Open Access

Nodal Solutions for a Class of Fourth-Order Two-Point Boundary Value Problems

Boundary Value Problems20102010:570932

DOI: 10.1155/2010/570932

Received: 18 February 2010

Accepted: 27 April 2010

Published: 31 May 2010

Abstract

We consider the fourth-order two-point boundary value problem https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq1_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq2_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq3_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq4_HTML.gif is a parameter, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq5_HTML.gif is given constant, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq6_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq7_HTML.gif on any subinterval of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq8_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq9_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq10_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq11_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq12_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq13_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq14_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq15_HTML.gif . By using disconjugate operator theory and bifurcation techniques, we establish existence and multiplicity results of nodal solutions for the above problem.

1. Introduction

The deformations of an elastic beam in equilibrium state with fixed both endpoints can be described by the fourth-order ordinary differential equation boundary value problem
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq16_HTML.gif is continuous, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq17_HTML.gif is a parameter. Since the problem (1.1) cannot transform into a system of second-order equation, the treatment method of second-order system does not apply to the problem (1.1). Thus, existing literature on the problem (1.1) is limited. In 1984, Agarwal and chow [1] firstly investigated the existence of the solutions of the problem (1.1) by contraction mapping and iterative methods, subsequently, Ma and Wu [2] and Yao [3, 4] studied the existence of positive solutions of this problem by the Krasnosel'skii fixed point theorem on cones and Leray-Schauder fixed point theorem. Especially, when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq18_HTML.gif , Korman [5] investigated the uniqueness of positive solutions of the problem (1.1) by techniques of bifurcation theory. However, the existence of sign-changing solution for this problem have not been discussed.

In this paper, applying disconjugate operator theory and bifurcation techniques, we consider the existence of nodal solution of more general the problem:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ2_HTML.gif
(1.2)

under the assumptions:

(H1) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq20_HTML.gif is a parameter, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq21_HTML.gif is given constant,

(H2) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq23_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq24_HTML.gif on any subinterval of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq25_HTML.gif ,

(H3) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq27_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq28_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq29_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ3_HTML.gif
(1.3)

for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq30_HTML.gif .

However, in order to use bifurcation technique to study the nodal solutions of the problem (1.2), we firstly need to prove that the generalized eigenvalue problem
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ4_HTML.gif
(1.4)
(where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq31_HTML.gif satisfies (H2)) has an infinite number of positive eigenvalues
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ5_HTML.gif
(1.5)
and each eigenvalue corresponding an essential unique eigenfunction https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq32_HTML.gif which has exactly https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq33_HTML.gif simple zeros in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq34_HTML.gif and is positive near 0. Fortunately, Elias [6] developed a theory on the eigenvalue problem
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ6_HTML.gif
(1.6)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ7_HTML.gif
(1.7)
and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq35_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq36_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq37_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq38_HTML.gif are called the quasi-derivatives of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq39_HTML.gif . To apply Elias's theory, we have to prove that (1.4) can be rewritten to the form of (1.6), that is, the linear operator
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ8_HTML.gif
(1.8)
has a factorization of the form
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ9_HTML.gif
(1.9)
on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq40_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq41_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq42_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq43_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq44_HTML.gif if and only if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ10_HTML.gif
(1.10)

This can be achieved under (H1) by using disconjugacy theory in [7].

The rest of paper is arranged as follows: in Section 2, we state some disconjugacy theory which can be used in this paper, and then show that (H1) implies the equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ11_HTML.gif
(1.11)

is disconjugacy on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq45_HTML.gif , moreover, we establish some preliminary properties on the eigenvalues and eigenfunctions of the generalized eigenvalue problem (1.4). Finally in Section 3, we state and prove our main result.

Remark 1.1.

For other results on the existence and multiplicity of positive solutions and nodal solutions for boundary value problems of ordinary differential equations based on bifurcation techniques, see Ma [812], An and Ma [13], Yang [14] and their references.

2. Preliminary Results

Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ12_HTML.gif
(2.1)

be https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq46_HTML.gif th-order linear differential equation whose coefficients https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq47_HTML.gif are continuous on an interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq48_HTML.gif .

Definition 2.1 (see [7, Definition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq49_HTML.gif , page 2]).

Equation (2.1) is said to be disconjugate on an interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq50_HTML.gif if no nontrivial solution has https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq51_HTML.gif zeros on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq52_HTML.gif , multiple zeros being counted according to their multiplicity.

Lemma 2.2 (see [7, Theorem https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq53_HTML.gif , page 3]).

Equation (2.1) is disconjugate on a compact interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq54_HTML.gif if and only if there exists a basis of solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq55_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ13_HTML.gif
(2.2)
on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq56_HTML.gif . A disconjugate operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq57_HTML.gif can be written as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ14_HTML.gif
(2.3)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq58_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ15_HTML.gif
(2.4)

and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq59_HTML.gif

Lemma 2.3 (see [7, Theorem https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq60_HTML.gif , page 9]).

