Let be a real or complex Banach space. We recall that a function is said to be discrete almost periodic if for any positive there exists a positive integer such that any set consisting of consecutive integers contains at least one integer with the property that
In the above definition is called an almost period of or an translation number. We denote by the set of discrete almost periodic functions.
Bochner's criterion: is a discrete almost periodic function if and only if (N) for any integer sequence , there exists a subsequence such that converges uniformly on as . Furthermore, the limit sequence is also a discrete almost periodic function.
The proof can be found in [38, Theorem 1.26, pages 4546]. Observe that functions with the property (N) are also called normal in literature (cf. [7, page 72] or [38]).
The above characterization, as well as the definition of continuous almost automorphic functions (cf. [21]), motivates the following definition.
Definition 2.1.
Let be a (real or complex) Banach space. A function is said to be discrete almost automorphic if for every integer sequence , there exists a subsequence such that
is well defined for each and
for each
Remark 2.2.

(i)
If is a continuous almost automorphic function in then is discrete almost automorphic.

(ii)
If the convergence in Definition 2.1 is uniform on then we get discrete almost periodicity.
We denote by the set of discrete almost automorphic functions. Such as the continuous case we have that discrete almost automorphicity is a more general concept than discrete almost periodicity; that is,
Remark 2.3.
Examples of discrete almost automorphic functions which are not discrete almost periodic were first constructed by Veech [39]. In fact, note that the examples introduced in [39] are not on the additive group but on its discrete subgroup A concrete example, provided later in [25, Theorem 1] by Bochner, is
where is any nonrational real number.
Discrete almost automorphic functions have the following fundamental properties.
Theorem 2.4.
Let be discrete almost automorphic functions; then, the following assertions are valid:

(i)
is discrete almost automorphic;

(ii)
is discrete almost automorphic for every scalar ;

(iii)
for each fixed in the function defined by is discrete almost automorphic;

(iv)
the function defined by is discrete almost automorphic;

(v)
; that is, is a bounded function;

(vi)
, where
Proof.
The proof of all statements follows the same lines as in the continuous case (see [21, Theorem 2.1.3]), and therefore is omitted.
As a consequence of the above theorem, the space of discrete almost automorphic functions provided with the norm
becomes a Banach space. The proof is straightforward and therefore omitted.
Theorem 2.5.
Let be Banach spaces, and let a discrete almost automorphic function. If is a continuous function, then the composite function is discrete almost automorphic.
Proof.
Let be a sequence in , and since there exists a subsequence of such that is well defined for each and for each Since is continuous, we have In similar way, we have therefore is in
Corollary 2.6.
If is a bounded linear operator on and is a discrete almost automorphic function, then , is also discrete almost automorphic.
Theorem 2.7.
Let and be discrete almost automorphic. Then defined by , is also discrete almost automorphic.
Proof.
Let be a sequence in . There exists a subsequence of such that is well defined for each and for each Also we have that is well defined for each and for each . The proof now follows from Theorem 2.4, and the identities
which are valid for all .
For applications to nonlinear difference equations the following definition, of discrete almost automorphic function depending on one parameter, will be useful.
Definition 2.8.
A function is said to be discrete almost automorphic in for each if for every sequence of integers numbers there exists a subsequence such that
is well defined for each , , and
for each and .
The proof of the following result is omitted (see [21, Section 2.2]).
Theorem 2.9.
If are discrete almost automorphic functions in for each in then the followings are true.

(i)
is discrete almost automorphic in for each in

(ii)
is discrete almost automorphic in for each in where is an arbitrary scalar.

(iii)
for each in .

(iv)
for each in where is the function in Definition 2.8.
The following result will be used to study almost automorphy of solution of nonlinear difference equations.
Theorem 2.10.
Let be discrete almost automorphic in for each in X, and satisfy a Lipschitz condition in uniformly in ; that is,
Suppose is discrete almost automorphic, then the function defined by is discrete almost automorphic.
Proof.
Let be a sequence in . There exists a subsequence of such that for all , and for each , . Also we have is well defined for each and for each . Since the function is Lipschitz, using the identities
valid for all we get the desired proof.
We will denote the space of the discrete almost automorphics functions in for each in .
Let denote the forward difference operator of the firstorder, that is, for each and , .
Theorem 2.11.
Let be a discrete almost automorphic function, then is also discrete almost automorphic.
Proof.
Since then by (i) and (iii) in Theorem 2.4, we have that is discrete almost automorphic.
More important is the following converse result, due to Basit [40, Theorem 1] (see also [17, Lemma 2.8]). Recall that is defined as the space of all sequences converging to zero.
Theorem 2.12.
Let be a Banach space that does not contain any subspace isomorphic to Let and assume that
is discrete almost automorphic. Then is also discrete almost automorphic.
As is well known a uniformly convex Banach space does not contain any subspace isomorphic to . In particular, every finitedimensional space does not contain any subspace isomorphic to . The following result will be the key in the study of discrete almost automorphic solutions of linear and nonlinear difference equations.
Theorem 2.13.
Let be a summable function, that is,
Then for any discrete almost automorphic function the function defined by
is also discrete almost automorphic.
Proof.
Let be a arbitrary sequence of integers numbers. Since is discrete almost automorphic there exists a subsequence of such that
is well defined for each and
for each . Note that
then, by Lebesgue's dominated convergence theorem, we obtain
In similar way, we prove
and then is discrete almost automorphic.
Remark 2.14.

(i)
The same conclusions of the previous results holds in case of the finite convolution
and the convolution

(ii)
The results are true if we consider an operator valued function such that
A typical example is , where satisfies .