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 pseudoalmost 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 pseudoalmost 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.