Slowly Oscillating Solutions of a Parabolic Inverse Problem: Boundary Value Problems
© F. Yang and C. Zhang. 2010
Received: 11 October 2010
Accepted: 20 December 2010
Published: 29 December 2010
The existence and uniqueness of a slowly oscillating solution to parabolic inverse problems for a type of boundary value problem are established. Stability of the solution is discussed.
It is well known that the space of almost periodic functions and some of its generalizations have many applications (e.g., [1–13] and references therein). However, little has been done for to inverse problems except for our work in [14–16]. Sarason in  studied the space of slowly oscillating functions. This is a -subalgebra of , the space of bounded, continuous, complex-valued functions on with the supremum norm . Compared with , is a quite large space (see [17–20]). What we are interested in is based on the belief that certainly has a variety of applications in many mathematical areas too. In , we studied slowly oscillating solutions of a parabolic inverse problem for Cauchy problems. In this paper, we devote such solutions for a type of boundary value problem.
Set . Let (resp., , where ) denote the -algebra of bounded continuous complex-valued functions on (resp., ) with the supremum norm. For (resp., ) and , the translate of by is the function (resp., , ).
A function is called slowly oscillating if for every , , the space of the functions vanishing at infinity. Denote by the set of all such functions.
A function is said to be slowly oscillating in and uniform on compact subsets of if for each and is uniformly continuous on for any compact subset . Denote by the set of all such functions. For convenience, such functions are also called uniformly slowly oscillating functions.
Let be a Banach space, and let be the space of bounded continuous functions from to . If we replace in (1) by , then we get the definition of .
As in , we always assume that is uniformly continuous.
The following two propositions come from [15, Section 1].
Let be such that is uniformly continuous on . Then .
The following proposition shows that the composite is also slowly oscillating.
Let . If and for all , then .
In the sequel, we will use the notations: , . means that is slowly oscillating in and uniformly for ; means that is slowly oscillating in and uniformly on .
be the fundamental solution of the heat equation .
2. A Type of Boundary Value Problem
In this section, we always assume the following: , , , , , , , , and , .
where ( ) are positive and increasing for and as .
To show the main results of this section, the following lemmas are needed. The first lemma is Lemma 3.1 on page 15 in .
Apply Lemma 2.1 to get the conclusion.
One sees that depends on only and is bounded near zero.
The existence and uniqueness of the solution comes from Theorem 5.3 on page 320 in .
By Lemma 2.1, one gets the desired inequality.
where is a constant. Since and are slowly oscillating, the right-hand sides of the inequality above approaches zero as . This means that . The proof is complete.
Consider the following problem.
Let , and let . We have the following two additional problems for and , respectively.
Problems 1, 2, and 3 are equivalent to each other.
the equivalence of Problems 2 and 3 can be proved similarly. The proof is complete.
where is determined by (2.37).
One can directly test that Problem 3 is equivalent to (2.37)-(2.38).
and therefore, .
Choose such that for , . Now, set . Then is a contraction from into itself, and therefore, has a unique fixed point. Thus, we have shown.
Let functions , , , and be as above. Then, for small , Problem 3 has a unique solution ( ) in with and .
Let be the solutions of Problem 3 in for the functions , , , and . Set , , , and . For the stability of the solution, we have the following.
where depends on , , , , , , , and .
The proof is complete.
Under the conditions in Theorem 2.6, the solution of Problem 3 is unique.
The research is supported by the NSF of China (no. 11071048).
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