- Research Article
- Open Access

# Existence of Positive Solutions for Nonlinear Eigenvalue Problems

- Sheng-Ping Wang
^{1}Email author, - Fu-Hsiang Wong
^{2}and - Fan-Kai Kung
^{2}

**2010**:961496

https://doi.org/10.1155/2010/961496

© Sheng-PingWang et al. 2010

**Received:**2 June 2009**Accepted:**2 February 2010**Published:**8 March 2010

## Abstract

## Keywords

- Differential Equation
- Real Number
- Partial Differential Equation
- Ordinary Differential Equation
- Positive Constant

## 1. Introduction

where and are nonnegative constants, and .

In the last thirty years, there are many mathematician considered the boundary value problem (BVP_{
λ
}) with
, 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]).

where
for
, then, there exists
such that the boundary value problem (BVP_{
λ
}) with
has a positive solution in
for
, while there is no such solution for
in which

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.

on , then is the Green's function of the differential equation in with respect to the boundary value condition .

(2) , is a cone in the Banach space with .

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 be defined as in . Then, we have the following results.

Lemma 1 C (see [10, Lemmas and ]).

Let be a real Banach space, and let be a cone. Assume that and is completely continuous. Then

where is the fixed point index of a compact map , such that for , with respect to .

## 2. Main Results

Now, we can state and prove our main result.

Theorem 2.1.

Proof.

_{ λ }) has a solution if, and only if, is the solution of the operator equation

where and . We now separate the rest proof into the following three steps.

Step 1.

Step 2.

Step 3.

It follows from Steps (1) and (2) and the property of the fixed point index (see, for example, [10, Theorem ]) that the proof is complete.

Remark 2.2.

There are many functions and positive constants satisfying (2.27). For example, Suppose that and . Let on , then on and

Remark 2.3.

Then, we have the following results.

It follows from Remark 2.2 that the hypothesis (2.2) of Theorem 2.1 is satisfied if .

which satisfies the hypothesis (2.1) of Theorem 2.1.

which satisfies the hypothesis (2.1) of Theorem 2.1.

Hence, we have the following two cases.

Case i.

Case ii.

By Cases (i), (ii) and Remark 2.2, we see that the hypothesis (2.2) of Theorem 2.1 is satisfied if .

We immediately conclude the following corollaries.

Corollary 2.4.

(BVP_{
λ
}) has at least one positive solution for
if one of the following conditions holds:

Proof.

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

Corollary 2.5.

Let

Proof.

_{ λ }) such that

Thus, we complete the proof.

Corollary 2.6.

Let

Proof.

_{ λ }) has two positive solutions and such that

Thus, we completed the proof.

## 3. Examples

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

Example 3.1.

If we take , then it follows from of Corollary 2.4 that (BVP.1) has a solution if .

Example 3.2.

If we take , then it follows from of Corollary 2.4 that (BVP.2) has a solution if .

Example 3.3.

Hence, it follows from Corollary 2.5 that (BVP.3) has two solutions if .

## Authors’ Affiliations

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