Abstract

We give very general characterizations for uniform exponential dichotomy of variational difference equations. We propose a new method in the study of exponential dichotomy based on the convergence of some associated series of nonlinear trajectories. The obtained results are applied to difference equations and also to linear skew-product flows.

1. Introduction

The stability theory of evolution equations received a remarkable contribution when Zabczyk proved the following result (see [1, Theorem ]).

Theorem 1.1. Let be a Banach space and let . If is a continuous strictly increasing convex function with such that for every there is such that then the spectral radius is less than .

The original proof was based on the Banach-Steinhaus Theorem and on the construction of an auxiliary sequence space associated with the function . As a consequence of this result it follows that a semigroup of linear operators on a Banach space is exponentially stable (i.e., there are such that , for all ) if and only if there is a continuous strictly increasing convex function with such that for every there is such that

Thus it was pointed out, for the first time in literature, that an asymptotic property, like stability, can be deduced from the convergence of a series of nonlinear trajectories.

A notable characterization for the stability of difference equations was obtained by Przyłuski and Rolewicz (see [2, Theorems and ]), where it is proved that a system of difference equations

on a Banach space is uniformly exponentially stable if and only if there is such that for every

This result can be considered the discrete-time counterpart of the stability theorem due to Datko (see [3, Theorem and Remark ]).

In recent years, a significant progress was made in the study of the qualitative properties of difference equations (see [4–26]). Stability of difference equations was studied from various perspectives, among which we mention the input-output techniques, the property that the trajectories lie in a certain space, the freezing method, the control type approaches (see [14–19]). The discrete input-output methods were extended to the case of exponential expansiveness (see [23]) as well as for uniform and exponential dichotomy (see [4–7, 9, 11, 12, 20–22, 25, 27]) and exponential trichotomy (see [8, 9, 24]).

Exponential dichotomy plays a very important role in the study of the asymptotic behavior of time-varying equations. Starting with classical works in this field (see [28–30]) many research studies have been done to define, characterize and extend diverse concepts of exponential dichotomy for various evolution equations (see [4–12, 20–22, 25, 27, 28, 31, 32]). In the last decades important results on exponential dichotomy in finite dimensional spaces were extended for the infinite dimensional case and valuable applications were provided (see [4–12, 27, 28, 31, 32]). The input-output methods or the so-called Perron type techniques were intensively used in the study of exponential dichotomy, the main idea being to associate with an evolution equation a linear control system and to characterize the existence of exponential dichotomy of the initial equation in terms of the solvability of the associated control system between various Banach sequence or function spaces (see [4–7, 9, 11, 12, 20–22, 25, 27, 28]). When one studies an asymptotic property via input-output methods, the solvability condition relative to the associated linear control system is expressed in terms of the upper boundedness of an associated linear operator or of a family of linear operators, and therefore the arguments involved are strongly connected with the qualitative theory of linear operators. In contrast with the theorems of Perron type, the Zabczyk and Przyluski-Rolewicz type conditions are more direct and the methods are not necessarily reduced only to the behavior of some associated linear operators. Recently, these techniques were applied to the study of exponential stability in [33, 34] and respectively, for exponential instability in [35].

Up to now, there is no Zabczyk-type result in literature concerning the general asymptotic property described by exponential dichotomy. This is motivated by the fact that due to the splitting on each fiber required by any dichotomy property (see e.g., [22, 27, 31]) the results concerning exponential stability and instability (see [33–35]) cannot be applied in order to deduce conditions for the existence of exponential dichotomy, because, generally, the dichotomy projections are not constant. In this framework, the natural question arises whether one can provide a characterization of Zabczyk type such that this is equivalent with the decomposition of the main space into a direct sum of two closed subspaces such that the behavior on these subspaces is modelled by exponential decay backward and forward in time.

The aim of this paper is to provide a complete resolution to this problem. We propose a new method in the study of exponential dichotomy of evolution equations and treat directly the case of variational difference equations. We do not only answer the above questions but also propose an approach at a greater level of generality than ever before. First, using constructive methods, we deduce necessary and sufficient conditions of nonlinear type for the existence of exponential dichotomy of variational difference equations, based on convergence conditions of some associated series. After that, the main results are applied to the study of the existence of exponential dichotomy for two main classes of systems: the difference equations and the linear skew-product flows. We show that the stability theorem of Zabczyk and also the Przyluski-Rolewicz theorem can be extended to the general case of exponential dichotomy and that the convergence of certain series of nonlinear trajectories may provide interesting information concerning the asymptotic behavior of the initial system.

2. Preliminary Results

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 . Denote by the identity operator on and by the set of all mappings with .

Throughout the paper denotes the set of real integers, is the set of all , and is the set of all , .

Definition 2.1. A mapping is called a discrete flow on if , for all and , for all .

Let . We consider the linear system of variational difference equations:

The discrete cocycle associated with the system () is

Remark 2.2. One has , for all .

Definition 2.3. The system () is said to be uniformly exponentially dichotomic if there are a family of projections and two constants and such that(i), for all ;(ii), for all , all and all ;(iii) for all , all and all ;(iv)for every , the restriction is an isomorphism.

Remark 2.4. (i) If for every ,, then from Definition 2.3 we obtain the concept of uniform exponential stability.
(ii) If, in Definition 2.3, for every ,, then we obtain the concept of uniform exponential expansiveness.

