# Periodic Systems Dependent on Parameters

- Min He
^{1}Email author

**2010**:796165

https://doi.org/10.1155/2010/796165

© Min He. 2010

**Received: **1 January 2010

**Accepted: **14 March 2010

**Published: **3 June 2010

## Abstract

This paper is concerned with a periodic system dependent on parameter. We study differentiability with respect to parameters of the periodic solution of the system. Applying a fixed point theorem and the results regarding parameters for -semigroups, we obtained some convenient conditions for determining differentiability with parameters of the periodic solution. The paper is concluded with an application of the obtained results to a periodic boundary value problem.

## 1. Introduction

This example motivates the discussion on the parameter properties of the general abstract periodic Cauchy Problem (1.3). Since the periodic system (1.3) depends on parameters, it is a natural need for investigating continuity and differentiability with respect to parameters of the solution of the system. Moreover, in applications, the differentiability with respect to parameter is often a typical and necessary condition for studying problems such as bifurcation and inverse problem [1]. It is worth mentioning that (1.1) indicates that the occurrence of parameters in the boundary conditions leads to the dependence of the domain of the operator on the parameters. We have developed some effective methods for dealing with this tricky phenomenon.

In our previous work [2], we have obtained results on continuity in parameters of (1.3). In this paper, we will discuss the differentiability with respect to parameters of solutions of (1.3).

It is clear that the differentiability with respect to parameter of semigroup will be the key for determining the differentiability with respect to parameter of the solution of (1.3). Some recent works [3, 4] have obtained fundamental results on the differentiability with respect to parameters of -semigroup. Applying these results together with some fixed point theorem, we are able to prove that (1.3) has a unique periodic solution, which is continuously (Frechét) differentiable with respect to parameter .

where for some , and is continuous in . After obtaining the differentiability results for (1.5), we, in Section 4, employ a fixed point theorem to attain the differentiability results of (1.3). Lastly, in Section 5, we will apply the obtained abstract results to the periodic boundary value problem (1.1) and use this example to illustrate the obtained results. One will see that the assumptions of the abstract theorems are just natural properties of (1.1).

## 2. Preliminaries

In this section, we state some existing theorems that will be used in later proofs. We start by giving the results on differentiability with respect to parameters.

Consider the abstract Cauchy problem (1.3), where is a closed linear operator on a Banach space and is a multiparameter ( is an open subset of a finite-dimensional normed linear space with norm ). Let be the -semigroup generated by the operator . For further information on -semigroup, see [5].

In [3], we obtained a general theorem on differentiability with respect to the parameter of -semigroup on the entire space . It is noticed that a major assumption of the theorem is that the resolvent is continuously (Frechét) differentiable with respect to . In a recent paper, Grimmer and He [4] have developed several ways to determine differentiability with respect to parameter of . Here, we include one of such theorems for reference.

Assumption Q.

Let be given. Then for each there exist bounded operators with bounded inverses and , such that .

Thus, having such a relationship for some implies a similar relationship at any other . Without loss of generality then, we may just consider the differentiating of the semigroup at .

Define , for , and assume that is invertible.

Theorem 2.1 (see [3]).

Assume Assumption Q and that

there are constants and such that

For each and are (Frechét) differentiable with respect to at .

Then for each , is (Frechét) differentiable with respect to at .

Theorem 2.2 (see [3]).

Assume the following

For each , there exists a constant such that

where is a smooth curve in running from to for some ,

Now we state a fixed point theorem from [6].

Definition 2.3 (see [6, page ]).

Theorem 2.4 (see [6, page ]).

If is a closed subset of a Banach space , is a subset of a Banach space , in is a uniform contraction on , and is continuous in for each fixed in , then the unique fixed point of in , is continuous in . Furthermore, if are the closures of open sets and has continuous first derivatives in , respectively, for , then has a continuous first derivative with respect to in .

Theorem 2.5 (see [7, page ]).

Theorem 2.6 (see [4]).

Let and be Banach spaces, and let Assume that

for each is continuously (Frechét) differentiable in . In particular, for , is the (Frechét) derivative of at , and is continuous in .

Lemma 2.7.

Proof.

The proof is standard and is omitted here.

## 3. Differentiability Results of (1.5)

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

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

then is continuously (Frechét) differentiable with respect to .

Proof.

has a unique solution, say , which is linear in .

is the unique solution of (3.2) for , which is continuous in .

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 .

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

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.

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.

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.

This completes the proof of Claim 2.

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 .

() 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 .

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 .)

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

there exists a constant such that

Then there exists a unique -periodic solution of (1.5), say , which is continuously (Frechét) differentiable with respect to for .

Proof.

where is a smooth curve in running from to for some ,

thus (3.9) is satisfied. Now the desired result follows from Theorem 3.4.

## 4. Differentiability Results of (1.3)

In this section, we discuss the general (1.3). Let .

Lemma 4.1.

Assume that Theorem 3.4( ) and ( ) are satisfied. Then is continuously (Frechét) differentiable with respect to for each .

Proof.

so is a uniform contraction. Also it is obvious that is continuous in by Theorem 3.4( ). Therefore from Theorem 2.4 it follows that there is a unique fixed point of , say .

is continuously (Frechét) differentiable w.r.t. .

on a Banach space , where for some and , and is continuous in .

Lemma 4.2.

Assume that Lemma 3.2( ) and Theorem 3.4( ) and (3.9) are satisfied and

is continuously (Frechét) differentiable with respect to .

which is continuously (Frechét) differentiable with respect to .

Proof.

which is continuously (Frechét) differentiable w.r.t. by Lemma 4.1.

Lemma 4.3.

Assume that Theorem 3.4( )-( ) and (3.9) and Lemma 4.2(K) are satisfied. In addition, assume that

Then the operator has a unique fixed point which is continuously (Frechét) differentiable with respect to .

Proof.

Therefore is a uniform contraction.

are continuous in . Therefore from Theorem 2.4 it follows that has a unique fixed point, say , which is continuously (Frechét) differentiable with respect to .

Now we present the main theorem for (1.3).

Theorem 4.4.

Assume that Theorem 3.4( )-( ) and (3.5), Lemmas 4.2(K), and 4.3( )–( ) are satisfied.

Then there exists a unique -periodic solution of (1.3), say , which is continuously (Frechét) differentiable with respect to for .

Proof.

This is an immediate result from Lemmas 4.2 and 4.3.

## 5. Application to a Periodic Boundary Value Problem

Consider the periodic boundary value problem (1.1) on the Banach space , where , for some and .

We now show that (5.4) satisfies all assumptions of Theorem 3.5.

Thus Assumptions Theorem 3.5( ) and ( ) are satisfied. Also, refer to [3, Section 4], we have shown that is continuously (Frechét) differentiable with respect to . So Assumption Theorem 3.5( ) is satisfied. Furthermore, from the expression of , it is obvious that is continuous in and is continuously (Frechét) differentiable with respect to , so Theorem 3.4( ) is satisfied. Now to apply Theorem 3.5, we only need to show that Theorem 3.4( ) is satisfied.

Thus Theorem 3.4( ) is satisfied. Now all the assumptions of Theorem 3.5 are satisfied, therefore (5.4) has a unique -periodic solution, say , which is continuously (Frechét) differentiable with respect to . Moreover, is the unique -periodic solution of (1.1) and it is continuously (Frechét) differentiable with respect to .

## Authors’ Affiliations

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