1. Introduction
The almost periodic functions were introduced by Bohr in 1925 and described phenomena that are similar to the periodic oscillations which can be observed in many fields, such as celestial mechanics, nonlinear vibration, electromagnetic theory, plasma physics, and engineering. An important generalization of the almost periodicity is the concept of the almost automorphy introduced in the literature [1] [2] [3] [4] by Bochner. In [5] , the author presents the theory of almost automorphic functions and their applications to differential equations.
The study of differential equations with piecewise constant argument (EPCA) is an important subject because these equations have the structure of continuous dynamical systems in intervals of unit length. Therefore they combine the properties of both differential and difference equations. There have been many papers studying DEPCA, see for instance [6] - [11] and the references therein.
Some papers deal with the existence of asymptotically ω-periodic solutions (see for instance [12] ), S-asymptotically ω-periodic solutions of DEPCA (see [13] ). Other articles deal with the existence of almost automorphic solutions of EPCA (see [14] [15] ). In this paper, we study the existence of almost automorphic solutions of the following differential equation with the piecewise constant argument of generalized (DEPCAG) type (see [16] [17] [18] ):
(1)
where
is a step function,
is continuous in
and
is continuous. More precisely, there exists a strictly increasing sequence of real numbers
, such that
as
and on each interval
,
is constant:
In order to give sufficient conditions of existence and uniqueness of almost automorphic solutions of Equation (1), we introduce the concept of
-almost automorphic functions that generalizes the one of
-almost automorphic ( [19] ) ones, where
is a subset of
. In this paper, in order to study the almost automorphic solutions of (1), we will not consider almost automorphic sequence, but we will use the theory of fixed point.
The paper is organized as follows. In Section 2, we recall definitions and properties of almost automorphic functions and introduce the concept of
-almost automorphic functions. In Section 3, we also study the existence and uniqueness of almost automorphic solutions of Equation (1) considering the concept of
-almost automorphic functions and using the Banach fixed point Theorem.
2. Almost Automorphic Functions with Respect to a Set
Let
denote a subset of
. For every non zero real number r we consider the function
such that for every
:
(2)
In particular for all
we have:
Definition 2.1. A subset A of
is said to be r-stable if it is invariant under the homothety of ratio r and center 0.
We give an example of such a set
and an associated function
.
Example 2.1. Let
be a discrete subgroup of
, then
for some (non negative) real
, and
is obviously r-stable for all non zero integer r. Set
where
is the integer part function and c is a constant; then it is easily seen that (2) is satisfied.
Proposition 2.1. The function
satisfies the following properties:
1)
,
.
2)
,
:
Proof. Substituting
for t in (2), gives (1); and (2) is obtained by induction from
where
and noticing that
. 
In all the sequel
denotes a real or complex Banach space.
Definition 2.2. A function
is said to be
-continuous if it is continuous in
, which is referred as an
-continuous function.
The set of all
-continuous functions
will be denoted by
and the set of those that are bounded by
. Clearly
is a closed subspace of the Banach space
of bounded continuous functions and then it is also a Banach space.
Definition 2.3. A bounded
-continuous function
is said to be almost automorphic with respect to the set
if for every real sequence s' valued in
, there are a subsequence s and a function
such that for all
:
and
. (3)
Such a function f is called
-almost automorphic and if the above limits are uniform, it is called
-almost periodic.
The set of all
-almost automorphic (resp. almost periodic) functions will be denoted by
(resp.
). Clearly
is a subspace of the Banach space
; we have the following:
Theorem 2.2. The space
is a Banach space.
Proof. We have just to show that
is a closed set in the Banach space
. For this purpose we use a diagonal process. Let
denote a sequence in
which converges to a function f in
and let s' be any sequence of elements of
. It follows from Definition 2.3 that there exists a subsequence s1 of s' and a function g1 such that (3) holds when we replace s and g with s1 and g1 respectively. Then, by induction, we can build a sequence
extracted from
, where
is a subsequence of s', and a sequence of functions
such that:
and
.