Green's function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq61_HTML.gif of the disconjugate Equation (2.3) and the two-point boundary value conditions
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ16_HTML.gif
(2.5)
satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ17_HTML.gif
(2.6)

Now using Lemmas 2.2 and 2.3, we will prove some preliminary results.

Theorem 2.4.

Let (H1) hold. Then

(i) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq62_HTML.gif is disconjugate on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq63_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq64_HTML.gif has a factorization
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ18_HTML.gif
(2.7)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq65_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq66_HTML.gif

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq67_HTML.gif if and only if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ19_HTML.gif
(2.8)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ20_HTML.gif
(2.9)

Proof.

We divide the proof into three cases.

Case 1.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq68_HTML.gif . The case is obvious.

Case 2.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq69_HTML.gif .

In the case, take
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ21_HTML.gif
(2.10)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq70_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq71_HTML.gif is a positive constant. Clearly, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq72_HTML.gif and then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ22_HTML.gif
(2.11)
It is easy to check that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq73_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq74_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq75_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq76_HTML.gif form a basis of solutions of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq77_HTML.gif . By simple computation, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ23_HTML.gif
(2.12)

Clearly, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq78_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq79_HTML.gif

By Lemma 2.2, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq80_HTML.gif is disconjugate on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq81_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq82_HTML.gif has a factorization
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ24_HTML.gif
(2.13)
and accordingly
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ25_HTML.gif
(2.14)

Using (2.14), we conclude that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq83_HTML.gif is equivalent to (2.8).

Case 3.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq84_HTML.gif .

In the case, take
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ26_HTML.gif
(2.15)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq85_HTML.gif .

It is easy to check that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq86_HTML.gif ,   https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq87_HTML.gif ,   https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq88_HTML.gif ,   https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq89_HTML.gif form a basis of solutions of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq90_HTML.gif . By simple computation, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ27_HTML.gif
(2.16)

From https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq91_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq92_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq93_HTML.gif , so https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq94_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq95_HTML.gif

By Lemma 2.2, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq96_HTML.gif is disconjugate on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq97_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq98_HTML.gif has a factorization
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ28_HTML.gif
(2.17)
and accordingly
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ29_HTML.gif
(2.18)

Using (2.18), we conclude that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq99_HTML.gif is equivalent to (2.8).

This completes the proof of the theorem.

Theorem 2.5.

Let (H1) hold and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq100_HTML.gif satisfy (H2). Then

(i) Equation (1.4) has an infinite number of positive eigenvalues
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ30_HTML.gif
(2.19)

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq101_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq102_HTML.gif

(iii) To each eigenvalue there corresponding an essential unique eigenfunction https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq103_HTML.gif which has exactly https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq104_HTML.gif simple zeros in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq105_HTML.gif and is positive near 0.

(iv) Given an arbitrary subinterval of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq106_HTML.gif , then an eigenfunction which belongs to a sufficiently large eigenvalue change its sign in that subinterval.

(v) For each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq107_HTML.gif , the algebraic multiplicity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq108_HTML.gif is 1.

Proof.

(i)–(iv) are immediate consequences of Elias [6, Theorems https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq109_HTML.gif ] and Theorem 2.4. we only prove (v).

Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ31_HTML.gif
(2.20)
with
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ32_HTML.gif
(2.21)
To show (v), it is enough to prove
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ33_HTML.gif
(2.22)
Clearly
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ34_HTML.gif
(2.23)
Suppose on the contrary that the algebraic multiplicity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq110_HTML.gif is greater than 1. Then there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq111_HTML.gif , and subsequently
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ35_HTML.gif
(2.24)
for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq112_HTML.gif . Multiplying both sides of (2.24) by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq113_HTML.gif and integrating from 0 to 1, we deduce that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ36_HTML.gif
(2.25)

which is a contradiction!

Theorem 2.6 (Maximum principle).

Let (H1) hold. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq114_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq115_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq116_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq117_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq118_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq119_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ37_HTML.gif
(2.26)

Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq120_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq121_HTML.gif .

Proof.

When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq122_HTML.gif , the homogeneous problem
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ38_HTML.gif
(2.27)
has only trivial solution. So the boundary value problem (2.26) has a unique solution which may be represented in the form
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ39_HTML.gif
(2.28)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq123_HTML.gif is Green's function.

By Theorem 2.4 and Lemma 2.3 (take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq124_HTML.gif ), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ40_HTML.gif
(2.29)

that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq125_HTML.gif

Using (2.28), when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq126_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq127_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq128_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq129_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq130_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq131_HTML.gif .

3. Statement of the Results

Theorem 3.1.

