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

  • Fenglin Yang1Email author and

    Affiliated with

    • Chuanyi Zhang1

      Affiliated with

      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 http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq4_HTML.gif -subalgebra of http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq6_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq7_HTML.gif with the supremum norm http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq8_HTML.gif . Compared with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq9_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq11_HTML.gif is based on the belief that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq13_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq14_HTML.gif (resp., http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq15_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq16_HTML.gif ) denote the http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq18_HTML.gif (resp., http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq19_HTML.gif ) with the supremum norm. For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq20_HTML.gif (resp., http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq21_HTML.gif ) and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq22_HTML.gif , the translate of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq23_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq24_HTML.gif is the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq25_HTML.gif (resp., http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq26_HTML.gif , http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq29_HTML.gif , http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq33_HTML.gif and uniform on compact subsets of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq34_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq35_HTML.gif for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq36_HTML.gif and is uniformly continuous on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq37_HTML.gif for any compact subset http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq38_HTML.gif . Denote by http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq40_HTML.gif be a Banach space, and let http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq42_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq43_HTML.gif . If we replace http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq44_HTML.gif in (1) by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq45_HTML.gif , then we get the definition of http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq48_HTML.gif be such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq49_HTML.gif is uniformly continuous on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq50_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq51_HTML.gif .

      For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq52_HTML.gif , suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq53_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq54_HTML.gif . Define http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq55_HTML.gif by
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq56_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq57_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq58_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq59_HTML.gif , then http://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: http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq61_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq62_HTML.gif . http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq63_HTML.gif means that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq64_HTML.gif is slowly oscillating in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq65_HTML.gif and uniformly for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq66_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq67_HTML.gif means that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq68_HTML.gif is slowly oscillating in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq69_HTML.gif and uniformly on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq70_HTML.gif .

      Let
      http://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:
      http://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: http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq71_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq72_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq73_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq74_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq75_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq76_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq77_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq78_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq79_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq80_HTML.gif .

      Let
      http://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:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ5_HTML.gif
      (2.3)

      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq81_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq82_HTML.gif ) are positive and increasing for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq83_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq84_HTML.gif as http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq86_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq87_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq88_HTML.gif be real, continuous functions on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq89_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq90_HTML.gif . If
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ6_HTML.gif
      (2.4)
      then
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ7_HTML.gif
      (2.5)

      Lemma 2.2.

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

      Proof.

      Replacing http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq98_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq99_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq100_HTML.gif , one gets
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq101_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq102_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq103_HTML.gif . Then the problem
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ12_HTML.gif
      (2.10)
      has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq104_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq105_HTML.gif is in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq106_HTML.gif and satisfies
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ13_HTML.gif
      (2.11)

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

      One sees that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq108_HTML.gif depends on http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq110_HTML.gif can be written as
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ14_HTML.gif
      (2.12)
      So,
      http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq112_HTML.gif . For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq113_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq114_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq115_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ16_HTML.gif
      (2.14)
      Note that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ17_HTML.gif
      (2.15)
      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq116_HTML.gif is a constant and
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ18_HTML.gif
      (2.16)
      So,
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ20_HTML.gif
      (2.18)

      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq117_HTML.gif is a constant. Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq118_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq120_HTML.gif . This means that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq122_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq123_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ21_HTML.gif
      (2.19)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ22_HTML.gif
      (2.20)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ23_HTML.gif
      (2.21)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ24_HTML.gif
      (2.22)
      One sees that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ25_HTML.gif
      (2.23)
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ27_HTML.gif
      (2.25)

      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq124_HTML.gif , and let http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq126_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq127_HTML.gif , respectively.

      Problem 2.

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

      Problem 3.

      Find functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq130_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq131_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ32_HTML.gif
      (2.30)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ33_HTML.gif
      (2.31)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ34_HTML.gif
      (2.32)
      http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq133_HTML.gif of Problem 1. Conversely, let ( http://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 ( http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ36_HTML.gif
      (2.34)
      Obviously, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq136_HTML.gif and satisfies (2.22). Also http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq137_HTML.gif satisfies (2.21) because http://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 http://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 http://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,
      http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq141_HTML.gif :
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ39_HTML.gif
      (2.37)
      Rewrite (2.33) as
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ40_HTML.gif
      (2.38)

      where http://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 http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq146_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq147_HTML.gif . That is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq148_HTML.gif has a fixed point in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq149_HTML.gif . Let
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ41_HTML.gif
      (2.39)
      Set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq150_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq151_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq152_HTML.gif , then, by Lemma 2.3, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq153_HTML.gif is in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq154_HTML.gif , and so, by (2.38), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq155_HTML.gif is in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq156_HTML.gif with
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ43_HTML.gif
      (2.41)
      Choose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq157_HTML.gif such that when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq158_HTML.gif , one has http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq159_HTML.gif . It follows that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ44_HTML.gif
      (2.42)
      Choose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq160_HTML.gif such that when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq161_HTML.gif , one has
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ45_HTML.gif
      (2.43)

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

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

      Choose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq167_HTML.gif such that for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq168_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq169_HTML.gif . Now, set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq170_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq171_HTML.gif is a contraction from http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq173_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq174_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq175_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq176_HTML.gif be as above. Then, for small http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq177_HTML.gif , Problem 3 has a unique solution ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq178_HTML.gif ) in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq179_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq180_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq181_HTML.gif .

      Let http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq183_HTML.gif for the functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq184_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq185_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq186_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq187_HTML.gif . Set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq188_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq189_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq190_HTML.gif , and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq192_HTML.gif , one has
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ48_HTML.gif
      (2.46)

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

      Proof.

      By (2.33),
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ49_HTML.gif
      (2.47)
      So,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ50_HTML.gif
      (2.48)
      Note that the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq202_HTML.gif is the solution of the problem
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq203_HTML.gif , one gets
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ53_HTML.gif
      (2.51)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_Equ54_HTML.gif
      (2.52)
      and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq204_HTML.gif is majorant of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F471491/MediaObjects/13661_2010_Article_928_IEq205_HTML.gif . One can specially assume that
      http://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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