In this section, we study (1.5), which is a special case of (1.3). We will prove that the uniques periodic solution of (1.5) is continuously (Frechét) differentiable with respect to parameter
. We first state a theorem from [2]. This result shows that (1.5) has a unique periodic solution which is continuous in parameter
.
Theorem 3.1 (see [2]).
Assume that
is continuous in
for each
, and
for some
and all
, and
for some integer
with
and all
.
Then there exists a unique
-periodic solution of (1.5), say
, which is continuous in
for
.
Now we will discuss differentiability with respect to parameter
of the periodic solution of (1.5). The following lemma presents a general result on the differentiability with respect to parameter of the fixed point of a parameter dependent operator.
Lemma 3.2.
Let
for each
, and let
be the fixed point of
for each
, which is continuous in
. Also, let
. If
has the first partial derivatives
and
which satisfy
is continuous in
for each
and
for all
, and
is continuous in
,
then
is continuously (Frechét) differentiable with respect to
.
Proof.
We begin by noting that the equation
has a unique solution, say
, which is linear in
.
It follows from Lemma 3.2(
) and Theorem 2.4 that
is the unique solution of (3.2) for
, which is continuous in
.
From the uniqueness, one observes that
for all scalars
and
. That is,
is linear in
and may be written as
, where
is a bounded linear operator for each
.
Now we show that
is the derivative of
.
Let
. Since
by hypothesis, one sees that
Note that there is a function
continuous in
and approaching zero as
such that
Now from (3.5) and since
is a solution of (3.2), we have
Thus
Since
and
are continuous, and
is compact, the right-hand side of this expression is
as
. Also, there is a
such that
for
, so
is bounded. Thus
as
.
Remark 3.3.
This proof is based on that of Theorem
from [6, page
].
Now we prove the main theorem of the section.
Theorem 3.4.
Assume that
for all
.
is continuously (Frechét) differentiable with respect to
for each
. Moreover for any
there is some
such that
is continuously (Frechét) differentiable with respect to
.
Then there exists a unique
-periodic solution of (1.5), say
, which is continuously (Frechét) differentiable with respect to
for
.
Proof.
First note that from Theorem 3.1, we have that
is the unique
-periodic solution of (1.5).
Now we want to show that
is continuously (Frechét) differentiable with respect to
by applying Lemma 3.2. To this end, we need to prove the following two claims first.
Claim 1.
is continuously (Frechét) differentiable with respect to
for
. In particular, for any
,
In fact, for any
and
with
,
The first two terms on the right go to
as
by Theorem 3.4(
) and (
). The last term on the right goes to
because
is continuous at
and the set
is compact, so (3.11) holds.
Now for each fixed
and
, and any
, and
, from Theorem 3.4(
)- (
) and (3.11), it is clear that
is continuous at
. This completes the proof of Claim 1.
Based on Claim 1, we have the following claim.
Claim 2.
In fact, from Theorem 2.5 it suffices to show that
W.l.o.g. assume that
. Let
be any point in
. Since
is continuous on
, there exists
such that
Now from (3.9), we have
Thus by a theorem from [8, page
], we have
Furthermore,
Now the left-hand side of (3.17) is
and the right-hand side of (3.17) is
where 
That is,
This completes the proof of Claim 2.
Next, consider the operator
We will apply Lemma 3.2 and Claims 1-2 to show that the operator
has
as the fixed point and
is continuously (Frechét) differentiable with respect to
.
Note that the operator
has the following properties.
(i)
is defined on the Banach space
.
(ii)
is a uniform contraction on
.
In fact, for all
and
,
()
is continuous in
for each fixed
. (For the detailed proof, see Theorem
from [2].)
Applying Theorem 2.4, we have that
is the fixed point of
.
Furthermore, it is clear that the first derivative of
with respect to 
satisfies Lemma 3.2(
). It is also clear from Claim 2 that the first derivative of
with respect to 
is continuous in
by Theorem 3.4(
) and (
). (Note that using the same argument as in the proof of Claim 1, we can show that
is continuous at
and thereby
is continuous at
.)
Finally applying Lemma 3.2, we have that
is continuously (Frechét) differentiable with respect to
. Using a similar argument as that in Claim 1 and Claim 2 we can show that
and
are continuously (Frechét) differentiable with respect to
. Thus,
is continuously (Frechét) differentiable with respect to
for
.
Now we present a theorem with an assumption on the resolvent of the operator
instead of the
-semigroup
.
Theorem 3.5.
Assume Theorem 3.4(
) and (
) and that
for some
for all
,
there exists a constant
such that 
for each
and each
is continuously (Frechét) differentiable with respect to
on
. Moreover, for any
there exists
such that
implies
where
is measurable and for 
Then there exists a unique
-periodic solution of (1.5), say
, which is continuously (Frechét) differentiable with respect to
for
.
Proof.
First note that from Theorem 2.2 we have, for each 
where
is a smooth curve in
running from
to
for some
, 
Moreover, since
is continuous in
, it is clear from (3.28) that
is continuous in
, so Theorem 3.4(
) is satisfied. Also it is clear from (3.28) that
is continuous in
. By Theorem 2.6, we have
Now by the Principle of Uniform Boundedness, there is a
such that
thus (3.9) is satisfied. Now the desired result follows from Theorem 3.4.