Open Access

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

Boundary Value Problems20102010:471491

DOI: 10.1155/2010/471491

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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq1_HTML.gif of almost periodic functions and some of its generalizations have many applications (e.g., [113] and references therein). However, little has been done for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq2_HTML.gif to inverse problems except for our work in [1416]. Sarason in [17] studied the space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq3_HTML.gif of slowly oscillating functions. This is a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq4_HTML.gif -subalgebra of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq5_HTML.gif , the space of bounded, continuous, complex-valued functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq6_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq7_HTML.gif with the supremum norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq8_HTML.gif . Compared with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq9_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq10_HTML.gif is a quite large space (see [1720]). What we are interested in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq11_HTML.gif is based on the belief that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq12_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq13_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq14_HTML.gif (resp., https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq15_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq16_HTML.gif ) denote the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq17_HTML.gif -algebra of bounded continuous complex-valued functions on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq18_HTML.gif (resp., https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq19_HTML.gif ) with the supremum norm. For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq20_HTML.gif (resp., https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq21_HTML.gif ) and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq22_HTML.gif , the translate of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq23_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq24_HTML.gif is the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq25_HTML.gif (resp., https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq26_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq27_HTML.gif ).

Definition 1.1.
  1. (1)

    A function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq28_HTML.gif is called slowly oscillating if for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq29_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq30_HTML.gif , the space of the functions vanishing at infinity. Denote by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq31_HTML.gif the set of all such functions.

     
  2. (2)

    A function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq32_HTML.gif is said to be slowly oscillating in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq33_HTML.gif and uniform on compact subsets of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq34_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq35_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq36_HTML.gif and is uniformly continuous on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq37_HTML.gif for any compact subset https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq38_HTML.gif . Denote by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq39_HTML.gif the set of all such functions. For convenience, such functions are also called uniformly slowly oscillating functions.

     
  3. (3)

    Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq40_HTML.gif be a Banach space, and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq41_HTML.gif be the space of bounded continuous functions from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq42_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq43_HTML.gif . If we replace https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq44_HTML.gif in (1) by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq45_HTML.gif , then we get the definition of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq46_HTML.gif .

     

As in [17], we always assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq47_HTML.gif is uniformly continuous.

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

Proposition 1.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq48_HTML.gif be such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq49_HTML.gif is uniformly continuous on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq50_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq51_HTML.gif .

For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq52_HTML.gif , suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq53_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq54_HTML.gif . Define https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq55_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ1_HTML.gif
(1.1)

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

Proposition 1.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq56_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq57_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq58_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq59_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq60_HTML.gif .

In the sequel, we will use the notations: https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq61_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq62_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq63_HTML.gif means that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq64_HTML.gif is slowly oscillating in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq65_HTML.gif and uniformly for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq66_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq67_HTML.gif means that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq68_HTML.gif is slowly oscillating in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq69_HTML.gif and uniformly on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq70_HTML.gif .

Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ2_HTML.gif
(1.2)

be the fundamental solution of the heat equation [21].

2. A Type of Boundary Value Problem

We will keep the notation in Section 1 and at the same time introduce the following new notation:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ3_HTML.gif
(2.1)

In this section, we always assume the following: https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq71_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq72_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq73_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq74_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq75_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq76_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq77_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq78_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq79_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq80_HTML.gif .

Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ4_HTML.gif
(2.2)

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

The following estimates are easily obtained:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ5_HTML.gif
(2.3)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq81_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq82_HTML.gif ) are positive and increasing for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq83_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq84_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq85_HTML.gif .

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.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq86_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq87_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq88_HTML.gif be real, continuous functions on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq89_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq90_HTML.gif . If
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ6_HTML.gif
(2.4)
then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ7_HTML.gif
(2.5)

Lemma 2.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq91_HTML.gif be a continuous function on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq92_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq93_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq94_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq95_HTML.gif are nondecreasing and nonnegative on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq96_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ8_HTML.gif
(2.6)
then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ9_HTML.gif
(2.7)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ10_HTML.gif
(2.8)

Proof.

Replacing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq97_HTML.gif in the two integrals of (2.6) by the expression on the right hand side in (2.6), changing the integral order of the resulting inequality and making use of the monotonicity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq98_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq99_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq100_HTML.gif , one gets
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ11_HTML.gif
(2.9)

Apply Lemma 2.1 to get the conclusion.

Lemma 2.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq101_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq102_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq103_HTML.gif . Then the problem
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ12_HTML.gif
(2.10)
has a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq104_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq105_HTML.gif is in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq106_HTML.gif and satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ13_HTML.gif
(2.11)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq107_HTML.gif .

One sees that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq108_HTML.gif depends on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq109_HTML.gif only and is bounded near zero.

Proof.

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

As in [22, 23], the solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq110_HTML.gif can be written as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ14_HTML.gif
(2.12)
So,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ15_HTML.gif
(2.13)

By Lemma 2.1, one gets the desired inequality.

Now we show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq111_HTML.gif . As in the proofs of Lemmas 2.1 and 2.3 in [15], one gets https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq112_HTML.gif . For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq113_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq114_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq115_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ16_HTML.gif
(2.14)
Note that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ17_HTML.gif
(2.15)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq116_HTML.gif is a constant and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ18_HTML.gif
(2.16)
So,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ19_HTML.gif
(2.17)
By Lemma 2.1, one has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ20_HTML.gif
(2.18)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq117_HTML.gif is a constant. Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq118_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq119_HTML.gif are slowly oscillating, the right-hand sides of the inequality above approaches zero as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq120_HTML.gif . This means that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq121_HTML.gif . The proof is complete.

