Roughness of (ℤ +, ℤ −)-Nonuniform Exponential Dichotomy for Difference Equations in Banach Spaces

In this paper we study the roughness of (ℤ +, ℤ −)-nonuniform exponential dichotomy for nonautonomous difference equations in the general context of infinite-dimensional spaces. An explicit form is given for each of the dichotomy constants of the perturbed equation in terms of the original ones. We emphasize that we do not assume any boundedness condition on the coefficients.


Introduction
The notion of uniform exponential dichotomy introduced by Perron in [1] for differential equations and by Li in [2] for difference equations plays an important role in the qualitative study of stable and unstable manifolds. In the nonautonomous setting, uniform exponential dichotomy is too restrictive and it is important to consider a more general behavior, for example, the nonuniform case, where a consistent contribution is due to Barreira and Valls [3,4]. Their study is motivated by ergodic theory and nonuniform hyperbolic theory. We also refer to [5,6].
One of the most important properties of exponential dichotomy (uniform or not) is its roughness (i.e., the manner in which exponential dichotomy varies under sufficiently small perturbations). The study of roughness of uniform exponential dichotomy has been intensively investigated (we mention in particular [7][8][9] and the references therein). For the roughness of nonuniform exponential dichotomy, we refer to [3,10] for continuous time and to [4] for discrete time.
The aim of this paper is to extend a discrete version of a roughness result from [11] to infinite-dimensional spaces. Another difference here is that we consider the general case of nonuniform exponential dichotomies and we do not assume any boundedness condition on the coefficients.

Preliminary Definitions
As usual, Z denotes the ring of integers. Also, we denote by Let be a real or complex Banach space and let B( ) be the Banach algebra of all bounded linear operators on . The norms on and on B( ) will be denoted by ‖ ⋅ ‖. For an arbitrary sequence ( ) ∈Z , in order to ease computation, we set by convention In this paper, we consider the following nonautonomous linear difference equation: where ( ) ∈Z ⊂ B( ) is a sequence of invertible operators on the Banach space . The corresponding (forward and backward) evolution operator of (3) is defined by The Scientific World Journal where is the identity operator on . It is easy to see that A ( , ) A ( , ) = A ( , ) , for , , ∈ Z. (5) Without the invertibility assumption on ∈ B( ), ∈ Z, neither the backward solutions to (3), nor the evolution operator A( , ) exists for < . Definition 1. We say that a sequence of projections ∈ B( ), ∈ Z (i.e., 2 = ), is an invariant projector for If = ( ) ∈Z is a sequence of projections on , then we denote by the complementary projection of , ∈ Z. We notice that if = ( ) ∈Z is an invariant projector for (3), then = ( ) ∈Z is also an invariant projector for (3). ( 2 ) ‖A( , ) ‖ ≤ | | − ( − ) , for ≤ in .

Remark 2. Relation (6) is equivalent to
The constant measures the nonuniformity of the dichotomy. In particular, when = 0, (3) is said to have a uniform exponential dichotomy. The projector = ( ) ∈Z will be called the dichotomy projector.
and there exist constants , > 0 and ≥ 0 such that the following estimates hold: If = 0, then we say that (3) has a (Z + , Z − )-uniform exponential dichotomy. This concept of exponential dichotomy was studied in [11] for differential equations in finite-dimensional spaces and in [12] for reversible evolution families in Banach spaces.
Note that the existence of (Z + , Z − )-nonuniform exponential dichotomy is equivalent to the existence of nonuniform exponential dichotomy on both Z + and Z − with dichotomy projectors + and − satisfying relation (9). In the particular case when + = − , for all ∈ Z, we obtain the concept of nonuniform exponential dichotomy on Z. This means that if (3) has a nonuniform exponential dichotomy on Z, then it also has a (Z + , Z − )-nonuniform exponential dichotomy.
Although relation (9) is replaced by the notion of (Z + , Z − )-nonuniform exponential dichotomy is different from the nonuniform version of exponential trichotomy defined in [13] for continuous time and in [14] for discrete time. This is due to the fact that no information is given regarding the boundedness of the projections + for ≤ 0 and − for ≥ 0 (for more details, we refer to [15]).

