces for an exponentially dichotomic linear skew-product flow. Various discrete-time characterizations for uniform exponential dichotomy of linear skew-product flows were obtained in [4, 6, 7]. In [4] Chow and Leiva introduced and characterized the concept of pointwise discrete dichotomy for a skew-product sequence over , with a Banach space and a compact Haussdorff space.
Let be a Banach space, let be a metric space, and let be a linear skew-product flow on , that is, , and , for all , and the mapping satisfies the following conditions: , (the identity operator on ), , for all and there are such that , for all . We associate with the variational discrete-time system
Using the consequences given by the pointwise behavior, Chow and Leiva established in [4] an important result concerning the exponential dichotomy of linear skew-product flows.
Theorem 1.1. Let be a linear skew-product flow on . Then is uniformly exponentially dichotomic if and only if the system () is uniformly exponentially dichotomic.A direct proof of the above theorem was presented in [6], without requiring continuity properties.
The impressive development of difference equations in the past few years (see, e.g., [1–4, 6–10, 15–25] and the references therein) led to important contributions at the study of the qualitative behavior of solutions of variational equations via discrete-time techniques. If , then one considers the linear system of variational difference equations
and associates to () the control system:
We denote by () the restriction of () to . One of the main results in [6] states that a linear skew-product flow is uniformly exponentially dichotomic if and only if the pair is admissible with respect to the system () and the space may be decomposed at every moment into a sum of stable fiber and unstable fiber, that is, , for all . Here, denotes the space of all sequences with and .
In [7], we introduced distinct concepts of admissibility considering as the input space -the space of all sequences of finite support and we gave a unified treatment considering both cases when the output space is or . The approach given in [7] relies on the unique solvability of the discrete-time system () between certain sequence spaces. The main results in [7] expressed the uniform exponential dichotomy of a discrete linear skew-product flow in terms of the uniform -admissibility () of one of the pairs and , with respect to the system ().
The natural question arises whether in the characterization of the exponential dichotomy the unique solvability of the associated discrete control system can be dropped. Another question is which are the connections between the solvability of the system () and the solvability of the system (). Using constructive techniques, we will provide complete answers for both questions and we will obtain new and optimal characterizations for uniform exponential dichotomy of variational difference equations. We denote by the space of sequences with finite support and . The admissibility concepts introduced in the present paper are new and simplify the solvability conditions in [6]. Specifically, the input space considered is the smallest possible input space, the solvability study is reduced to the behavior on the half-line and the boundedness condition is imposed only on the unstable fiber. Moreover we extend the applicability area to the general context of discrete variational systems.
Another purpose of this paper is to provide a complete study concerning the relation between the discrete admissibility on the whole line and the corresponding discrete admissibility on the half-line. Our main strategy will be to provide an almost exhaustive analysis, providing the connections between admissibility of concrete pairs of sequence spaces defined on and the uniform exponential dichotomy of a general system of discrete variational equations, emphasizing the appropriate techniques for each case considered therein. Finally, applying the main results, we will deduce new characterizations for uniform exponential dichotomy of linear skew-product flows, in terms of the uniform -admissibility of the pairs and .
2. Uniform Exponential Dichotomy of Variational Difference Equations
Let be a real or complex Banach space, let be a metric space and let . The norm on and on -the Banach algebra of all bounded linear operators on will be denoted by . Notations
Let denote the set of the integers, let be the set of the nonnegative integers () and let be the set of the nonpositive integers (). For every , let denote the characteristic function of . If let be the linear space of all sequences with the property that the set is finite and let .
If let denote the set of all sequences with the property that , which is a Banach space with respect to the norm
Let be the set of all bounded sequences , which is a Banach space with respect to the norm . If , then -the set of all sequences with , is a closed linear subspace of . For , we set . Similarly, .
Definition 2.1. A mapping is called discrete flow on if and , for all .Let . We consider the linear system of variational difference equations (). The discrete cocycle associated with the system () is
where is the identity operator on . It is easy to see that , for all .
Definition 2.2. The system () is said to be uniformly exponentially dichotomic if there exist a family of projections and two constants and such that the following properties hold:(i), for all ;(ii), for all and all ;(iii), for all and all ;(iv)the restriction is an isomorphism, for all .
For every we denote by the linear space of all sequences with the property , for all .
Let . For every we consider the stable space
and the unstable space
We note that and are linear subspaces, for every .Remark 2.3. If the system () is uniformly exponentially dichotomic and
then the family of projections given by Definition 2.2 is uniquely determined and
Moreover, for every , we have that and , for all (see [7, Proposition 2.1]).We associate to the system () the input-output control system ().
Definition 2.4. Let . The pair is said to be uniformly -admissible for the system () if the following assertions hold: (i)for every there is a unique solution of the system () corresponding to the input sequence ;(ii)there is such that , for all .
For the proof of the next result we refer to [7, Theorem 3.6].
