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.