- Open Access
Existence of positive solutions for singular fourth-order three-point boundary value problems
© Sun and Zhu; licensee Springer 2013
- Received: 24 July 2012
- Accepted: 20 February 2013
- Published: 8 March 2013
In this article, we consider the boundary value problem , , subject to the boundary conditions and . In this setting, and are constants and is a parameter. By imposing a sufficient structure on the nonlinearity , we deduce the existence of at least one positive solution to the problem. The novelty in our setting lies in the fact that may be singular at and . Our results here are achieved by making use of the Krasnosel’skii fixed point theorem. We conclude with examples illustrating our results and the improvements that they present.
MSC:34B15, 34B25, 34B18.
- positive solution
- boundary value problems
where , are constants, is a parameter, may be singular at and/or . Here, by a positive solution we mean a function which is positive on and satisfies problem (1.1).
where is a closed set with measure zero and the nonlinear term may be singular for . The author showed the existence of n positive solutions by constructing a suitable integral equation and applying fixed point theorems on a cone.
The author obtained the existence and nonexistence of positive solutions by applying the Guo-Krasnosel’skii fixed point theorem and Schauder’s fixed point theorem.
where the nonlinear term may be singular at , and . The author presented the existence of a positive solution by using the fixed point index theorem and the properties of Green’s function.
where is a constant.
Inspired and motivated by the works mentioned above, we deal with the existence and nonexistence of positive solutions to problem (1.1) by making use of the fixed point theorem together with the properties of Green’s function. The main features of the paper are as follows. Firstly, we apply the Taylor expansion formula to prove a lemma, and then we give a comparison lemma and construct a special cone. Secondly, we present the existence of positive solutions for problem (1.1). To our best knowledge, no paper has considered problem (1.1). The arguments are based upon the fixed point theorem for the special cone.
The paper is organized as follows. In Section 2, we give some properties of Green’s function associated with problem (1.1) and construct a suitable cone and transform problem (1.1) into an integral equation. In Section 3, we discuss the existence of at least one positive solution for problem (1.1).
Let be a Banach space of all continuous functions with the norm , .
Throughout the paper, we assume that
(H1) is continuous.
The proof is complete. □
It is well known that problem (1.1) has a positive solution if and only if u is a fixed point of A.
Lemma 2.3 Suppose that (H1)∼(H3) hold. Then .
The proof is complete. □
Lemma 2.4 Suppose that (H1)∼(H3) hold. Then is completely continuous.
Proof For any , we have .
By the Arzela-Ascoli theorem, we know that is completely continuous.
It shows that a completely continuous operator converges to an operator A uniformly on . Hence A is continuous.
Hence A is uniformly bounded.
Therefore A is equicontinuous. Consequently, A is completely continuous. □
, and , , or
, and , .
Then T has a fixed point in .
Theorem 3.1 Suppose that (H1), (H2) and (H3) hold. In addition, assume that the following conditions hold:
Then problem (1.1) has at least one positive solution for λ small enough, and problem (1.1) has no positive solution for λ large enough.
From (H4), there exists a constant such that for .
By Lemma 2.5 we know that problem (1.1) has at least one positive solution.
Hence as .
which is a contradiction. The proof is complete. □
Remark 3.1 The conclusion of Theorem 3.1 also holds if .
Theorem 3.2 Suppose that (H1), (H2) and (H3) hold. In addition, assume that
Then problem (1.1) has at least one positive solution for any .
It follows from Lemma 2.5 that problem (1.1) has at least one positive solution. □
Now, we give examples to illustrate the main results in the paper.
Then problem (4.1) has at least one positive solution if and .
Take . Notice, for any fixed , that and for and .
Obviously, conditions (H1)∼(H3) are satisfied.
Thus, the existence of a positive solution follows from Theorem 3.1 if and .
Then problem (4.3) has at least one positive solution.
Take . Notice, for any fixed , that and .
Obviously, conditions (H1)∼(H3) are satisfied.
Thus, the existence of a positive solution follows from Theorem 3.2.
The authors would like to express their thanks to the editor of the journal and the anonymous referees for their careful reading of the first draft of the manuscript and making many helpful comments and suggestions which improved the presentation of the paper. The authors were supported financially by the Foundation of Shanghai Municipal Education Commission (Grant No. DYL201105).
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