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Existence of Positive Solutions for Nonlinear Eigenvalue Problems
Boundary Value Problems volume 2010, Article number: 961496 (2010)
Abstract
We use a fixed point theorem in a cone to obtain the existence of positive solutions of the differential equation, ,
, with some suitable boundary conditions, where
is a parameter.
1. Introduction
We consider the existence of positive solutions of the following two-point boundary value problem:

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]).
Assume that

is an increasing function with respect to . If there exists a constant
such that

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.
(1)Let

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.
for
and
)
for
and
)
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
(1) if

(2)
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.
Suppose that there exist two distinct positive constants ,
and a function
with
and
such that


Then (BVP
λ
) has a positive solution with
between
and
if

where

Proof.
It is clear that (BVP
λ
) has a solution if, and only if,
is the solution of the operator equation

It follows from the definition of in our observation
and Lemma B that

Hence, , which implies
. Furthermore, it is easy to check that
is completely continuous. If there exists a
such that
, then we obtain the desired result. Thus, we may assume that

where and
. We now separate the rest proof into the following three steps.
Step 1.
It follows from the definitions of and
that, for
,

which implies

Hence, by (2.5),

which implies

Hence

We now claim that

In fact, if there exist and
such that
then, by (2.11),

which gives a contradiction. This proves that (2.13) holds. Thus, by Lemma C,

Step 2.
First, we claim that

Suppose to the contrary that there exist and
such that

It is clear that (2.17) is equivalent to

Since and
it follows that there exists a
such that

Let

Then . From
on
, we see that
on
on
and
on
. It follows from

and on
that

Hence,

Thus

This contradiction implies

Therefore, by Lemma C,

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.
It follows from the conclusion of Theorem 2.1 that the positive constant and nonnegative function
satisfy

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

Remark 2.3.
We now define

A simple calculation shows that

Then, we have the following results.
(i)Suppose that . Taking
, there exists
(
can be chosen small arbitrarily) such that

Hence,

It follows from Remark 2.2 that the hypothesis (2.2) of Theorem 2.1 is satisfied if .
(ii)Suppose that . Taking
, there exists
(
can be chosen large arbitrarily) such that

Hence,

which satisfies the hypothesis (2.1) of Theorem 2.1.
(iii)Suppose that . Taking
, there exists
(
can be chosen small arbitrarily) such that

Hence,

which satisfies the hypothesis (2.1) of Theorem 2.1.
(iv)Suppose that . Taking
, there exists a
(
can be chosen large arbitrarily) such that

Hence, we have the following two cases.
Case i.
Assume that is bounded, say

for some constant . Taking
(since
can be chosen large arbitrarily,
can be chosen large arbitrarily, too),

Case ii.
Assume that is unbounded, then there exist a
(
can be chosen large arbitrarily) and
such that

It follows from and (2.37) that

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

on
for some
and
.
Then, for , (BVP
λ
) has at least two positive solutions
and
such that

Proof.
It follows from Remark 2.3 that there exist two real numbers satisfying

Hence, by Theorem 2.1 and Remark 2.2, we see that for each , there exist two positive solutions
and
of (BVP
λ
) such that

Thus, we complete the proof.
Corollary 2.6.
Let

on
, for some
.
Then, for , (BVP
λ
) has at least two positive solutions
and
such that

Proof.
It follows from Remark 2.3 that there exist two real numbers satisfying

Hence, by Theorem 2.1 and Remark 2.2, we see that, for each , (BVP
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.
Consider the following boundary value problem:

Clearly,

If we take , then it follows from
of Corollary 2.4 that (BVP.1) has a solution if
.
Example 3.2.
Consider the following boundary value problem:

Clearly,

If we take , then it follows from
of Corollary 2.4 that (BVP.2) has a solution if
.
Example 3.3.
Consider the following boundary value problem:

Clearly, if we take and
,

Hence, it follows from Corollary 2.5 that (BVP.3) has two solutions if .
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Wang, SP., Wong, FH. & Kung, FK. Existence of Positive Solutions for Nonlinear Eigenvalue Problems. Bound Value Probl 2010, 961496 (2010). https://doi.org/10.1155/2010/961496
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Keywords
- Differential Equation
- Real Number
- Partial Differential Equation
- Ordinary Differential Equation
- Positive Constant