Open Access

Existence of Positive Solutions for Nonlinear Eigenvalue Problems

Boundary Value Problems20102010:961496

DOI: 10.1155/2010/961496

Received: 2 June 2009

Accepted: 2 February 2010

Published: 8 March 2010

Abstract

We use a fixed point theorem in a cone to obtain the existence of positive solutions of the differential equation, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq1_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq2_HTML.gif , with some suitable boundary conditions, where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq3_HTML.gif is a parameter.

1. Introduction

We consider the existence of positive solutions of the following two-point boundary value problem:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ1_HTML.gif
(BVP λa)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq4_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq5_HTML.gif are nonnegative constants, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq6_HTML.gif .

In the last thirty years, there are many mathematician considered the boundary value problem (BVP λ ) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq7_HTML.gif , see, for example, Chu et al. [1], Chu et al. [2], Chu and Zhau [3], Chu and Jiang [4], Coffman and Marcus [5], Cohen and Keller [6], Erbe [7], Erbe et al. [8], Erbe and Wang [9], Guo and Lakshmikantham [10], Iffland [11], Njoku and Zanolin [12], Santanilla [13].

In 1993, Wong [14] showed the following excellent result.

Theorem 1 A (see [14]).

Assume that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ2_HTML.gif
(1.1)
is an increasing function with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq8_HTML.gif . If there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq9_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ3_HTML.gif
(1.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq10_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq11_HTML.gif , then, there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq12_HTML.gif such that the boundary value problem (BVP λ ) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq13_HTML.gif has a positive solution in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq14_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq15_HTML.gif , while there is no such solution for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq16_HTML.gif in which https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq17_HTML.gif

Seeing such facts, we cannot but ask "whether or not we can obtain a similar conclusion for the boundary value problem (BVP λ )." We give a confirm answer to the question.

First, We observe the following statements.

(1)Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ4_HTML.gif
(1.3)

on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq18_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq19_HTML.gif is the Green's function of the differential equation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq20_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq21_HTML.gif with respect to the boundary value condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq22_HTML.gif .

(2) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq23_HTML.gif , is a cone in the Banach space with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq24_HTML.gif .

In order to discuss our main result, we need the follo wing useful lemmas which due to Lian et al. [15] and Guo and Lakshmikantham [10], respectively.

Lemma 1 B (see [10]).

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq25_HTML.gif be defined as in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq26_HTML.gif . Then, we have the following results.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq28_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq29_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq30_HTML.gif )

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq32_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq33_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq34_HTML.gif )

Lemma 1 C (see [10, Lemmas https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq35_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq36_HTML.gif ]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq37_HTML.gif be a real Banach space, and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq38_HTML.gif be a cone. Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq39_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq40_HTML.gif is completely continuous. Then

(1) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq41_HTML.gif if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ5_HTML.gif
(1.4)

(2) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq42_HTML.gif

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq43_HTML.gif is the fixed point index of a compact map https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq44_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq45_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq46_HTML.gif , with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq47_HTML.gif .

2. Main Results

Now, we can state and prove our main result.

Theorem 2.1.

Suppose that there exist two distinct positive constants https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq48_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq49_HTML.gif and a function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq50_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq51_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq52_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ6_HTML.gif
(2.1)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ7_HTML.gif
(2.2)
Then (BVP λ ) has a positive solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq53_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq54_HTML.gif between https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq55_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq56_HTML.gif if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ8_HTML.gif
(2.3)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ9_HTML.gif
(2.4)

Proof.

It is clear that (BVP λ ) has a solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq57_HTML.gif if, and only if, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq58_HTML.gif is the solution of the operator equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ10_HTML.gif
(2.5)
It follows from the definition of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq59_HTML.gif in our observation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq60_HTML.gif and Lemma B that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ11_HTML.gif
(2.6)
Hence, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq61_HTML.gif , which implies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq62_HTML.gif . Furthermore, it is easy to check that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq63_HTML.gif is completely continuous. If there exists a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq64_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq65_HTML.gif , then we obtain the desired result. Thus, we may assume that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ12_HTML.gif
(2.7)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq66_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq67_HTML.gif . We now separate the rest proof into the following three steps.

Step 1.

It follows from the definitions of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq68_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq69_HTML.gif that, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq70_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ13_HTML.gif
(2.8)
which implies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ14_HTML.gif
(2.9)
Hence, by (2.5),
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ15_HTML.gif
(2.10)
which implies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ16_HTML.gif
(2.11)
Hence
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ17_HTML.gif
(2.12)
We now claim that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ18_HTML.gif
(2.13)
In fact, if there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq71_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq72_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq73_HTML.gif then, by (2.11),
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ19_HTML.gif
(2.14)
which gives a contradiction. This proves that (2.13) holds. Thus, by Lemma C,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ20_HTML.gif
(2.15)

Step 2.

First, we claim that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ21_HTML.gif
(2.16)
Suppose to the contrary that there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq74_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq75_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ22_HTML.gif
(2.17)
It is clear that (2.17) is equivalent to
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ23_HTML.gif
(2.18)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq76_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq77_HTML.gif it follows that there exists a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq78_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ24_HTML.gif
(2.19)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ25_HTML.gif
(2.20)
Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq79_HTML.gif . From https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq80_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq81_HTML.gif , we see that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq82_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq83_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq84_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq85_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq86_HTML.gif . It follows from
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ26_HTML.gif
(2.21)
and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq87_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq88_HTML.gif that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ27_HTML.gif
(2.22)
Hence,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ28_HTML.gif
(2.23)
Thus
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ29_HTML.gif
(2.24)
This contradiction implies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ30_HTML.gif
(2.25)
Therefore, by Lemma C,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ31_HTML.gif
(2.26)

Step 3.

