The Solution of Two-Point Boundary Value Problem of a Class of Duffing-Type Systems with Non- http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq1_HTML.gif Perturbation Term

  • Jiang Zhengxian1 and

    Affiliated with

    • Huang Wenhua1Email author

      Affiliated with

      Boundary Value Problems20092009:287834

      DOI: 10.1155/2009/287834

      Received: 14 June 2009

      Accepted: 10 August 2009

      Published: 19 August 2009

      Abstract

      This paper deals with a two-point boundary value problem of a class of Duffing-type systems with non- http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq2_HTML.gif perturbation term. Several existence and uniqueness theorems were presented.

      1. Introduction

      Minimax theorems are one of powerful tools for investigation on the solution of differential equations and differential systems. The investigation on the solution of differential equations and differential systems with non- http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq3_HTML.gif perturbation term using minimax theorems came into being in the paper of Stepan A.Tersian in 1986 [1]. Tersian proved that the equation http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq4_HTML.gif exists exactly one generalized solution under the operators http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq5_HTML.gif related to the perturbation term http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq6_HTML.gif being selfadjoint and commuting with the operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq7_HTML.gif and some other conditions in [1]. Huang Wenhua extended Tersian's theorems in [1] in 2005 and 2006, respectively, and studied the existence and uniqueness of solutions of some differential equations and differential systems with non- http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq8_HTML.gif perturbation term [24], the conditions attached to the non- http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq9_HTML.gif perturbation term are that the operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq10_HTML.gif related to the term is self-adjoint and commutes with the operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq11_HTML.gif (where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq12_HTML.gif is a selfadjoint operator in the equation http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq13_HTML.gif ). Recently, by further research, we observe that the conditions imposed upon http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq14_HTML.gif can be weakened, the self-adjointness of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq15_HTML.gif can be removed and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq16_HTML.gif is not necessarily commuting with the operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq17_HTML.gif .

      In this note, we consider a two-point boundary value problem of a class of Duffing-type systems with non- http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq18_HTML.gif perturbation term and present a result as the operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq19_HTML.gif related to the perturbation term is not necessarily a selfadjoint and commuting with the operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq20_HTML.gif . We obtain several valuable results in the present paper under the weaker conditions than those in [24].

      2. Preliminaries

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq21_HTML.gif be a real Hilbert space with inner product http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq22_HTML.gif and norm http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq23_HTML.gif , respectively, let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq24_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq25_HTML.gif be two orthogonal closed subspaces of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq26_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq27_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq28_HTML.gif denote the projections from http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq29_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq30_HTML.gif and from http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq31_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq32_HTML.gif , respectively. The following theorem will be employed to prove our main theorem.

      Theorem 2.1 ([2]).

      Let   http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq33_HTML.gif   be a real Hilbert space, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq34_HTML.gif   an everywhere defined functional with Gâteaux derivative   http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq35_HTML.gif   everywhere defined and hemicontinuous. Suppose that there exist two closed subspaces   http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq36_HTML.gif   and   http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq37_HTML.gif   such that   http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq38_HTML.gif and two nonincreasing functions   http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq39_HTML.gif   satisfying
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ1_HTML.gif
      (2.1)
      and
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ2_HTML.gif
      (2.2)
      for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq40_HTML.gif , and
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ3_HTML.gif
      (2.3)

      for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq41_HTML.gif . Then

      (a) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq42_HTML.gif has a unique critical point http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq43_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq44_HTML.gif ;

      (b) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq45_HTML.gif .

      We also need the following lemma in the present work. To the best of our knowledge, the lemma seems to be new.

      Lemma 2.2.

      Let   http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq46_HTML.gif   and   http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq47_HTML.gif   be two diagonalization   http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq48_HTML.gif   matrices, let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq49_HTML.gif   and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq50_HTML.gif be the eigenvalues of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq51_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq52_HTML.gif , respectively, where each eigenvalue is repeated according to its multiplicity. If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq53_HTML.gif commutes with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq54_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq55_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq56_HTML.gif is a diagonalization matrix and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq57_HTML.gif are the eigenvalues of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq58_HTML.gif .

      Proof.

      Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq59_HTML.gif is a diagonalization http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq60_HTML.gif matrix, there exists an inverse matrix http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq61_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq62_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq63_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq64_HTML.gif are the distinct eigenvalues of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq65_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq66_HTML.gif are the http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq67_HTML.gif identity matrices. And since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq68_HTML.gif , that is,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ4_HTML.gif
      (2.4)
      we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ5_HTML.gif
      (2.5)
      Denote http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq69_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq70_HTML.gif are the submatrices such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq71_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq72_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq73_HTML.gif are defined, then, by (2.5),
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ6_HTML.gif
      (2.6)
      Noticed that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq74_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq75_HTML.gif , and hence
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ7_HTML.gif
      (2.7)
      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq76_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq77_HTML.gif are the same order square matrices. Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq78_HTML.gif is a diagonalization http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq79_HTML.gif matrix, there exists an invertible matrix http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq80_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ8_HTML.gif
      (2.8)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq81_HTML.gif are the eigenvalues of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq82_HTML.gif .

