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Existence of positive solutions for singular fourth-order three-point boundary value problems
Advances in Difference Equations volume 2013, Article number: 51 (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.
In this paper, we consider the following nonlinear singular fourth-order three-point boundary value problem:
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).
The theory of boundary value problems for ordinary differential equations arises in different areas of applied mathematics, physics and so on. The existence of positive solutions for boundary value problems has become an important area of investigation and received a great deal of attention in recent years (see [1–20] and the references cited therein). In , by making use of the fixed point theorem and degree theory, Bai and Wang proved the existence, uniqueness and multiplicity of positive solutions for the following fourth-order two-point boundary value problem:
In , Yao studied the following nonlinear fourth-order ordinary differential equation:
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.
In , Sun considered the following third-order boundary value problems:
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.
In , Zhang and Wang studied the following nonlinear singular fourth-order boundary value problem:
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.
In , by applying the Krasnosel’skii fixed point theorem, Graef, Qian and Yang established the existence and nonexistence of positive solutions for the following fourth-order three-point boundary value problem:
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).
2 Preliminary lemmas
Let be a Banach space of all continuous functions with the norm , .
Throughout the paper, we assume that
(H1) is continuous.
(H2) There exists a continuous function such that
(H3) There exists a continuous function such that
Lemma 2.1 Suppose that and . Then the linear boundary value problem
has a unique positive solution, which can be expressed by
Proof In fact, if is a solution of problem (2.1), by the Taylor expansion formula, we have
which together with the boundary condition implies and
The proof is complete. □
Lemma 2.2 For all , we have
Proof If , then
If , then
Define a cone by
then K is a positive cone in . Denote
Fix . Define an operator by
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 .
Proof From (H2) and (H3), we know that
On the other hand, for any , we have , , and
The proof is complete. □
Lemma 2.4 Suppose that (H1)∼(H3) hold. Then is completely continuous.
Proof For any , we have .
Then, from (H2) and (H3), we have for and . Let
It is easy to see that is a continuous function on and is bounded on any bounded set. Define
By the Arzela-Ascoli theorem, we know that is completely continuous.
Let . For , we know that
It shows that a completely continuous operator converges to an operator A uniformly on . Hence A is continuous.
Suppose that is a bounded set, then there exists such that for any . From (H3), we know that for . Then we have
Hence A is uniformly bounded.
On the other hand, for any , we know that
Therefore A is equicontinuous. Consequently, A is completely continuous. □
Let E be a Banach space, and let be a cone in E. Assume that and are open subsets of E with and . Let be a completely continuous operator such that either
, and , , or
, and , .
Then T has a fixed point in .
3 Main results
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.
Proof For small enough, let
From (H4), there exists a constant such that for .
Let , . For any , we get
On the other hand, let
From (H5), there exists such that for . Let , and let . For any and , we know that and
which implies that
By Lemma 2.5 we know that problem (1.1) has at least one positive solution.
For λ large enough, we prove that problem (1.1) has no positive solution. Otherwise, there exists with such that problem (1.1) has a positive solution , then we get
Hence as .
Again from (H5), there exist and such that
where M is defined by (3.2). Let n be large enough. Choose such that . Thus, we get
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 .
Proof From (H7), there exist constants and such that
where M is defined by (3.2). Let . For any , we know that
On the other hand, from (H6), there exists a constant such that for , where N is defined by (3.1). Since is continuous on , there exists such that for . Choose
Let . For any , we get
Thus, for any and , we know that
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.
Consider the following boundary value problem:
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.
Now, for any fixed , (H4) and (H5) follow immediately from
Thus, the existence of a positive solution follows from Theorem 3.1 if and .
Consider the following boundary value problem:
Then problem (4.3) has at least one positive solution.
Take . Notice, for any fixed , that and .
Obviously, conditions (H1)∼(H3) are satisfied.
Now, for any fixed , (H6) and (H7) follow immediately from
Thus, the existence of a positive solution follows from Theorem 3.2.
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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).
The authors declare that they have no competing interests.
The authors declare that the work was realized in collaboration with same responsibility. All authors read and approved the final manuscript.