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 45-46]. 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 first-order, 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 finite-dimensional 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
.