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq83_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq84_HTML.gif is an invertible matrix such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq85_HTML.gif and
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ9_HTML.gif
      (2.9)

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq86_HTML.gif is a diagonalization matrix and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq87_HTML.gif are the eigenvalues of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq88_HTML.gif .

      The proof of Lemma 2.2 is fulfilled.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq89_HTML.gif denote the usual inner product on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq90_HTML.gif and denote the corresponding norm by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq91_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq92_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq93_HTML.gif denote the inner product on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq94_HTML.gif . It is known very well that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq95_HTML.gif is a Hilbert space with inner product

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ10_HTML.gif
      (2.10)

      and norm http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq96_HTML.gif , respectively.

      Now, we consider the boundary value problem

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ11_HTML.gif
      (2.11)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq97_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq98_HTML.gif is a real constant diagonalization http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq99_HTML.gif matrix with real eigenvalues http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq100_HTML.gif (each eigenvalue is repeated according to its multiplicity), http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq101_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq102_HTML.gif is a potential Carathéodory vector-valued function , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq103_HTML.gif is continuous, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq104_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq105_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq106_HTML.gif .

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq107_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq108_HTML.gif , then (2.11) may be written in the form

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ12_HTML.gif
      (2.12)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq109_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq110_HTML.gif . Clearly, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq111_HTML.gif is a potential Carathéodory vector-valued function, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq112_HTML.gif . Clearly, if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq113_HTML.gif is a solution of (2.12), http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq114_HTML.gif will be a solution of (2.11).

      Assume that there exists a real bounded diagonalization http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq115_HTML.gif matrix http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq116_HTML.gif such that for a.e. http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq117_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq118_HTML.gif

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ13_HTML.gif
      (2.13)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq119_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq120_HTML.gif commutes with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq121_HTML.gif and is possessed of real eigenvalues http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq122_HTML.gif . In the light of Lemma 2.2, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq123_HTML.gif is a diagonalization http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq124_HTML.gif matrix with real eigenvalues http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq125_HTML.gif (each eigenvalue is repeated according to its multiplicity). Assume that there exist positive integers http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq126_HTML.gif such that for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq127_HTML.gif

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ14_HTML.gif
      (2.14)

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq128_HTML.gif be http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq129_HTML.gif linearly independent eigenvectors associated with the eigenvalues http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq130_HTML.gif and let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq131_HTML.gif be the orthonormal vectors obtained by orthonormalizing to the eigenvectors http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq132_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq133_HTML.gif . Then for every http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq134_HTML.gif

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ15_HTML.gif
      (2.15)

      And let the set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq135_HTML.gif be a basis for the space http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq136_HTML.gif , then for every http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq137_HTML.gif ,

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ16_HTML.gif
      (2.16)

      It is well known that each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq138_HTML.gif can be represented by the absolutely convergent Fourier series

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ17_HTML.gif
      (2.17)

      Define the linear operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq139_HTML.gif

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ18_HTML.gif
      (2.18)

      Clearly, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq140_HTML.gif is a selfadjoint operator and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq141_HTML.gif is a Hilbert space for the inner product

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ19_HTML.gif
      (2.19)

      and the norm induced by the inner product is

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ20_HTML.gif
      (2.20)

      Define

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ21_HTML.gif
      (2.21)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ22_HTML.gif
      (2.22)

      Clearly, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq142_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq143_HTML.gif are orthogonal closed subspaces of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq144_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq145_HTML.gif .

      Define two projective mappings http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq146_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq147_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq148_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq149_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq150_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq151_HTML.gif is a selfadjoint operator.

      Using the Riesz representation theorem , we can define a mapping http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq152_HTML.gif by

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ23_HTML.gif
      (2.23)

      We observe that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq153_HTML.gif in (2.23) is defined implicity. Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq154_HTML.gif in (2.23), we have

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ24_HTML.gif
      (2.24)

      Clearly, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq155_HTML.gif and hence http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq156_HTML.gif is defined implicity by (2.24). It can be proved that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq157_HTML.gif is a solution of (2.11) if and only if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq158_HTML.gif satisfies the operator equation

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ25_HTML.gif
      (2.25)

      3. The Main Theorems

      Now, we state and prove the following theorem concerning the solution of problem (2.11).

      Theorem 3.1.

      Assume that there exists a real diagonalization http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq159_HTML.gif matrix http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq160_HTML.gif with real eigenvalues http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq161_HTML.gif satisfying (2.14) and commuting with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq162_HTML.gif . Denote
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ26_HTML.gif
      (3.1)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ27_HTML.gif
      (3.2)
      If
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ28_HTML.gif
      (3.3)
      problem (2.11) has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq163_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq164_HTML.gif satisfies http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq165_HTML.gif , and
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ29_HTML.gif
      (3.4)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq166_HTML.gif is a functional defined in (2.24) and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq167_HTML.gif .