Remark 2.5. A remarkable progress in the study of the asymptotic behavior of variational equations modelled by cocycles over flows was done due to the work of Pliss and Sell (see [32]). The authors presented a complete study concerning significant robustness properties of linear skew-product semiflows on , where is a Banach space and is a metric space. For important applications we also refer to the book of Sell and You (see [31]). In [32, Lemma ] (see item (3)), the authors obtained the structure of the dichotomy projections of a linear skew-product semiflow on , where is an invariant set in and also proved the invariance properties of the stable set and unstable set with respect to the linear skew-product flow (see item (4)). The structure of the dichotomy projections represents an important step in the study of an asymptotic property like dichotomy or trichotomy, because this anticipates the expectations concerning the decomposition of the central space at every point.

In what follows, as consequences of Definition 2.3, we will deduce the structure of the dichotomy projections of a system of variational difference equations, using certain invariant subspaces of the space . Compared with [32] where is deduced a global decomposition of the central space , we will consider a decomposition of the space via each fiber determined by the points of the space . For the sake of clarity we will present short proofs of these auxiliary results.

For every , we denote by the linear space of all sequences with the property that

For every we define the stable subspace

and the unstable subspace

Remark 2.6. For every , the mapping is called the trajectory determined by and the mapping with is called a negative continuation started at .

Lemma 2.7. The following assertions hold:(i), for all ;(ii), for all .

Proof. The assertion (i) is immediate. To prove (ii), let and let . Then, there is with and . Let Then , for all , . Taking , it follows that and . This shows that .
Conversely, let . Then, there is with and . Let and let , . An easy computation shows that and . This implies that , so .

Lemma 2.8. If the system () is uniformly exponentially dichotomic with respect to the family of projections , then .

Proof. Let be given by Definition 2.3 and let . For every and every we have that which implies that Let be such that . Setting and , from relation (2.7) it follows that , for all . This implies that , for all , so , for all and the proof is complete.

Lemma 2.9. If the system () is uniformly exponentially dichotomic with respect to the families of projections , then

Proof. Let be given by Definition 2.3. From Lemma 2.8 we have that . Let . Obviously, . Conversely, if , then and we successively deduce that For in (2.9) it follows that , so .
Let . Since the system () is uniformly exponentially dichotomic, for every , the restriction is an isomorphism. We define , and using the inequality (iii) from Definition 2.3 it follows that , for all . An easy computation shows that . Then, using the above estimation we obtain that . Conversely, let . Then, there is with and . From
we deduce that , so .

Remark 2.10. According to Lemma 2.9 we conclude that for variational difference equations, the family of dichotomy projections is uniquely determined.

3. Exponential Dichotomy of Variational Difference Equations

Let be a real or complex Banach space, let be a metric space and let be a discrete flow on .

Let . We consider the linear system of variational difference equations:

In all what follows, we denote by the set of all continuous nondecreasing functions with and , for all ,

Theorem 3.1. If there are and such that then there are such that

Proof. Let and let .
Let and let with . For every we have that
Since is nondecreasing, from relation (3.3) we deduce that Since from relation (3.4), it follows that . In addition, we observe that , for all . Then, setting and taking into account that does not depend on or we obtain that Let be such that
Let and let with . Using relation (3.5) and Lemma 2.7(i) we deduce that
which implies that From relations (3.6) and (3.8) it necessarily follows that Taking into account that does not depend on or it follows that Let and let . Let and let . Then, there are and such that . Using relations (3.5) and (3.10) and Lemma 2.7(i) we successively deduce that and the proof is complete.

Theorem 3.2. If there are and such that(i), for all and all ;(ii), for all and all , then the following assertions hold:(i)for every , the restriction is an isomorphism;(ii)there are such that

Proof. (i) Let . From Lemma 2.7(ii) we have that the restriction is correctly defined and it is surjective. To prove the injectivity, let . Then, from our first hypothesis, we deduce that . Since , this implies that , and so is also injective.
(ii) Let and let .
Let and let with . From (i) we have that , for all .
Let . Then
We set and since is nondecreasing, from relation (3.13) we obtain that This inequality implies that . Since does not depend on or we deduce that Let be such that Let and let with . Using Lemma 2.7 and relation (3.15) we have that Then, we obtain that From relations (3.16) and (3.18) it follows that . Since does not depend on or we deduce that Let and let . Let and let . Then, there are and such that . From relations (3.15) and (3.19) and using Lemma 2.7 we have that

The main result of this section is the following theorem.

Theorem 3.3. The system () is uniformly exponentially dichotomic if and only if , for all and there are and such that the following assertions hold:(i), for all and all ;(ii), for all and all ;(iii), for all and all .

Proof. Necessity
Let be the family of projections and let be two constants given by Definition 2.3. From Lemma 2.9 we have that and , so , for all .Taking , it is easy to observe that the inequalities (i)–(iii) hold for .Sufficiency
From Theorems 3.1 and 3.2 we have that there are such that From relations (3.21) and (3.22) it follows that Step 1. We prove that is closed, for all .Let and let with as . For every we have that For in (3.24), since is continuous, we deduce that which implies that From (3.26) we obtain that , as . In particular, there is such that , for all . Since is nondecreasing it follows that , for all . This implies that , so .Step 2. We show that is closed, for all .Let and let with as . Then, for every there is with and .
It is easy to see that , for every and every . Then, from relation (3.22) we have that
From relation (3.27) we deduce that for every , the sequence is convergent. Taking from , for all , it follows that . Moreover Let . For in (3.27) we have that which implies that Since , from relation (3.31), we obtain that , and so is closed.
According to our hypothesis, relation (3.23), and Steps 1 and 2 it follows that
Then, for every there is a projection with and . From Lemma 2.7 we have that , for all . Finally, from relations (3.21) and (3.22) and Theorem 3.2(i) we conclude that the system () is uniformly exponentially dichotomic.