Let
and take
. For
, we have:
Since
converges to f in
, there is
such that:
Therefore, if
,
and
, it follows that:
The condition
implies that
; then
is a subsequence of both
and
. Then,
and
converge to
and
respectively as
. Consequently there exists
depending on p and q such that:
, if
. Thus, given
and
, we have found
such that
if
. This means that
is a Cauchy sequence of real numbers. Thus
converges to a bounded measurable function g. On the other hand, if
and
, we can write:
.
For each
,
converges to 0 as
, it follows that the diagonal sequence
also converges to 0. Since the sequence
converges uniformly to f and
converges to
, it follows that
. It remains to show that
converges to
. It is sufficient to prove that the sequence
converges uniformly since we can deal as before where we proved that
. To do that, we keep the above notation with
and
. Then, from (3) we have:
Let
. Using the uniform convergence of the sequence
, we get
such that
for
and all
. From the definition of the limit, there is
depending on p and q such that:
It follows that:
(4)
Now replacing t by
in (4) yields
for
and all
. The uniform convergence of
is thus established. Then f belongs to
proving the theorem. 
Proposition 2.3. Let
be r-stable and
. If
(resp.
), then
(resp.
). If
(resp.
) and
, the same conclusion holds.
Proof. We keep the notation of Definition 2.3. For the given sequence s' we consider the sequence rs'. Then we get an associate subsequence rs together with a function g. It follows from (3) and the properties of the function
that
converges to
and
converges to
. These convergences are uniform if it is the case in (3). This proves the first part of the Proposition; the second part can be deduced straightforwardly. 
We associate to the subset
the following property:
(P1) There is a bounded set
in
such that all real t can be written as
where
and
.
Then we have the following:
Proposition 2.4. Let
satisfy (P1) and let f be an
-almost automorphic (resp.
-almost periodic) function. If f is uniformly continuous, then f is almost automorphic (resp. almost periodic).
Proof. As above we use the notation of Definition 2.3. Since
is a compact set we may assume that
for each
with
,
and
. Then we have:
The uniform continuity of f shows that the first term on the right side tends to zero. Since f is
-almost automorphic, it follows that the second term also converges to zero. On the other hand, f being uniformly continuous, the same holds for g. Then writing:
shows that
converges to
which proves that f is almost automorphic. The almost periodic case follows straightforwardly. 
Remark 2.1. We note that
satisfies the condition (P1): it suffices to take
, since for every real number x,
.
Definition 2.4. A continuous function
is said to be almost automorphic in
for each
, if for every sequence of real numbers
, there exists a subsequence
such that for each
and
,
and
.
Then we have the following result.
Theorem 2.5. ( [5] , Theorem 2.2.5) If f is almost automorphic in
for each
and if f is Lipschitzian in x uniformly in t, then g satisfies the same Lipschitz condition in x uniformly in t.
Using the above theorem we obtain:
Theorem 2.6. Let
be almost automorphic in
for each
. Assume that f satisfies a Lipschitz condition in x uniformly in
. Let also
be almost automorphic. Then the function
defined by
is
-almost automorphic.
Proof. Let
be a sequence of
. Using Proposition 2.3, we can extract a subsequence
such that:
1)
for each
and
,
2)
for each
and
,
3)
for each
,
4)
for each
,
5)
for each
,
6)
for each
.
Consider the function
defined by
. From the Lipchitz condition on f, there exists a constant
such that:
We deduce from (1) and (5) that
Similarly, we have:
Then we deduce from (2) and (6) that
Now, we show that the function
is bounded. Since f is almost automorphic in t, then
. Then we have
We deduce that for every

Remark 2.2. Let
satisfy the conditions of the previous theorem. We have that the function
defined by
is bounded.