Theorem 2.5. Let be such that . The following assertions hold: (i)if the pair is uniformly -admissible for the system (), then the system () is uniformly exponentially dichotomic;(ii) if and , then the system () is uniformly exponentially dichotomic if and only if the pair is uniformly -admissible for the system ().
We consider the input-output system
The main question arises whether in the characterization of the exponential dichotomy the unique solvability of the associated discrete equation can be dropped. Another question is which are the connections between the solvability of the system () and the solvability of the system (). In what follows we will give complete answers for both questions, our study focusing on these central purposes.
Definition 2.6. Let . The pair is said to be uniformly -admissible for the system () if there is such that the following assertions hold:(i) for every there is solution of the system () corresponding to ;(ii)if and is a solution of () corresponding to with the property that , for every , then
Lemma 2.7. Let . If the pair is uniformly -admissible for the system () then , for all
Proof. Let . We consider the sequence From hypothesis, there is solution of the system () corresponding to .
Let . From for all and since we deduce that . In addition, considering the sequence we have that and . This shows that . It follows that Since was arbitrary, the proof is complete.
The first main result of this section is as follows.
Theorem 2.8. Let . The following assertions are equivalent: (i)the pair is uniformly -admissible for the system ();(ii)the pair is uniformly -admissible for the system () and , for all
Proof. (i)(ii) Let be given by Definition 2.4. Let . Then the sequence belongs to . From the uniform -admissibility of the pair it follows that there is a unique solution of () corresponding to . Moreover
Taking
we have that is a solution of () corresponding to .
Let be a solution of () corresponding to with the property that , for all . Then for every , there is with and
Considering the sequence
we obtain that and
This implies that
is solution of the system () corresponding to . From the uniqueness, we deduce that . Then, using (2.8) we have that
Taking into account that does not depend on or we obtain that the pair is uniformly -admissible for the system ().
In addition, from Lemma 2.7 we have that , for all .
(ii)(i) Let be given by Definition 2.6.
Let . Then, there is such that , for all .
Consider the sequence Then . From the uniform -admissibility of the pair there is solution of the system () corresponding to .
Let . Since there is and such that . Let
Then and
From it follows that there is with and
Let
Then and from (2.16) and (2.17) we obtain that
We define
Then, using (2.19) we have that
which implies that
This shows that is a solution of the system () corresponding to .
Let be a solution of () corresponding to and let . Then
Let . Considering
we have that for all , so is a solution of the system () corresponding to . From (2.23) it follows that , for all . Then, from the uniform -admissibility of the pair we obtain that . In particular for all Since was arbitrary it follows that , for all , so was uniquely determined.
It remains to verify that condition (ii) in Definition 2.4 is fulfilled. Let and let be the solution of () corresponding to the input . Let be such that , for all . We consider the sequence and let
Then we have that is the solution of the system () corresponding to . From
we deduce that
Let . Denoting by we deduce that
Moreover, since , for all , it follows that .
This implies that , for all . From the uniform -admissibility of the pair we obtain that
Since , from (2.29) we deduce that
which is equivalent with
and, respectively, with
Since relation (2.32) holds for all with the property that , for all , for in (2.32) we obtain that , for all . Taking into account that does not depend on or , we conclude that the pair is uniformly -admissible for the system () and the proof is complete.
The second main result of this section is as follows.
Theorem 2.9. Let be such that . The following assertions hold: (i)if the pair is uniformly -admissible for the system () and for all then the system () is uniformly exponentially dichotomic;(ii)if and , then the system () is uniformly exponentially dichotomic if and only if the pair is uniformly -admissible for the system () and , for all .
Proof. This follows from Theorems 2.5 and 2.8.
Remark 2.10. Naturally, the question arises whether, generally, the uniform -admissibility of the pair for the system () and the property that , for all , implies the uniform exponential dichotomy of the system (). The answer is negative, as the following example shows.
Example 2.11. Let be a Banach space and let . On we consider the norm . Let
Let and let . For every we consider the operator
The discrete cocycle associated with the system
is given by
for all .
Step 1. We prove that , for all .
Let . Using (2.35) it is easy to see that . We prove that . Indeed, if , by defining , we observe that and . This shows that , so .
Conversely, let . Then there is with . If , in particular, we have that
Since we have that . Then, from (2.36) we obtain that
As in (2.37) it follows that . Then . This shows that . So, , for all .
Step 2. We prove that the pair is uniformly -admissible for the system (). Let and let . If , then we define
Since there is such that , for all . Then
From (2.39) it follows that , so, in particular, . We define and an easy computation shows that is a solution of the system () corresponding to .
Let be a solution of () corresponding to with the property that , for all .
Let . From and Step 1 it follows that . This implies that , so
Inductively, we obtain that
so
Since , for all , we have that
Taking into account that , using relation (2.43) we deduce that
For in (2.44) we obtain that . Then, from (2.43) it follows that , for all . Moreover, if , from
we deduce that . Inductively, it follows that
In particular, this implies that
From relations (2.42) and (2.47) we obtain that , for all . This shows that the pair is uniformly -admissible for the system ().
