DICHOTOMY AND ALMOST AUTOMORPHIC SOLUTION OF DIFFERENCE SYSTEM

We study almost automorphic solutions of recurrence relations with values in a Banach space V for quasilinear almost automorphic difference systems. Its linear part is a constant bounded linear operator Λ defined on V satisfying an exponential dichotomy. We study the existence of almost automorphic solutions of the non-homogeneous linear difference equation and to quasilinear difference equation. Assuming global Lipschitz type conditions, we obtain Massera type results for these abstract systems. The case where the eigenvalues λ verify |λ| = 1 is also treated. An application to differential equations with piecewise constant argument is given.

When system (1.2) has a summable dichotomy (see [35,36]) with Green function G, then: (1.5) is the unique bounded solution of (1.2).Thus (1.5) could be the unique almost automorphic solutions of (1.2).We would like to exploit this point.For f : Z → V almost automorphic sequence, perhaps the more simple equation (1.2), that is, with A = I identity: automorphic sequence, by the following result of Basit ([7, Theorem 1]) (see also [27,Lemma 2.8]).
Theorem 1. (Basit [7]) Let V be a Banach space that does not contain any subspace isomorphic to c 0 .If f : Z → V is an almost automorphic sequence, then every bounded solution y : Z → V of equation (1.6) is an almost automorphic sequence.
As it is well known a uniformly convex Banach space, every finite-dimensional normed space and a Hilbert space does not contain any subspace isomorphic to c 0 .
About introduction of theory of continuous almost authomorphic functions can be found in [8,11].Contributions on this theory can be found, for example in [6,20], [43]- [51], [19,42], [29,Chapter 4].Those contributions include topics like almost automorphic functions with values in Banach spaces, with values in fuzzynumber-type and on groups.Applications cover, studies in linear and nonlinear evolution equations, integro-differential, functional-differential equations and dynamical systems.
There are several types of differential equations, as those with impulsive effect, which connect sequences and functions, see Perestyuk-Samoilenko [32], Halanay-Wexler [22].An other important class is the differential equations with piecewise constant argument as: where [•] is the integer part function.For these equations it holds that y : R → V is almost automorphic if and only if the sequence y : Z → V is almost automorphic, see section 5 and Huang et al. [24].Recently this has been established for an abstract situation by Ming-Dat [28].
EJQTDE, 2013 No. 32, p. 2 In this paper, we first review some important properties of almost automorphic sequences, and then we study the existence of almost automorphic solutions of linear difference equations (1.2) and (1.3).In section 2, we expose some basic and related properties about the theory of almost automorphic functions.In section 3, we establish the existence of almost automorphic solutions of non-homogeneous linear difference equation.In section 4, we discuss the existence of almost automorphic solutions of nonlinear difference equations (1.3), where A is a bounded operator defined on a Banach space V .In Section 5, we show an application to where A and B are constant p × p complex matrices and h : R → V p is an almost automorphic function.

Preliminaries
Let V be a real or complex Banach space.We recall that function f : Z → V is said to be Bochner almost periodic sequence if and only if for any integer sequence (k ′ n ), there exists a subsequence (k n ) such that f (k + k n ) converges uniformly on Z as n → ∞.Furthermore, the limit sequence is also an almost periodic sequence.We denote by AP (Z, V ) the set of almost periodic sequences.See [4,15].
The pointwise convergence motivates the following definition.
Definition 1.Let V be a (real or complex) Banach space.A function f : Z → V is said to be almost automorphic sequence if for every integer sequence (k ′ n ), there exists a subsequence (k n ) such that are well defined for each k ∈ Z.
As in the continuous case we have that f ∈ AA (Z, V ) implies that f is a bounded function and sup k∈Z f (k) = sup k∈Z f (k) and for fixed l Examples of almost automorphic sequences which are not almost periodic sequences were firstly constructed by Veech [41], the examples are not on the additive group R but on its discrete subgroup Z.A concrete example of an almost automorphic function, provided later in [11, Theorem 1] by Bochner, is: We denote by AA (Z, V ) the vectorial space of almost automorphic sequence in V .Clearly AP (Z, V ) ⊂ AA (Z, V ) and the norm: The following is a fundamental Lemma Lemma 1.Let B(V ) the Banach space of linear bounded functions of V into V and v ∈ l 1 (Z, B (V )), i.e. an operator valued sequence v : For f ∈ AA (Z, V ) the convolution sequence defined by In particular; this is the case for In similar way, we prove lim and then φ ∈ AA (Z, V ).
Remark 1.For m, n ∈ Z fixed and f ∈ AA(Z, V ), the sequences For applications to nonlinear difference equations the following definition, of almost automorphic sequences depending on one parameter, will be useful.
are well defined for each k ∈ Z, x ∈ V .
We will denote AA (Z × V, V ) the vectorial space of the almost automorphic sequences in k ∈ Z for each x ∈ V .
Important composition results are Theorem 2. Let V , W be Banach spaces, and let g : ) such that the translation limits in (2.1) exists for both L and φ simultaneously.On the other hand g is continuous, we have lim n→∞ g (φ In similar way, we have lim n→∞ g φ (k n } such that the translations limits in (2.6) exists, for every x ∈ V , for the function g and also the translation limits in (2.1) exists for both L and φ simultaneously (see proof of Theorem 1).Then, applying (2.7) and those limits (2.6) for g ( The conclusion follows.

