Let
To present the main results of this paper, we need the following assumption.
(H)
is such that there exists
such that
, for all
.
Remark 2.1. (1)
is a translation invariant Banach space. For every
, one has
too. Set
, then
satisfies (H), and therefore there exist a great number of functions satisfying the assumption (H). (2) Reference [5] uses an assumption similar to (H) implicitly.
Let
. We have the following lemma.
Lemma 2.2.
Under the assumption (H), one has
.
Proof.
By (H), there exists
such that
,
. Let
and
, it is easy to verify that
,
,
,
,
,
,
, and
,
, for all
that is,
,
. Set
, similarly we can obtain
,
, and
, for all
that is,
.
Lemma 2.3.
Suppose that
is a Banach space,
denotes the set of bounded linear operators from
to
,
and
, then
is bounded invertible and
where
,
is an identical operator.
The proof of Lemma 2.3 can be found in any book of functional analysis. We remark that if
is a linear operator and its inverse exists, then
is also a linear operator.
To get w-solutions or solutions of (1.2), we start with its corresponding difference equations.
Suppose that
is an ow-solution of (1.2), then
satisfies the three conditions in Definition 1.3. By a process of integrating (1.2) two times in
as in [6–10], we can easily get
Similarly if
is an ew-solution of (1.2), by the process of integrating (1.2) two times in
, we get
These lead to the difference equations
From the analysis above, one sees that if
is an ow-solution (resp., ew-solution) of (1.2), then one gets (2.5) (resp., (2.6)); if
is a solution of (1.2), then one gets both (2.5) and (2.6). Conversely, we will show that the w-solutions or solutions of (1.2) are obtained via the solutions of (2.5) and (2.6). In order to get the solutions of (2.5) and (2.6), we will consider the following difference equations:
Notice that for any sequences
, and
, one has
. Especially,
In virtue of studying (2.7) and (2.8), we have the following theorem.
Theorem 2.4.
Under the assumption (H), (2.7) (resp., (2.8)) has a unique solution
(resp.,
).
Proof.
As the proof of [7, Theorem 9], define
by
, where
is the Banach space consisting of all bounded sequences
in
with
. Notice Lemmas 1.4 and 2.3, we know that (2.7) has a unique solution
. By the process of proving Lemma 2.2, we have
that is,
where
(this follows in the same way as [7, Theorem 9]). Therefore, (2.7) has a unique solution
.
Similarly, (2.8) has a unique solution
and
, that is,
where
. Therefore, (2.8) has a unique solution
. This completes the proof.
Remark 2.5. (i) In Theorem 2.4, since
and
, we can easily get
It must be stressed that (2.9) and (2.10) are important, since they can guarantee the continuity of the w-solutions or solutions of (1.2) constructed in Theorems 2.6, 2.7, and 2.8.
(ii) Let
with
satisfying (2.9)
, and
with
satisfying (2.10)
. Notice the fact that the solution of (2.7) (resp., (2.8)) must be a solution of (2.5) (resp., (2.6)), it is false conversely. So, suppose the assumption (H) holds, it follows from Theorem 2.4 that (2.5) (resp., (2.6)) has solution
(resp.,
). Moreover, such solutions may not be unique. See Example 3.1 at the end of this paper.
In the following, we focus on seeking the almost-periodic w-solutions or solutions of (1.2) via the almost-periodic sequence solutions of (2.5) and (2.6). As mentioned above, it is due to
that, to get almost-periodic w-solutions or solutions of (1.2), we have to use a way quite different from [6–10]. Our main idea is to construct solutions or w-solutions of (1.2) piecewise. It seems that this is a new technique.
Without loss of generality, suppose that
(resp.,
) is an arbitrary solution of (2.5) (resp., (2.6)). To prove the following theorems, we need to introduce some notations firstly:
where
and
. It can be easily verified that 
For the existence of the almost-periodic ow-solution of (1.2), we have the following.
Theorem 2.6.
Under the assumption (H), (1.2) has an ow-solution
such that
.
Proof.
Under the assumption (H), define
as
where
From (2.9), it follows that
is continuous on
and
,
. Moreover, for
,
, one has
; for
,
, one has
. By simple calculation, for
,
, we have
Note that
, this implies that the one-sided derivatives
exist at 
. Since
, the second-order derivatives
are continuous at
,
. Therefore,
is an ow-solution of (1.2) such that
,
. Furthermore, it is easy to check that
is almost periodic, we omit the details. The proof is complete.
For the existence of the almost-periodic ew-solution of (1.2), we have the following.
Theorem 2.7.
Under the assumption (H), (1.2) has an ew-solution
such that
.
Proof.
Under the assumption (H), define
as
where
From (2.10), it follows that
is continuous on
and
,
. The rest of the proof is similar to that of Theorem 2.6, we omit the details.
For the existence of almost-periodic solution of (1.2), we have the following.
Theorem 2.8.
Under the assumption (H), if
is the common solution of (2.5) and (2.6), then (1.2) has a solution
such that
,
. If
replaces
, the conclusion is still true.
Proof.
Since
and
are solutions of (2.5) and (2.9) respectively, and they are also solutions of (2.6) and (2.10), respectively, it follows from Theorems 2.6 and 2.7 that, by simple calculation, the almost-periodic ow-solution
constructed as the proof of Theorem 2.6 with
,
, is the same as the almost-periodic ew-solution
constructed as the proof of Theorem 2.7 with
,
. This implies
is an almost-periodic solution of (1.2) such that
,
. If
replaces
, the proof is similar, we omit the details.
Remark 2.9.
As mentioned above, an ow-solution of (1.2) is not equivalent to an ew-solution of (1.2), and a solution of (1.2) is an ow-solution of (1.2) as well as an ew-solution of (1.2). See the examples in Section 3.
The following theorem is usually used for judging whether or not a w-solution of (1.2) is a solution of (1.2).
Theorem 2.10.
Suppose that
is a solution of (1.2), then
Proof.
If
is a solution of (1.2), then
must be common solution of (2.5) and (2.6). Moreover, (2.6) is equivalent to
Substituting
into both the above equation and (2.5), then add the resulting equations to get the result.