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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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ1_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ2_HTML.gif)
is an increasing function with respect to . If there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ3_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ4_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ5_HTML.gif)
(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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ6_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ7_HTML.gif)
Then (BVP
λ
) has a positive solution with
between
and
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ8_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ9_HTML.gif)
Proof.
It is clear that (BVP
λ
) has a solution if, and only if,
is the solution of the operator equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ10_HTML.gif)
It follows from the definition of in our observation
and Lemma B that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ11_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ12_HTML.gif)
where and
. We now separate the rest proof into the following three steps.
Step 1.
It follows from the definitions of and
that, for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ13_HTML.gif)
which implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ14_HTML.gif)
Hence, by (2.5),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ15_HTML.gif)
which implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ16_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ17_HTML.gif)
We now claim that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ18_HTML.gif)
In fact, if there exist and
such that
then, by (2.11),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ19_HTML.gif)
which gives a contradiction. This proves that (2.13) holds. Thus, by Lemma C,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ20_HTML.gif)
Step 2.
First, we claim that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ21_HTML.gif)
Suppose to the contrary that there exist and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ22_HTML.gif)
It is clear that (2.17) is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ23_HTML.gif)
Since and
it follows that there exists a
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ24_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ25_HTML.gif)
Then . From
on
, we see that
on
on
and
on
. It follows from
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ26_HTML.gif)
and on
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ27_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ28_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ29_HTML.gif)
This contradiction implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ30_HTML.gif)
Therefore, by Lemma C,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ31_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ32_HTML.gif)
There are many functions and positive constants
satisfying (2.27). For example, Suppose that
and
. Let
on
, then
on
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ33_HTML.gif)
Remark 2.3.
We now define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ34_HTML.gif)
A simple calculation shows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ35_HTML.gif)
Then, we have the following results.
(i)Suppose that . Taking
, there exists
(
can be chosen small arbitrarily) such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ36_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ37_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ38_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ39_HTML.gif)
which satisfies the hypothesis (2.1) of Theorem 2.1.
(iii)Suppose that . Taking
, there exists
(
can be chosen small arbitrarily) such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ40_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ41_HTML.gif)
which satisfies the hypothesis (2.1) of Theorem 2.1.
(iv)Suppose that . Taking
, there exists a
(
can be chosen large arbitrarily) such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ42_HTML.gif)
Hence, we have the following two cases.
Case i.
Assume that is bounded, say
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ43_HTML.gif)
for some constant . Taking
(since
can be chosen large arbitrarily,
can be chosen large arbitrarily, too),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ44_HTML.gif)
Case ii.
Assume that is unbounded, then there exist a
(
can be chosen large arbitrarily) and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ45_HTML.gif)
It follows from and (2.37) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ46_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq130_HTML.gif)
![](http://media.springernature.com/full/springer-static/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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq134_HTML.gif)
on
for some
and
.
Then, for , (BVP
λ
) has at least two positive solutions
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ47_HTML.gif)
Proof.
It follows from Remark 2.3 that there exist two real numbers satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ48_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ49_HTML.gif)
Thus, we complete the proof.
Corollary 2.6.
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq148_HTML.gif)
on
, for some
.
Then, for , (BVP
λ
) has at least two positive solutions
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ50_HTML.gif)
Proof.
It follows from Remark 2.3 that there exist two real numbers satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ51_HTML.gif)
Hence, by Theorem 2.1 and Remark 2.2, we see that, for each , (BVP
λ
) has two positive solutions
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ52_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ53_HTML.gif)
Clearly,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ54_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ55_HTML.gif)
Clearly,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ56_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ57_HTML.gif)
Clearly, if we take and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ58_HTML.gif)
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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DOI: https://doi.org/10.1155/2010/961496