Throughout the rest of this paper, for
, we denote by
the set of all the functions
satisfying that there exists a function
such that
and, for any compact set
, we denote by
the set of all the functions
such that (1.7) is replaced by
In addition, we denote by
the norm of
and
.
Lemma 2.1.
Let
,
be compact, and
. Then
.
Proof.
For all
, there exist
such that
Since
, for the above
, there exists a relatively dense set
such that
for all
,
, and
. On the other hand, since
, there exists a function
such that (2.1) holds.
Fix
,
. For each
, there exists
such that
. Thus, we have
for each
and
, which gives that
Now, by Minkowski's inequality and (2.4), we get
which means that
.
Theorem 2.2.
Assume that the following conditions hold:
-
(a)
with
, and
with
.
-
(b)
, and there exists a set
with
such that
is compact in
.
Then there exists
such that
.
Proof.
Since
, there exists
such that
. Let
Then
and
. On the other hand, since
, there is a function
such that (2.1) holds.
It is easy to see that
is measurable. By using (2.1), for each
, we have
Thus,
.
Next, let us show that
. By Lemma 2.1,
. In addition, we have
. Thus, for all
, there exists a relatively dense set
such that
for all
and
. By using (2.11), we deduce that
for all
and
. Thus,
.
Lemma 2.3.
Let
be compact,
, and
. Then
, where
Proof.
Noticing that
is a compact set, for all
, there exist
such that
Combining this with
, for all
, there exists
such that
for all
and
. Thus, we get
which yields that
On the other hand, since
, for the above
, there exists
such that, for all
,
This together with (2.17) implies that
Hence,
.
Theorem 2.4.
Assume that
and the following conditions hold:
-
(a)
with
and
. Moreover,
with
;
-
(b)
with
and
, and there exists a set
with
such that
is compact in
.
Then there exists
such that
.
Proof.
Let
be as in the proof of Theorem 2.2. In addition, let
, where
It follows from Theorem 2.2 that
, that is,
.
Next, let us show that
. For
, we have
where
was used. For
, since
, by Lemma 2.3, we know that
which yields
that is,
. Now, we get
.
Next, let us discuss the existence and uniqueness of pseudo-almost periodic solutions for the following abstract semilinear evolution equation in
:
Theorem 2.5.
Assume that
and the following conditions hold:
-
(a)
with
and
. Moreover,
with
-
(b)
the evolution family
generated by
has an exponential dichotomy with constants
, dichotomy projections
,
, and Green's function
;
-
(c)
for all
, for all
, and for all
there exists a relatively dense set
such that
and
for all
and
with
.
Then (2.25) has a unique pseudo-almost periodic mild solution provided that
Proof.
Let
, where
and
. Then
and
is compact in
. By the proof of Theorem 2.4, there exists
such that
.
Let
where
and
. Denote
where
By [13, Theorem 2.3] we have
. In addition, by a similar proof to that of [2, Theorem 3.2], one can obtain that
. So
maps
into
. For
, by using the Hölder's inequality, we obtain
for all
, which yields that
has a unique fixed point
and
This completes the proof.
Remark 2.6.
For some general conditions which can ensure that the assumption (c) in Theorem 2.5 holds, we refer the reader to [17, Theorem 4.5]. In addition, in the case of
and
generating an exponential stable semigroup
, the assumption (c) obviously holds.