Step 3. We prove that the system () is not uniformly exponentially dichotomic.
Supposing that the system () is uniformly exponentially dichotomic, there exists a family of projections and two constants given by Definition 2.2. Then
According to Remark 2.3 and Step 1 we have that , for all . Then (2.48) yields , for all . In particular, for , from the above inequality we obtain that , for all , which is absurd. This shows that the system () is not uniformly exponentially dichotomic.
Definition 2.12. Let . The pair is said to be uniformly for the system () if the following assertions hold:(i)for every there is a unique solution of the system () corresponding to ;(ii)there is such that , for all .
For the proof of next theorem we refer to [7, Theorem 3.7].
Theorem 2.13. Let . The following assertions hold: (i)if the pair is uniformly -admissible for the system (), then the system () is uniformly exponentially dichotomic;(ii)if , then the system () is uniformly exponentially dichotomic if and only if the pair is uniformly -admissible for the system ().
For every we consider the subspaces
Definition 2.14. Let . The pair is said to be uniformly for the system () if there is such that the following assertions hold:(i)for every there is solution of the system () corresponding to ;(ii)if and is a solution of () corresponding to with the property that for every , then for all .
Working with the subspaces and using similar arguments with those in Lemma 2.7 and Theorem 2.8 we obtain the following.
Theorem 2.15. Let . The following assertions are equivalent: (i)the pair is uniformly -admissible for the system ();(ii) the pair is uniformly -admissible for the system () and for all
As a consequence of Theorems 2.13 and 2.15 we deduce the following result.
Theorem 2.16. Let . The following assertions hold: (i)if the pair is uniformly -admissible for the system () and for all , then the system () is uniformly exponentially dichotomic;(ii)if , then the system () is uniformly exponentially dichotomic if and only if the pair is uniformly -admissible for the system () and for all
Remark 2.17. It is easy to see that the system presented in Example 2.11 has the property that for all . Using similar arguments with those from Example 2.11, we deduce that the pair is uniformly -admissible for the system (). But, for all that, the system () is not uniformly exponentially dichotomic.
Remark 2.18. As an immediate consequence of Theorem 2.16 we obtain the main result of the paper [6]. The techniques involved in the proofs from [6] are different from those presented above.
3. Applications for Uniform Exponential Dichotomy of Linear Skew-Product Flows
In this section we apply the results obtained in the previous section in order to deduce characterizations for uniform exponential dichotomy of linear skew-product flows.
Let be a real or complex Banach space, let be a metric space and let . A mapping is called a flow on if and , for all .
Definition 3.1. A pair is called a linear skew-product flow if is a flow on and the mapping satisfies the following conditions:(i), (the identity operator), for all ;(ii), for all ;(iii)there are and such that , for all .
Definition 3.2. A linear skew-product flow is said to be uniformly exponentially dichotomic if there are a family of projections and two constants such that the following properties hold:(i), for all ;(ii), for all and ;(iii), for all and ;(iv)for every , the operator is invertible.
Remark 3.3. Let be a linear skew-product flow on . We associate with the variational discrete-time system (). The discrete cocycle associated with the system () is .If is a linear skew-product flow on , we consider the system
As consequences of the main results from the previous section we deduce the following.
Theorem 3.4. Let be a linear skew-product flow on and let be such that . The following assertions hold: (i) if the pair is uniformly -admissible for the system () and for all then is uniformly exponentially dichotomic;(ii)if , then is uniformly exponentially dichotomic if and only if the pair is uniformly -admissible for the system () and , for all .
Proof. This follows from Theorems 2.9 and 1.1.
Theorem 3.5. Let be a linear skew-product flow on and let . Then is uniformly exponentially dichotomic if and only if the pair is uniformly -admissible for the system () and for all
Proof. This follows from Theorems 2.16 and 1.1.
4. Conclusions
In order to study the existence of exponential dichotomy of a variational difference equation (), we associate to () the linear control system:
which is denoted by () if and by () if . The uniform exponential dichotomy of () may be expressed either in terms of the unique solvability of the system () or using the solvability of the system () in certain hypotheses, provided (in both cases) that the norm of the solution of the control system satisfies a boundedness condition with respect to the norm of the input sequence. In the first case the smaller input space is , while in the second case the smaller input space may be considered . When the associated control system is on the half-line, then the uniqueness of solution may be dropped. In this case, the space should be the sum between the stable and the unstable space at every point , but the boundedness condition may hold only for output sequences starting in the unstable space. Our study is explicitly done for the case when the output space is an -space as well as when this a -space, emphasizing the particular properties of each case. In base of Theorem 1.1 the main results are applicable not only to the general case of variational difference equations but also to the class of linear skew-product flows in infinite-dimensional spaces, without requiring measurability or continuity properties.
Acknowledgments
The author would like to thank the referee for thoroughly reading of the paper and for helpful suggestions and comments. The work is supported by the PN II Exploratory Research Grant ID 1081.