Almost Automorphic Solutions of Non-Homogueneous Difference Systems
Difference equations usually describe the evolution of certain phenomena over the course of the time.In this section we deal with those equations known as the first-order difference equations.These equations naturally apply to various fields, like biology (the study of competitive species in population dynamics), physics (the study of motion of interacting bodies), the study of control systems, neurology, and electricity: see [4,17], [21]- [25], [31]- [40].Consider the following system of first order linear difference equations where A is a complex matrix or, more generally, a bounded linear operator defined on a Banach space V and f ∈ AA (Z, V ).We wish to obtain several Massera types theorems under dichotomy conditions.Moreover, the case where the eigenvalues λ satisfying |λ| = 1 is also considered.
Let P be a projection matrix and define G the Green matrix associate to P by EJQTDE, 2013 No. 32, p. 6 Lemma 2. If the constant p × p-matrix A has a (µ 1 , µ 2 )-exponential dichotomy and f ∈ B (Z, V p ) then the linear non-homogeneous system (3.2) has the unique solution y ∈ B (Z, V p ) given by (3.3) Proof.The sequence y given by (3.3) is bounded satisfying (3.4) and (3.1).Indeed Theorem 4. If the constant p × p matrix A has a (µ 1 , µ 2 ) exponential dichotomy and f ∈ AA (Z, V p ), then the solution y in (3.3) is the unique AA (Z, V p ) of the linear non-homogeneous system (3.1).Moreover, We will prove that Γ 1 f and Γ 2 f belongs to AA (Z, V p ) .Let y = Γ 1 f and ( mn ) a sequence in Z. f ∈ AA (Z, V p ) implies that there exists a subsequence (m n ) ⊂ ( mn ) such that f (k) = lim n→∞ f (k + m n ) exists for k ∈ Z EJQTDE, 2013 No. 32, p. 7 and Since the , by using Lebesgue's domination theorem As a consequence, we have for the scalar abstract case: Theorem 5. Let V be a Banach space and f ∈ AA (Z, V ), then there exists a unique solution y ∈ AA (Z, V ) of (3.5) given by For |λ| = 1 we have: Theorem 6.Let V be a Banach space which does not contain any subspace isomorphic to c 0 .Let f ∈ AA (Z, V ) and |λ| = 1.Then a solution y of (3.5) is bounded if and only if ) there exists a unique bounded solution, namely that corresponding to (3.6 If A ∈ B (V ) is a general bounded operator, Lemma 1 implies: Theorem 7. Let V be a Banach space, and let A ∈ B (V ) such that A = 1 and f ∈ AA (Z, V ).Then there is a solution y ∈ AA (Z, V ) of (3.1) given by: For any constant matrix A, there exists a nonsingular matrix T such that T AT −1 = B is an upper triangular matrix.This procedure, called "Method of Reduction", was used in the discrete case earlier by Agarwal (cf.[4, Theorem 2.10.1]).In the continuous case, Corduneanu [15, Theorem 6.2.2] used it in the existence of AP (R, C p ) solutions and N'Guerekata [30, Remark 6.2.2] with AA (R, C p ) solutions.See also [26].Theorem 8. Suppose A is a constant p × p complex matrix with eigenvalues λ such as |λ| = 1.Then for any function f ∈ AA (Z, V p ) there is a unique solution y ∈ AA (Z, V p )of (3.1).
Theorem 5 implies that the pth component v p (n) of the solution v (n) satisfies an equation as (3.5) and hence any bounded solution v p ∈ AA (Z, C p ). Then substituting v p (n) in the (p − 1)th equation of (3.6) we obtain again an equation of the form (3.5) for v p−1 (n), and so on.The proof is completed.EJQTDE, 2013 No. 32, p. 9 Now, we study the case when all the eigenvalues {λ i } Assume that v satisfies the upper triangular system (3.6).So, by Theorem 7 the p-th coordinate v p ∈ AA (Z, V ) and it is given by for some η p ∈ V .Replacing this expression in the (p − 1)th equation in (3.6), we have v p−1 ∈ AA (Z, V ) and for some η p−1 ∈ V : Then F p−1 (v p ) ∈ B (Z, V ) if and only if η p = 0 and hence So, when the eigenvalues {λ i } p i=1 of a matrix A satisfy |λ i | = 1, 1 ≤ i ≤ p we have Theorem 9. Let V be a Banach space with does not contain any subspace isomorphic to c 0 .Let {λ i } p i=1 be the eigenvalues of A satisfying |λ i | = 1.Then every bounded solution of (3.4) y ∈ AA (Z, V ).When all these λ i are distinct, these solutions have the form: In the general case, a formula for the bounded solutions can be also obtained with an infinity of solutions, so much as V r , where r is the number of different eigenvalues λ i.
Proof.If {λ i } p i=1 are distinct, the transformed system (3.6) is now diagonal and by Theorem 6 and (3.7) we obtain (3.9) .In the general case, we use the previous analysis and the solutions of the form (3.8).Theorem 10.Let V be a Banach space with does not contain any subspace isomorphic to c 0 and let {λ i } p i=1 the eigenvalues of the p × p constant matrix A. Then every bounded solution y of (3.4) satisfies y ∈ AA (Z, V p ).Moreover, a formula for the almost automorphic solutions can be explicated with an infinity of solutions so much as V r , where r is the number of different eigenvalues λ i with λ i = 1.
Finally, we can also prove the following result.Theorem 12. Let V be a Banach space that does not contain any subspace isomorphic to c 0 .Assume that the set formed by λ in the spectrum of A with |λ| = 1 is countable.If f ∈ AA (Z, V ) , then each bounded solution of (3.5) y ∈ AA (Z, V ).

