Periodic Systems Dependent on Parameters
© Min He. 2010
Received: 1 January 2010
Accepted: 14 March 2010
Published: 3 June 2010
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.
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 . 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 , 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).
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 .
In , 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  have developed several ways to determine differentiability with respect to parameter of . Here, we include one of such theorems for reference.
Theorem 2.1 (see ).
Assume Assumption Q and that
Theorem 2.2 (see ).
Assume the following
Now we state a fixed point theorem from .
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 ).
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 . This result shows that (1.5) has a unique periodic solution which is continuous in parameter .
Theorem 3.1 (see ).
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.
This proof is based on that of Theorem from [6, page ].
Now we prove the main theorem of the section.
Based on Claim 1, we have the following claim.
This completes the proof of Claim 2.
() is continuous in for each fixed . (For the detailed proof, see Theorem from .)
thus (3.9) is satisfied. Now the desired result follows from Theorem 3.4.
4. Differentiability Results of (1.3)
Now we present the main theorem for (1.3).
This is an immediate result from Lemmas 4.2 and 4.3.
5. Application to a Periodic Boundary Value Problem
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 .
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