This paper is concerned with the existence, multiplicity, and nonexistence of positive solutions for nonhomogeneous m-point boundary value problems with two parameters. The proof is based on the fixed-point theorem, the upper-lower solutions method, and the fixed-point index.
MSC: 34B10, 34B18.
Keywords:nonhomogeneous BVP; positive solutions; upper-lower solutions; fixed-point theorem; fixed point index
Many authors have studied the existence, nonexistence, and multiplicity of positive solutions for multipoint boundary value problems by using the fixed-point theorem, the fixed point index theory, and the lower and upper solutions method. We refer the readers to the references [1-4]. Recently, Hao, Liu and Wu  studied the existence, nonexistence, and multiplicity of positive solutions for the following nonhomogeneous boundary value problems:
where , (), , , may be singular at and/or . They showed that there exists a positive number such that the problem has at least two positive solutions for , at least one positive solution for and no solution for by using the Krasnosel’skii-Guo fixed-point theorem, the upper-lower solutions method, and the topological degree theory.
Inspired by the above references, the purpose of this paper is to study the following more general nonhomogeneous boundary value problems:
Theorem 1.1Assume the following conditions hold:
If, then there exists a bounded and continuous curve Γ separatinginto two disjoint subsetsandsuch that (1) has at least two positive solutions for, one positive solution for, and no solution for. Moreover, letbe the parametric representation of Γ, where
For the proof of Theorem 1.1, we also need the following lemmas.
is given by
Moreover, the Green function satisfies the following properties:
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 with the norm . A function is said to be a solution of (1) if satisfies (1). Moreover, from Lemma 2.2, it is clear to see that is a solution of (1) is equivalent to the fixed point of the operator T defined as
The proof procedure of Lemma 2.3 is standard, so we omit it.
On the other hand, we also have
3 Proof of Theorem 1.1
where L satisfies
This is a contradiction.
where M satisfies
This is a contradiction. □
Proof Let be the solution of Eq. (1) at , then be the upper solution of (1) at with . Since or , is not a solution of (1), but it is the lower solution of (1) at . Therefore, by Lemma 2.4, we obtain the result. □
which implies that is an upper solution of (3) at . On the other hand, 0 is a lower solution of (1) and . By (H3), 0 is not a solution of (1). Hence, (1) has a positive solution at , Lemma 3.2 now implies the conclusion of Lemma 3.3. □
Define a set S by
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 asuch (1) has a positive solution atfor all, no solution atfor all. Similarly, there exists asuch that (1) has a positive solution atfor all, and no solution atfor all.
Lemma 3.7Assume (H1)-(H5) hold. Then there exists a continuous curve Γ separatinginto two disjoint subsetsandsuch thatis bounded andis unbounded, Eq. (1) has at least one solution for, and no solution for. The functionis nonincreasing, that is, if
Define the set
where L satisfies
Then it follows from Lemma 3.1,
Furthermore, we have
By the additivity of the fixed-point index,
Example Consider the following boundary value problem:
The authors declare that they have no competing interests.
In this paper, the author studies the existence, multiplicity, and nonexistence of positive solutions for nonhomogeneous m-point boundary value problems with two parameters. The proof is based on the upper-lower solutions method and fixed-point index. All authors typed, read, and approved the final manuscript.
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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