3. A Differential Equation with a General Piecewise Constant Argument
We consider the differential Equation (1) where
is a step function,
is continuous in
and
is continuous. Thus, in the sequel
. Moreover, in addition to (P1), we consider the two following conditions:
(P2)
,
and
.
(P3)
is almost automorphic in
for each
and f satisfies a Lipschitz condition in x uniformly in
.
We give a consequence of (P1) that will be useful for the sequel.
Proposition 3.1. Assume that (P1) is satisfied, then there exists a bounded set
in
such that:
,
.
Proof. Assume that (P1) is satisfied. For each
there exists
, such that
. Hence, we have
and then:
Since
is a step function, it is bounded on each bounded subset of
. Therefore,
is a bounded set such that
for all
. The proposition is thus proved. 
Definition 3.1. A solution of (1) is a function
defined on
for which the following conditions hold:
1)
is continuous on
.
2) The derivative
exists at each point
, with possible exception at the points
, where one-sided derivatives exist.
3) The Equation (1) is satisfied on each interval
,
.
Theorem 3.2. Let f satisfy (P2) and (P3). Then the solution of (1) satisfies:
Proof. Considering the integral of
, on
, we obtain:

Lemma 3.3. Assume that (P2) and (P3) are satisfied and that
is an
-almost automorphic operator. Then
, maps
into itself.
Proof. Let
be a sequence of elements of
. We have from (P2) that
and
for
. Then, there exists a subsequence
of
such that:
1)
for each
,
2)
for each
,
3)
for each
,
4)
for each
,
5)
for each
,
6)
for each
,
7)
for each
,
8)
for each
.
We put
and
Then, we have
.
Using a change of variable and (P2), we find
, which can be written as
Now, using
, we can write
Hence, using the Lebesgue Dominated convergence theorem, we deduce that
Similarly, taking into account (P2), we get
Since
, it follows that
Hence, using the Lebesgue Dominated convergence theorem, we deduce that

We set
, where
is the bounded subset of
introduced in Proposition 3.1. Note that, if
, then
.
Theorem 3.4. Assume that (P1), (P2) and (P3) are satisfied and that
is constant on the interval
. If
then (1) has a unique
-almost automorphic solution which is also the unique almost automorphic solution of (1).
Proof. First Step
We define the nonlinear operator Γ by the expression
According to Theorem 2.6, the function
belongs to
. According to Lemma 3.3 the operator Γ maps
into itself. Since
for all
, we have:
This proves that Γ is a contraction. We conclude that Γ has a unique fixed point in
. We denote by z the unique
-almost automorphic solution of (1).
Second Step
We show that z is an almost automorphic solution of (1). Since z is
-almost automorphic, using Proposition 2.3, it suffices to prove that z is uniformly continuous. Consider the set
of possible points of discontinuity of
. We have
and then
for all
. If we set
it follows that
for all
. Therefore, since z is continuous and D is countable, the mean value Theorem (see [20] , Theorem 8.5.2) asserts that
, for all
with
. This means that z is lipschitzian and then uniformly continuous. Thus, z is an almost automorphic function.
The function z is necessarily the unique almost automorphic solution of (1). In fact, an almost automorphic function is also
-almost automorphic and (1) has a unique such solution. The theorem is thus proved. 
Corollary 3.5. Let
be a
-almost automorphic operator and assume that (P3) is satisfied. If
then the following equation:
has a unique
-almost automorphic solution which is also his unique almost automorphic solution.
Proof. We have
,
and
. 
Corollary 3.6. Suppose that
is a
-almost automorphic operator and that (P3) is satisfied. If
then the following equation:
has a unique
-almost automorphic solution which is also its unique almost automorphic solution.
Proof. We have that
. Then
is constant on each interval
where
. We observe also that
If
where
, then
,
and
. All real t can be written as
where
and
. 
Acknowledgements
The authors would like to thank the referee for his valuable remarks.