      Proof.

      First, by virtue of (2.21) and (2.22), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ30_HTML.gif
      (3.5)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ31_HTML.gif
      (3.6)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ32_HTML.gif
      (3.7)

      Denote http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq168_HTML.gif

      By (2.24), (2.13), (3.5), (3.6), (3.7), (3.1), and (3.2), for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq169_HTML.gif , let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq170_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq171_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq172_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq173_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq174_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq175_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ33_HTML.gif
      (3.8)
      for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq176_HTML.gif , let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq177_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq178_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq179_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq180_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq181_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq182_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ34_HTML.gif
      (3.9)

      By (3.3), http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq183_HTML.gif Clearly, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq184_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq185_HTML.gif are nonincreasing. Now, all the conditions in the Theorem 2.1 are satisfied. By virtue of Theorem 2.1, there exists a unique http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq186_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq187_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq188_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq189_HTML.gif is a functional defined implicity in (2.24) and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq190_HTML.gif . http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq191_HTML.gif is just a unique solution of (2.12) and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq192_HTML.gif is exactly a unique solution of (2.11). The proof of Theorem 3.1 is completed.

      Now, we assume that there exists a positive integer http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq193_HTML.gif such that

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ35_HTML.gif
      (3.10)

      for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq194_HTML.gif Define

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ36_HTML.gif
      (3.11)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ37_HTML.gif
      (3.12)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ38_HTML.gif
      (3.13)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ39_HTML.gif
      (3.14)

      Replace the condition (2.14) by (3.10) and replace (2.21), (2.22), (3.1), and (3.2) by (3.11), (3.12), (3.13), and (3.14), respectively. Using the similar proving techniques in the Theorem 3.1, we can prove the following theorem.

      Theorem 3.2.

      Assume that there exists a real diagonalization http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq195_HTML.gif matrix http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq196_HTML.gif with real eigenvalues http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq197_HTML.gif satisfying (2.13) and (3.10) and commuting with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq198_HTML.gif . If the functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq199_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq200_HTML.gif defined in (3.13) and (3.14) satisfy (3.3), problem (2.11) has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq201_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq202_HTML.gif satisfies http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq203_HTML.gif and (3.4).

      It is also of interest to the case of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq204_HTML.gif .

      Corollary 3.3.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq205_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq206_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq207_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq208_HTML.gif be as in (2.11). Assume that there exists a real diagonalization http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq209_HTML.gif matrix http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq210_HTML.gif with real eigenvalues http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq211_HTML.gif satisfying (2.13) and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq212_HTML.gif Denote
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ40_HTML.gif
      (3.15)
      If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq213_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq214_HTML.gif satisfy (3.3), the problem
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ41_HTML.gif
      (3.16)
      has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq215_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq216_HTML.gif satisfies http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq217_HTML.gif and (3.4), where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq218_HTML.gif is a functional defined in
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ42_HTML.gif
      (3.17)

      Corollary 3.4.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq219_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq220_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq221_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq222_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq223_HTML.gif be as in Corollary 3.3. The eigenvalues of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq224_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq225_HTML.gif satisfy http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq226_HTML.gif Denote
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ43_HTML.gif
      (3.18)

      If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq227_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq228_HTML.gif satisfy (3.3), problem (3.16) has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq229_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq230_HTML.gif satisfies http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq231_HTML.gif and (3.4), where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq232_HTML.gif is a functional defined in (3.17).

      If there exists a http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq233_HTML.gif functional http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq234_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq235_HTML.gif , then (2.13) should be

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ44_HTML.gif
      (3.19)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq236_HTML.gif is just a Hessian of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq237_HTML.gif . In this case, the following corollary follows from Theorem 3.1.

      Corollary 3.5.

      Let the eigenvalues of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq238_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq239_HTML.gif satisfy (2.14). If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq240_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq241_HTML.gif defined in (3.1) and (3.2) satisfy (3.3), problem (2.11)(where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq242_HTML.gif ) has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq243_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq244_HTML.gif satisfies http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq245_HTML.gif and (3.4).

      Using the similar techniques of the present paper, we can also investigate the two-point boundary value problem

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ45_HTML.gif
      (3.20)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq246_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq247_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq248_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq249_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq250_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq251_HTML.gif are as in problem (2.11). The corresponding results are similar to the results in the present paper.

      The special case of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq252_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq253_HTML.gif in problem (3.20) has been studied by Zhou Ting and Huang Wenhua [5]. Zhou and Huang adopted the techniques different from this paper to achieve their research.

      Authors’ Affiliations

      (1)
      School of Sciences, Jiangnan University

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      Copyright

      © J. Zhengxian and H. Wenhua. 2009

      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.