Roughness
In this section, we study the roughness of (Z + , Z − )-nonuniform exponential dichotomy for difference equations in Banach spaces. Thus, we investigate the existence of (Z + , Z − )nonuniform exponential dichotomy for the perturbed linear difference equation where each is a bounded linear operator on such that + is invertible, ∈ Z. The corresponding evolution operator of (12) is given bŷ Lemma 6. Let , , > 0. If then the constants are well defined and positive. Moreover,̃1 ≤̃2.
The Scientific World Journal 3 Proof. We first prove that Indeed, since we have that 2 < sinh , which is equivalent to Hence, cosh 2 − (1 + 2 sinh ) > 0. Now it is easy to see that and thus the constants̃1 and̃2 are well defined. To show that̃1 > 0, we need to prove that An easy computation shows that this is equivalent to ( − 1)(1 − −2 ) > 0, which obviously holds. On the other hand, since 1 − cosh < 0, we obtain that and thus̃2 > 0. Sincẽ we have that̃1 ≤̃2. This completes the proof of the lemma.
We set̃1 The roughness of nonuniform exponential dichotomy on both Z + and Z − was studied in [4]. The authors obtained the following results.
We point out that the assumption < in the theorems above is not a restrictive one since the notion of nonuniform exponential dichotomy is apparently motivated by ergodic theory which states that the nonuniform part in the dichotomies of the "most" equations is arbitrarily small (is as small as desired in comparison to the Lyapunov exponents).
In the theorem below, we establish the roughness of (Z + , Z − )-nonuniform exponential dichotomy. (3) has a (Z + , Z − )nonuniform exponential dichotomy with < and the sequence ( ) ∈Z satisfies

Theorem 9. If the difference equation
for some sufficiently small > 0, then the perturbed equation (12) also has a (Z + , Z − )-nonuniform exponential dichotomy.
Also, we obtain that for ≥ 0. Setting ( ) = + ( , 0) + 0 , for ≥ 0, we have that ∈ C 0 + and 0 + = . Therefore, must coincide with 0 + . This is equivalent to In particular, for = 0, we obtain the desired identity (37), which implies that̂+ The same arguments hold for = Z − . For a more thorough approach, we include the proofs. We set Consider the Banach space with the norm According to Lemma 10 in [4], the equation Furthermore, from the proof of Lemma 19 in [4], we have that by Lemma 11 in [4], we deduce that̂− is a projection and The Scientific World Journal 5 we obtain that Moreover, we have that Using the same arguments as in Lemma 10 in [4], we prove that̂− Indeed, if we consider the Banach space with the norm then the operator 0 − : has a unique fixed point 0 − ∈ C 0 − , given by On the other hand, we obtain that for ≤ 0. Setting ( ) = − ( , 0) − 0 , for ≤ 0, we have that ∈ C 0 − and 0 − = . Thus, In particular, for = 0, we obtain (56). This clearly leads tô We set In the following, we will prove that is an invertible operator. Using (32), we obtain Similarly, by (50), we havê Combining the inequalities above, we deduce that Hence, provided that > 0 is sufficiently small, the operator is invertible. Therefore, we can define the projections Using (9), (37), and (56), we obtain Similarly, it follows that − 0 =̂− 0 . Now we are ready to computẽ+ The Scientific World Journal and using similar arguments, we havê Repeating the arguments in the proofs of Theorems 7 and 8, the above equalities prove that the perturbed equation has a nonuniform exponential dichotomy on both Z + and Z − with projections̃+ =Â ( , 0)̃+ 0Â (0, ) , ∈ Z. Furthermore, sincẽ for every ∈ Z, we obtain the desired statement.
The previous result is a discrete version of Theorem 5 in [11] in the general context of nonuniform exponential dichotomies in infinite-dimensional spaces.