Periodic Systems Dependent on Parameters

One of the fundamental subjects in dynamic systems is the boundary value problem. When studying boundary value problems of differential and integrodifferential equations, we often encounter the problems involving parameters. Take, for example, a periodic boundary value problem

(1.1)

on the Banach space , where and are both -periodic and continuously differentiable. It appears that the boundary conditions contain four scalars and . Because these scalars may vary as the environment of the system changes, they are considered as parameters. Reforming (1.1) (for details, see Section 5), we have the periodic boundary value problem

(1.2)

where . Clearly is -periodic.

Furthermore, when (1.2) is written as a matrix equation (for details, see Section 5), its associated abstract Cauchy problem has the following form:

(1.3)

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).

According to the semigroup theory, when generates a -semigroup , the weak solution of (1.3) can be expressed in terms of the -semigroup :

(1.4)

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 .

We now give the outline of the approaches and contents of the paper. The general approach is that we first prove some theorems for the general periodic system (1.3). Then, by applying these results, we derive a theorem concerning (1.2) and thereby we obtain differentiability with respect to the parameter of the solution of (1.1). The paper begins with the preliminary section, which presents some differentiability results, a fixed point theorem, and related theorems. These results will be used in proving our theorems in later sections. In order to obtain results for (1.3), we, in Section 3, first study a special case of (1.3)

(1.5)

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).

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 .

Note that if , then

(2.1)

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

(2.2)

There is such that for all .

There is such that for all .

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 some for all .

For each , there exists a constant such that

(2.3)

for each and each is continuously (Frechét) differentiable with respect to on . Moreover, for any , there exists some ball centered at , say such that implies

(2.4)

where , is measurable and for

(2.5)

Then for each is continuously (Frechét) differentiable with respect to on for . In particular, for

(2.6)

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 ]).

Suppose that   is a subset of a Banach space is a subset of a Banach space , and is a family of operators taking . The operator is said to be a uniform contraction on if and there is a such that

(2.7)

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 ]).

Let be a continuous mapping of an open subset of into . is continuously (Frechét) differentiable in if and only if f is (Frechét) differentiable at each point with respect to the th ( ) variable, and the mapping (of into ) is continuous in . Then at each point of , the derivative of is given by

(2.8)

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 .

Then, for each , there is a constant such that

(2.9)

Lemma 2.7.

Let . If , then exists, and

(2.10)

Moreover, .

Proof.

The proof is standard and is omitted here.

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

(3.1)

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

(3.2)

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

It follows from Lemma 3.2( ) and Theorem 2.4 that

(3.3)

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

From the uniqueness, one observes that

(3.4)

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

(3.5)

Note that there is a function continuous in and approaching zero as such that

(3.6)

Now from (3.5) and since is a solution of (3.2), we have

(3.7)

Thus

(3.8)

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     

(3.9)

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

(3.10)

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 ,

(3.11)

In fact, for any and with ,

(3.12)

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

(3.13)

W.l.o.g. assume that . Let be any point in . Since is continuous on , there exists such that

(3.14)

Now from (3.9), we have

(3.15)

Thus by a theorem from [8, page ], we have

(3.16)

Furthermore,

(3.17)

Now the left-hand side of (3.17) is

(3.18)

and the right-hand side of (3.17) is

(3.19)

where

That is,

(3.20)

This completes the proof of Claim 2.

Next, consider the operator

(3.21)

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 ,

(3.22)

() 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

(3.23)

satisfies Lemma 3.2( ). It is also clear from Claim 2 that the first derivative of with respect to

(3.24)

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,

(3.25)

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

(3.26)

where is measurable and for

(3.27)

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

(3.28)

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

(3.29)

Now by the Principle of Uniform Boundedness, there is a such that

(3.30)

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

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.

First note that from Theorem 3.4( ), we see that exists by Lemma 2.7. Also,

(4.1)

Next consider the operator defined on :

(4.2)

Then we have

(4.3)

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 .

Furthermore, since

(4.4)

which clearly satisfy Lemma 3.2( ) and ( ), so by Lemma 3.2, we have that

(4.5)

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

Let .

Consider the equation

(4.6)

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 .

Then there exists a unique -periodic solution of (4.6), say , which is continuously (Frechét) differentiable with respect to for . Also

(4.7)

which is continuously (Frechét) differentiable with respect to .

Proof.

Let . Then . Also it is obvious that satisfies Theorem 3.4( ). Therefore by Theorem 3.4, there is a unique -solution of (4.6) which is continuously (Frechét) differentiable with respect to . In particular, is continuously (Frechét) differentiable with respect to . Moreover, using the same argument as that in the proof of Theorem 3.4 we see that

(4.8)

Thus

(4.9)

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

Define by

(4.10)

Lemma 4.3.

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

is continuous in for each , and

(4.11)

for some and all .

where is continuous in and .

is continuous in .

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

Proof.

It is clear that is a Banach space. Since , then, by the continuity of , there is such that implies

(4.12)

Now for ,

(4.13)

(4.14)

Hence,

(4.15)

Therefore is a uniform contraction.

Furthermore, is continuous in for fixed , and also

(4.16)

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.

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

Let

(5.1)

where

(5.2)

Then (1.1) becomes

(5.3)

where .

Assume and let and . Then the associated abstract Cauchy problem is

(5.4)

on , where

(5.5)

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

It is well known that the operator generates an analytic semigroup. The resolvent of satisfies, for all and ,

(5.6)

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.

In fact, for ,

(5.7)

where , and , and depend on . Moreover,

(5.8)

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 .