# Slowly Oscillating Solutions of a Parabolic Inverse Problem: Boundary Value Problems

- Fenglin Yang
^{1}Email author and - Chuanyi Zhang
^{1}

**2010**:471491

**DOI: **10.1155/2010/471491

© F. Yang and C. Zhang. 2010

**Received: **11 October 2010

**Accepted: **20 December 2010

**Published: **29 December 2010

## Abstract

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.

## 1. Introduction

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 [17] 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 [15], 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., , ).

- (1)
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.

- (2)
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.

- (3)
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 [17], we always assume that is uniformly continuous.

The following two propositions come from [15, Section 1].

Proposition 1.2.

Let be such that is uniformly continuous on . Then .

The following proposition shows that the composite is also slowly oscillating.

Proposition 1.3.

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 [21].

## 2. A Type of Boundary Value Problem

In this section, we always assume the following: , , , , , , , , and , .

be Green's function for the boundary value problems [22, 23].

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 [24].

Lemma 2.1.

Lemma 2.2.

Proof.

Apply Lemma 2.1 to get the conclusion.

Lemma 2.3.

where .

One sees that depends on only and is bounded near zero.

Proof.

The existence and uniqueness of the solution comes from Theorem 5.3 on page 320 in [25].

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.

Problem 1.

*Find functions*

*and*

*such that*

Let , and let . We have the following two additional problems for and , respectively.

Problem 2.

*Find functions*

*and*

*such that*

Problem 3.

*Find functions*

*and*

*such that*

Lemma 2.4.

Problems 1, 2, and 3 are equivalent to each other.

Proof.

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.

Theorem 2.5.

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.

Theorem 2.6.

where depends on , , , , , , , and .

Proof.

The proof is complete.

Corollary 2.7.

Under the conditions in Theorem 2.6, the solution of Problem 3 is unique.

## Declarations

### Acknowledgment

The research is supported by the NSF of China (no. 11071048).

## Authors’ Affiliations

## References

- Agarwal RP, de Andrade B, Cuevas C:
**Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations.***Nonlinear Analysis: Real World Applications*2010,**11**(5):3532-3554. 10.1016/j.nonrwa.2010.01.002MathSciNetView ArticleMATH - Andres J, Bersani AM, Grande RF:
**Hierarchy of almost-periodic function spaces.***Rendiconti di Matematica e delle sue Applicazioni*2006,**26**(2):121-188.MathSciNetMATH - Basit B, Zhang C:
**New almost periodic type functions and solutions of differential equations.***Canadian Journal of Mathematics*1996,**48**(6):1138-1153. 10.4153/CJM-1996-059-9MathSciNetView ArticleMATH - Berglund JF, Junghenn HD, Milnes P:
*Analysis on Semigroups: Function Spaces, Compactifications, Representations,, Canadian Mathematical Society Series of Monographs and Advanced Texts*. John Wiley & Sons, New York, NY, USA; 1989:xiv+334.MATH - Bourgain J:
**Construction of approximative and almost periodic solutions of perturbed linear Schrödinger and wave equations.***Geometric and Functional Analysis*1996,**6**(2):201-230. 10.1007/BF02247885MathSciNetView ArticleMATH - Corduneanu C:
*Almost Periodic Functions, Interscience Tracts in Pure and Applied Mathematics, no. 2*. 1st edition. John Wiley & Sons, New York, NY, USA; 1968:x+237. - Corduneanu C:
*Almost Periodic Functions*. 2nd edition. John Wiley & Sons, New York, NY, USA; 1989.MATH - Diagan T:
*Pseudo Almost Periodic Functions in Banach spaces*. Nova Science, New York, NY, USA; 2007. - Fink AM:
*Almost Periodic Differential Equations, Lecture Notes in Mathematics*.*Volume 377*. Springer, Berlin, Germany; 1974:viii+336. - Hino Y, Naito T, Nguyen Van Minh , Shin JS:
*Almost Periodic Solutions of Differential Equations in Banach Spaces, Stability and Control: Theory, Methods and Applications*.*Volume 15*. Taylor & Francis, London, UK; 2002:viii+250.MATH - N'Guérékata GM:
*Topics in Almost Automorphy*. Springer, New York, NY, USA; 2005:xii+168.MATH - Shen W:
**Travelling waves in time almost periodic structures governed by bistable nonlinearities. I. Stability and uniqueness.***Journal of Differential Equations*1999,**159**(1):1-54. 10.1006/jdeq.1999.3651MathSciNetView ArticleMATH - Zhang C:
*Almost Periodic Type Functions and Ergodicity*. Science Press, Beijing, China; Kluwer Academic Publishers, Dordrecht, The Netherlands; 2003:xii+355.View ArticleMATH - Zhang C, Yang F:
**Remotely almost periodic solutions of parabolic inverse problems.***Nonlinear Analysis: Theory, Methods & Applications*2006,**65**(8):1613-1623. 10.1016/j.na.2005.10.036MathSciNetView ArticleMATH - Yang F, Zhang C:
**Slowly oscillating solutions of parabolic inverse problems.***Journal of Mathematical Analysis and Applications*2007,**335**(2):1238-1258. 10.1016/j.jmaa.2007.01.098MathSciNetView ArticleMATH - Zhang C, Yang F:
**Pseudo almost periodic solutions to parabolic boundary value inverse problems.***Science in China. Series A*2008,**51**(7):1203-1214. 10.1007/s11425-008-0006-2MathSciNetView ArticleMATH - Sarason D:
**Remotely almost periodic functions.**In*Proceedings of the Conference on Banach Algebras and Several Complex Variables (New Haven, Conn., 1983), Providence, RI, USA, Contemp. Math.*.*Volume 32*. American Mathematical Society; 1984:237-242. - Zhang C:
**New limit power function spaces.***IEEE Transactions on Automatic Control*2004,**49**(5):763-766. 10.1109/TAC.2004.825970View ArticleMathSciNet - Zhang C, Meng C:
**-algebra of strong limit power functions.***IEEE Transactions on Automatic Control*2006,**51**(5):828-831. 10.1109/TAC.2006.875016MathSciNetView Article - Zhang C:
**Strong limit power functions.***The Journal of Fourier Analysis and Applications*2006,**12**(3):291-307. 10.1007/s00041-006-6004-2MathSciNetView ArticleMATH - Friedman A:
*Partial Differential Equations of Parabolic Type*. Prentice-Hall, Englewood Cliffs, NJ, USA; 1964:xiv+347. - Guo B:
*Inverse Problem of Parabolic Partial Differential Equations*. Science and Technology Press of Heilongjiang Province, Harbin, China; 1988. - Zauderer E:
*Partial Differential Equations of Applied Mathematics, Pure and Applied Mathematics*. John Wiley & Sons, New York, NY, USA; 1983:xiii+779.MATH - Hale JK, Verduyn Lunel SM:
*Introduction to Functional-Differential Equations, Applied Mathematical Sciences*.*Volume 99*. Springer, New York, NY, USA; 1993:x+447.View ArticleMATH - Ladyzenskaja OA, Solonnikov VA, Ural'ceva NN:
*Linear and Quasi-Linear Equations of Parabolic Type*. American Mathematical Society, Providence, RI, USA; 1968.

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.