It follows from Steps (1) and (2) and the property of the fixed point index (see, for example, [10, Theorem https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq89_HTML.gif ]) that the proof is complete.

Remark 2.2.

It follows from the conclusion of Theorem 2.1 that the positive constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq90_HTML.gif and nonnegative function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq91_HTML.gif satisfy
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ32_HTML.gif
(2.27)

There are many functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq92_HTML.gif and positive constants https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq93_HTML.gif satisfying (2.27). For example, Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq94_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq95_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq96_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq97_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq98_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq99_HTML.gif and

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ33_HTML.gif
(2.28)

Remark 2.3.

We now define
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ34_HTML.gif
(2.29)
A simple calculation shows that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ35_HTML.gif
(2.30)

Then, we have the following results.

(i)Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq100_HTML.gif . Taking https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq101_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq102_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq103_HTML.gif can be chosen small arbitrarily) such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ36_HTML.gif
(2.31)
Hence,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ37_HTML.gif
(2.32)

It follows from Remark 2.2 that the hypothesis (2.2) of Theorem 2.1 is satisfied if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq104_HTML.gif .

(ii)Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq105_HTML.gif . Taking https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq106_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq107_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq108_HTML.gif can be chosen large arbitrarily) such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ38_HTML.gif
(2.33)
Hence,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ39_HTML.gif
(2.34)

which satisfies the hypothesis (2.1) of Theorem 2.1.

(iii)Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq109_HTML.gif . Taking https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq110_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq111_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq112_HTML.gif can be chosen small arbitrarily) such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ40_HTML.gif
(2.35)
Hence,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ41_HTML.gif
(2.36)

which satisfies the hypothesis (2.1) of Theorem 2.1.

(iv)Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq113_HTML.gif . Taking https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq114_HTML.gif , there exists a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq115_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq116_HTML.gif can be chosen large arbitrarily) such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ42_HTML.gif
(2.37)

Hence, we have the following two cases.

Case i.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq117_HTML.gif is bounded, say
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ43_HTML.gif
(2.38)
for some constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq118_HTML.gif . Taking https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq119_HTML.gif (since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq120_HTML.gif can be chosen large arbitrarily, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq121_HTML.gif can be chosen large arbitrarily, too),
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ44_HTML.gif
(2.39)

Case ii.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq122_HTML.gif is unbounded, then there exist a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq123_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq124_HTML.gif can be chosen large arbitrarily) and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq125_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ45_HTML.gif
(2.40)
It follows from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq126_HTML.gif and (2.37) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ46_HTML.gif
(2.41)

By Cases (i), (ii) and Remark 2.2, we see that the hypothesis (2.2) of Theorem 2.1 is satisfied if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq127_HTML.gif .

We immediately conclude the following corollaries.

Corollary 2.4.

(BVP λ ) has at least one positive solution for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq128_HTML.gif if one of the following conditions holds:

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq130_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq132_HTML.gif

Proof.

It follows from Remark 2.3 and Theorem 2.1 that the desired result holds, immediately.

Corollary 2.5.

Let

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq134_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq136_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq137_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq138_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq139_HTML.gif .

Then, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq140_HTML.gif , (BVP λ ) has at least two positive solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq141_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq142_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ47_HTML.gif
(2.42)

Proof.

It follows from Remark 2.3 that there exist two real numbers https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq143_HTML.gif satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ48_HTML.gif
(2.43)
Hence, by Theorem 2.1 and Remark 2.2, we see that for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq144_HTML.gif , there exist two positive solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq145_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq146_HTML.gif of (BVP λ ) such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ49_HTML.gif
(2.44)

Thus, we complete the proof.

Corollary 2.6.

Let

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq148_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq150_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq151_HTML.gif , for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq152_HTML.gif .

Then, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq153_HTML.gif , (BVP λ ) has at least two positive solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq154_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq155_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ50_HTML.gif
(2.45)

Proof.

It follows from Remark 2.3 that there exist two real numbers https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq156_HTML.gif satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ51_HTML.gif
(2.46)
Hence, by Theorem 2.1 and Remark 2.2, we see that, for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq157_HTML.gif , (BVP λ ) has two positive solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq158_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq159_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ52_HTML.gif
(2.47)

Thus, we completed the proof.

3. Examples

To illustrate the usage of our results, we present the following examples.

Example 3.1.

Consider the following boundary value problem:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ53_HTML.gif
(BVP.1)
Clearly,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ54_HTML.gif
(3.1)

If we take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq160_HTML.gif , then it follows from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq161_HTML.gif of Corollary 2.4 that (BVP.1) has a solution if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq162_HTML.gif .

Example 3.2.

Consider the following boundary value problem:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ55_HTML.gif
(BVP.2)
Clearly,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ56_HTML.gif
(3.2)

If we take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq163_HTML.gif , then it follows from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq164_HTML.gif of Corollary 2.4 that (BVP.2) has a solution if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq165_HTML.gif .

Example 3.3.

Consider the following boundary value problem:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ57_HTML.gif
(BVP.3)
Clearly, if we take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq166_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq167_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ58_HTML.gif
(3.3)

Hence, it follows from Corollary 2.5 that (BVP.3) has two solutions if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq168_HTML.gif .

Authors’ Affiliations

(1)
Holistic Education Center, Cardinal Tien College of Healthcare and Management
(2)
Department of Mathematics, National Taipei University of Education

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© Sheng-PingWang et al. 2010

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