Consider the following problem.

Problem 1.

Find functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq122_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq123_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ21_HTML.gif
(2.19)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ22_HTML.gif
(2.20)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ23_HTML.gif
(2.21)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ24_HTML.gif
(2.22)
One sees that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ25_HTML.gif
(2.23)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ26_HTML.gif
(2.24)
It follows from (2.24) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ27_HTML.gif
(2.25)

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq124_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq125_HTML.gif . We have the following two additional problems for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq126_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq127_HTML.gif , respectively.

Problem 2.

Find functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq128_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq129_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ28_HTML.gif
(2.26)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ29_HTML.gif
(2.27)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ30_HTML.gif
(2.28)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ31_HTML.gif
(2.29)

Problem 3.

Find functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq130_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq131_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ32_HTML.gif
(2.30)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ33_HTML.gif
(2.31)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ34_HTML.gif
(2.32)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ35_HTML.gif
(2.33)

Lemma 2.4.

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

Proof.

The existence and uniqueness of the solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq132_HTML.gif of Problem 2 can be easily obtained from that of the solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq133_HTML.gif of Problem 1. Conversely, let ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq134_HTML.gif ) be the solution of Problem 2. We show that Problem 1 has a unique solution ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq135_HTML.gif ). The uniqueness comes from the uniqueness of (2.19)–(2.21). For the existence, let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ36_HTML.gif
(2.34)
Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq136_HTML.gif and satisfies (2.22). Also https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq137_HTML.gif satisfies (2.21) because https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq138_HTML.gif . By (2.23) and (2.27), one sees that (2.20) is true. Finally, we show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq139_HTML.gif satisfies (2.19) and therefore, along with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq140_HTML.gif , constitutes a solution of Problem 1. In fact,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ37_HTML.gif
(2.35)
Thus, we have shown the equivalence of Problems 1 and 2. Replacing (2.34) by the function
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ38_HTML.gif
(2.36)

the equivalence of Problems 2 and 3 can be proved similarly. The proof is complete.

By Lemma 2.4, to solve Problem 1, we only need to solve Problem 3. By (2.30)–(2.32), we have the integral equation about https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq141_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ39_HTML.gif
(2.37)
Rewrite (2.33) as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ40_HTML.gif
(2.38)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq142_HTML.gif is determined by (2.37).

One can directly test that Problem 3 is equivalent to (2.37)-(2.38).

Note that for a given https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq143_HTML.gif , Lemma 2.3 shows that (2.30)–(2.32) (or equivalently, (2.37)) have a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq144_HTML.gif . Thus, (2.38) does define an operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq145_HTML.gif . Therefore, we only need to show that the integral (2.38) has a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq146_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq147_HTML.gif . That is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq148_HTML.gif has a fixed point in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq149_HTML.gif . Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ41_HTML.gif
(2.39)
Set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq150_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq151_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq152_HTML.gif , then, by Lemma 2.3, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq153_HTML.gif is in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq154_HTML.gif , and so, by (2.38), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq155_HTML.gif is in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq156_HTML.gif with
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ42_HTML.gif
(2.40)
Equation (2.37) gives the estimate
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ43_HTML.gif
(2.41)
Choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq157_HTML.gif such that when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq158_HTML.gif , one has https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq159_HTML.gif . It follows that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ44_HTML.gif
(2.42)
Choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq160_HTML.gif such that when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq161_HTML.gif , one has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ45_HTML.gif
(2.43)

and therefore, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq162_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq163_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq164_HTML.gif . By (2.38), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq165_HTML.gif . Note that the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq166_HTML.gif is the solution of the problem
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ46_HTML.gif
(2.44)
So, by Lemma 2.3, one has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ47_HTML.gif
(2.45)

Choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq167_HTML.gif such that for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq168_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq169_HTML.gif . Now, set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq170_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq171_HTML.gif is a contraction from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq172_HTML.gif into itself, and therefore, has a unique fixed point. Thus, we have shown.

Theorem 2.5.

Let functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq173_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq174_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq175_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq176_HTML.gif be as above. Then, for small https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq177_HTML.gif , Problem 3 has a unique solution ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq178_HTML.gif ) in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq179_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq180_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq181_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq182_HTML.gif be the solutions of Problem 3 in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq183_HTML.gif for the functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq184_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq185_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq186_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq187_HTML.gif . Set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq188_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq189_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq190_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq191_HTML.gif . For the stability of the solution, we have the following.

Theorem 2.6.

For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq192_HTML.gif , one has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ48_HTML.gif
(2.46)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq193_HTML.gif depends on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq194_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq195_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq196_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq197_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq198_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq199_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq200_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq201_HTML.gif .

Proof.

By (2.33),
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ49_HTML.gif
(2.47)
So,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ50_HTML.gif
(2.48)
Note that the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq202_HTML.gif is the solution of the problem
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ51_HTML.gif
(2.49)
Using a formula similar to (2.37) and Lemma 2.2 for the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq203_HTML.gif , one gets
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ52_HTML.gif
(2.50)
Applying Lemma 2.2 and (2.48), one gets the desired conclusion with
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ53_HTML.gif
(2.51)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ54_HTML.gif
(2.52)
and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq204_HTML.gif is majorant of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq205_HTML.gif . One can specially assume that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ55_HTML.gif
(2.53)

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

(1)
Department of Mathematics, Harbin Institute of Technology

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© F. Yang and C. Zhang. 2010

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