Open Access

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

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- https://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- https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq5_HTML.gif related to the perturbation term https://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 https://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- https://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- https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq9_HTML.gif perturbation term are that the operator https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq11_HTML.gif (where https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq15_HTML.gif can be removed and https://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 https://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- https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq22_HTML.gif and norm https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq23_HTML.gif , respectively, let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq24_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq25_HTML.gif be two orthogonal closed subspaces of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq26_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq27_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq28_HTML.gif denote the projections from https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq29_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq30_HTML.gif and from https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq31_HTML.gif to https://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   https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq33_HTML.gif   be a real Hilbert space, https://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   https://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   https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq36_HTML.gif   and   https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq37_HTML.gif   such that   https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq38_HTML.gif and two nonincreasing functions   https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq39_HTML.gif   satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ1_HTML.gif
(2.1)
and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ2_HTML.gif
(2.2)
for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq40_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ3_HTML.gif
(2.3)

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

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

(b) https://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   https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq46_HTML.gif   and   https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq47_HTML.gif   be two diagonalization   https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq48_HTML.gif   matrices, let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq49_HTML.gif   and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq50_HTML.gif be the eigenvalues of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq51_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq53_HTML.gif commutes with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq54_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq55_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq56_HTML.gif is a diagonalization matrix and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq57_HTML.gif are the eigenvalues of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq58_HTML.gif .

Proof.

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

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

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

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq86_HTML.gif is a diagonalization matrix and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq87_HTML.gif are the eigenvalues of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq89_HTML.gif denote the usual inner product on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq90_HTML.gif and denote the corresponding norm by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq91_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq92_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq93_HTML.gif denote the inner product on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq94_HTML.gif . It is known very well that https://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

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

and norm https://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

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

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq97_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq98_HTML.gif is a real constant diagonalization https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq99_HTML.gif matrix with real eigenvalues https://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), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq101_HTML.gif https://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 , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq103_HTML.gif is continuous, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq104_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq105_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq106_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq107_HTML.gif , https://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

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

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq109_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq110_HTML.gif . Clearly, https://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, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq112_HTML.gif . Clearly, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq113_HTML.gif is a solution of (2.12), https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq115_HTML.gif matrix https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq116_HTML.gif such that for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq117_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq118_HTML.gif

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

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq119_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq120_HTML.gif commutes with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq121_HTML.gif and is possessed of real eigenvalues https://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, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq123_HTML.gif is a diagonalization https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq124_HTML.gif matrix with real eigenvalues https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq126_HTML.gif such that for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq127_HTML.gif

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

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq128_HTML.gif be https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq130_HTML.gif and let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq132_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq133_HTML.gif . Then for every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq134_HTML.gif

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

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

https://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 https://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

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

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

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

Clearly, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq140_HTML.gif is a selfadjoint operator and https://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

https://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

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

Define

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

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

Define two projective mappings https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq146_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq147_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq148_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq149_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq150_HTML.gif , then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq152_HTML.gif by

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

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

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

Clearly, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq155_HTML.gif and hence https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq158_HTML.gif satisfies the operator equation

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq159_HTML.gif matrix https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq160_HTML.gif with real eigenvalues https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq161_HTML.gif satisfying (2.14) and commuting with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq162_HTML.gif . Denote
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ26_HTML.gif
(3.1)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ27_HTML.gif
(3.2)
If
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq163_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq164_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq165_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ29_HTML.gif
(3.4)

where https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ30_HTML.gif
(3.5)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ31_HTML.gif
(3.6)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ32_HTML.gif
(3.7)

Denote https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq169_HTML.gif , let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq170_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq171_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq172_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq173_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq174_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq175_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ33_HTML.gif
(3.8)
for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq176_HTML.gif , let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq177_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq178_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq179_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq180_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq181_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq182_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ34_HTML.gif
(3.9)

By (3.3), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq183_HTML.gif Clearly, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq184_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq186_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq187_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq188_HTML.gif where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq190_HTML.gif . https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq193_HTML.gif such that

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

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

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ36_HTML.gif
(3.11)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ37_HTML.gif
(3.12)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ38_HTML.gif
(3.13)
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq195_HTML.gif matrix https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq196_HTML.gif with real eigenvalues https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq198_HTML.gif . If the functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq199_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq201_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq202_HTML.gif satisfies https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq204_HTML.gif .

Corollary 3.3.

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

Corollary 3.4.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq219_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq220_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq221_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq222_HTML.gif , and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq224_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq225_HTML.gif satisfy https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq226_HTML.gif Denote
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_Equ43_HTML.gif
(3.18)

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq227_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq229_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq230_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq231_HTML.gif and (3.4), where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq233_HTML.gif functional https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq234_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq235_HTML.gif , then (2.13) should be

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

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq236_HTML.gif is just a Hessian of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq238_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq239_HTML.gif satisfy (2.14). If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq240_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq242_HTML.gif ) has a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq243_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq244_HTML.gif satisfies https://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

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

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq246_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq247_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq248_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq249_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq250_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F287834/MediaObjects/13661_2009_Article_837_IEq252_HTML.gif and https://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

References

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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.