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Periodic Systems Dependent on Parameters
Journal of Inequalities and Applications volume 2010, Article number: 796165 (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
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

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

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:

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
:

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)

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
.
Note that if , then

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

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

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

where , is measurable and for

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

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

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

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

Lemma 2.7.
Let . If
, then
exists, and

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

Next consider the operator defined on :

Then we have

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

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

is continuously (Frechét) differentiable w.r.t. .
Let .
Consider the equation

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

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

Thus

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

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

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

Now for ,


Hence,

Therefore is a uniform contraction.
Furthermore, is continuous in
for fixed
, and also

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
.
Let

where

Then (1.1) becomes

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

on , where

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
,

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 ,

where , and
, and
depend on
. Moreover,

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
.
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He, M. Periodic Systems Dependent on Parameters. J Inequal Appl 2010, 796165 (2010). https://doi.org/10.1155/2010/796165
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DOI: https://doi.org/10.1155/2010/796165
Keywords
- Banach Space
- Weak Solution
- Open Subset
- Periodic Boundary
- Fixed Point Theorem