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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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Keywords
- Banach Space
- Weak Solution
- Open Subset
- Periodic Boundary
- Fixed Point Theorem