Let (H1), (H2), and (H3) hold. Assume that for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq132_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ41_HTML.gif
(3.1)

Then there are at least https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq133_HTML.gif nontrivial solutions of the problem (1.2). In fact, there exist solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq134_HTML.gif such that for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq135_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq136_HTML.gif has exactly https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq137_HTML.gif simple zeros on the open interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq138_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq139_HTML.gif and there exist solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq140_HTML.gif such that for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq141_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq142_HTML.gif has exactly https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq143_HTML.gif simple zeros on the open interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq144_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq145_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq146_HTML.gif with the norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq147_HTML.gif Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ42_HTML.gif
(3.2)

with the norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq148_HTML.gif Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq149_HTML.gif is completely continuous, here https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq150_HTML.gif is given as in (2.20).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq151_HTML.gif be such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ43_HTML.gif
(3.3)
here https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq152_HTML.gif . Clearly
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ44_HTML.gif
(3.4)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ45_HTML.gif
(3.5)
then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq153_HTML.gif is nondecreasing and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ46_HTML.gif
(3.6)
Let us consider
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ47_HTML.gif
(3.7)

as a bifurcation problem from the trivial solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq154_HTML.gif .

Equation (3.7) can be converted to the equivalent equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ48_HTML.gif
(3.8)

Further we note that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq155_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq156_HTML.gif near 0 in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq157_HTML.gif .

In what follows, we use the terminology of Rabinowitz [15].

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq158_HTML.gif under the product topology. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq159_HTML.gif denote the set of function in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq160_HTML.gif which have exactly https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq161_HTML.gif interior nodal (i.e., nondegenerate) zeros in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq162_HTML.gif and are positive near https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq163_HTML.gif , set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq164_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq165_HTML.gif . They are disjoint and open in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq166_HTML.gif . Finally, let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq167_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq168_HTML.gif .

The results of Rabinowitz [13] for (3.8) can be stated as follows: for each integer https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq169_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq170_HTML.gif , there exists a continuum https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq171_HTML.gif of solutions of (3.8), joining https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq172_HTML.gif to infinity in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq173_HTML.gif . Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq174_HTML.gif .

Notice that we have used the fact that if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq175_HTML.gif is a nontrivial solution of (3.7), then all zeros of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq176_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq177_HTML.gif are simply under (H1), (H2), and (H3).

In fact, (3.7) can be rewritten to
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ49_HTML.gif
(3.9)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ50_HTML.gif
(3.10)

clearly https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq178_HTML.gif satisfies (H2). So Theorem 2.5(iii) yields that all zeros of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq179_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq180_HTML.gif are simple.

Proof of Theorem 3.1.

We only need to show that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ51_HTML.gif
(3.11)
Suppose on the contrary that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ52_HTML.gif
(3.12)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ53_HTML.gif
(3.13)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq181_HTML.gif   joins https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq182_HTML.gif to infinity in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq183_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq184_HTML.gif is the unique solutions of (3.7) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq185_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq186_HTML.gif , there exists a sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq187_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq188_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq189_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq190_HTML.gif . We may assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq191_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq192_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq193_HTML.gif . From the fact
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ54_HTML.gif
(3.14)
we have that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ55_HTML.gif
(3.15)

Furthermore, since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq194_HTML.gif is completely continuous, we may assume that there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq195_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq196_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq197_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq198_HTML.gif .

Since
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ56_HTML.gif
(3.16)
we have from (3.15) and (3.6) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ57_HTML.gif
(3.17)
that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ58_HTML.gif
(3.18)
By (H2), (H3), and (3.17) and the fact that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq199_HTML.gif , we conclude that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq200_HTML.gif , and consequently
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_Equ59_HTML.gif
(3.19)

By Theorem 2.6, we know that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq201_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq202_HTML.gif . This means https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq203_HTML.gif is the first eigenvalue of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq204_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq205_HTML.gif is the corresponding eigenfunction. Hence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq206_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq207_HTML.gif is open and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq208_HTML.gif , we have that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq209_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq210_HTML.gif large. But this contradict the assumption that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq211_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq212_HTML.gif , so (3.12) is wrong, which completes the proof.

Declarations

Acknowledgments

This work is supported by the NSFC (no. 10671158), the Spring-sun program (no. Z2004-1-62033), SRFDP (no. 20060736001), the SRF for ROCS, SEM (2006[ https://static-content.springer.com/image/art%3A10.1155%2F2010%2F570932/MediaObjects/13661_2010_Article_938_IEq213_HTML.gif ]), NWNU-KJCXGC-SK0303-23, and NWNU-KJCXGC-03-69.

Authors’ Affiliations

(1)
Department of Mathematics, Northwest Normal University
(2)
College of Physical Education, Northwest Normal University

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Copyright

© J. Xu and X. Han. 2010

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