Almost Automorphic Solutions of Nonlinear Difference Systems
Now we study the existence of almost automorphic solutions to the equation (4.1) where A is a bounded linear operator defined on a Banach space V and g ∈ AA (Z × V, V ).One of the main results in this section is the following theorem for the quasilinear case: Theorem 13.Assume that the constant p × p matrix A has a (µ 1 , µ 2 )-exponential dichotomy and g = g (k, y) ∈ AA (Z × V p , V p ) satisfies the Lipschitz condition For φ ∈ AA (Z, V p ) since g (k, x) satisfies (4.2), we obtain by Theorem 3 that g (•, φ (•)) ∈ AA (Z, V p ) .Define the operator Γ : AA (Z, V p ) → AA (Z, V p ) by So Γ is well defined thanks to Theorem 4. Now given φ 1 , φ 2 ∈ AA (Z, V p ), we have then by (4.4) the function Γ is a contraction.Then there exist a unique y ∈ AA (Z, V p ) such that Γy = y.That is, y satisfies (4.3) and hence y is solution of (4.1).
Then in the scalar abstract case: In the particular case g (k, x) = L (k) g 1 (x) we obtain the following Corollary.

Theorem 11 .
Let V be a Banach space.Suppose f ∈ AA (Z, V ) and A = N k=1 λ k P k where the complex numbers λ k are mutually distinct with |λ k | = 1, and (P k ) 1≤k≤N forms a complex system N k=1 P k = I of mutually disjoint projections on V. Then the unique bounded solution y of (3.1) is in AA (Z, V ) .Proof.Let k ∈ {1, . . ., N } be fixed.By Corollary 1 we have P k f ∈ AA (Z, V ), since P k is bounded.Applying the projection P k to (3.1) we obtain (3.10)P k y (n + 1) = P k Ay (n) + P k f (n) .Therefore, by Theorem 8, we getP k y ∈ AA (Z, V ) we conclude that y (n) = N k=1 P k y (n) ∈ AA (Z, V )as a finite sum of almost automorphic sequences.This is an explicit result of the general theorem obtained by Minh et al. [27, Theorem 2.4] for every Banach space.
and by Basit's Theorem A, F ∈ AA (Z, V ) if and only if it is bounded.So, in this case every solution y of (3.5) is in AA (Z, V ).EJQTDE, 2013 No. 32, p. 8