Abstract
This paper is concerned with the existence, multiplicity, and nonexistence of positive solutions for nonhomogeneous mpoint boundary value problems with two parameters. The proof is based on the fixedpoint theorem, the upperlower solutions method, and the fixedpoint index.
MSC: 34B10, 34B18.
Keywords:
nonhomogeneous BVP; positive solutions; upperlower solutions; fixedpoint theorem; fixed point index1 Introduction
Many authors have studied the existence, nonexistence, and multiplicity of positive solutions for multipoint boundary value problems by using the fixedpoint theorem, the fixed point index theory, and the lower and upper solutions method. We refer the readers to the references [14]. Recently, Hao, Liu and Wu [5] studied the existence, nonexistence, and multiplicity of positive solutions for the following nonhomogeneous boundary value problems:
where
Inspired by the above references, the purpose of this paper is to study the following more general nonhomogeneous boundary value problems:
where λ, μ are positive parameters,
Theorem 1.1Assume the following conditions hold:
(H1)
(H2)
(H3)
And either
(H4) There exist constants
(H5)
If
Then on
For the proof of Theorem 1.1, we also need the following lemmas.
Lemma 1.2[6]
LetEbe a Banach space, Ka cone inEand Ω bounded open inE. Let
Lemma 1.3[6]
LetEbe a Banach space andKa cone inE. For
2 Preliminaries
Lemma 2.1[5]
Assume that
is given by
Moreover, the Green function satisfies the following properties:
(i)
(ii)
Lemma 2.2Assume that (H1)(H5) hold. If
Proof Integrating both sides of (1) from 0 to t twice and applying the boundary conditions, then we can obtain
Furthermore, by Lemma 2.1, we can obtain
□
Let E denote the Banach space
In addition, define a cone
where
Lemma 2.3If (H1)(H3) hold, then
The proof procedure of Lemma 2.3 is standard, so we omit it.
Now, we will establish the classical lower and upper solutions method for our problem.
As usual, we say that
Similarly, we define the upper solution
Lemma 2.4Let
Proof Define
It is clear to see that
For any
On the other hand, we also have
From above inequalities, we obtain that
Therefore, by Schauder’s fixed theorem, the operator T has a fixed point
3 Proof of Theorem 1.1
Lemma 3.1Assume (H1)(H5) hold and Σ be a compact subset of
Proof Suppose on the contrary that there exists a sequence
Then
Since Σ is compact, the sequence
and at least
Case (I). If
where L satisfies
Since
This is a contradiction.
Case (II). If
where M satisfies
Since
This is a contradiction. □
Lemma 3.2Assume (H1)(H4) hold. If (1) has a positive solution at
Proof Let
Lemma 3.3Assume (H1)(H5) hold. Then there exists
Proof Let
It is clear to see that
which implies that
Define a set S by
Then it follows from Lemma 3.3 that
Lemma 3.4Assume (H1)(H5) hold. Then
Proof Let
by (H4). Furthermore, we can obtain that
□
Lemma 3.5Assume (H1)(H5) hold. Then every chain inShas a unique supremum inS.
Lemma 3.6Assume (H1)(H5) hold. Then there exists a
Lemma 3.7Assume (H1)(H5) hold. Then there exists a continuous curve Γ separating
then
Lemma 3.8Let
Proof From (H4), there exists constant
Then by the uniform continuity of f and g on a compact set, there exist
for all
Let
and
From above inequalities, it is clear to see that
Proof of Theorem 1.1 From above lemmas, we need only to show the existence of the second positive solution
of (1) for
Let
Define the set
Then D is bounded open set in E and
for all
Now for some fixed λ and μ, it follows from assumption (H4) that there exists a
where L satisfies
Let
Then it follows from Lemma 3.1,
Moreover, for
Furthermore, we have
Thus,
By the additivity of the fixedpoint index,
which yields
Hence, T has at least one fixed point in
Example Consider the following boundary value problem:
where
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
In this paper, the author studies the existence, multiplicity, and nonexistence of positive solutions for nonhomogeneous mpoint boundary value problems with two parameters. The proof is based on the upperlower solutions method and fixedpoint index. All authors typed, read, and approved the final manuscript.
Acknowledgements
The authors would like to thank the referees for valuable comments and suggestions for improving this paper. The first author is supported financially by the Fundamental Research Funds for the Central Universities.
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