Abstract
We are interested in the following singular boundary value problem:
where
Keywords:
singularity; global continuous theorem; solution of boundedness; fixed point index; positive solution1 Introduction
We are concerned with the second order nonlocal boundary value problem:
where
Integral boundary conditions and multipoint boundary conditions for differential equations come from many areas of applied mathematics and physics [17]. Recently, singular boundary value problems have been extensively considered in a lot of literature [1,2,5,8], since they model many physical phenomena including gas diffusion through porous media, nonlinear diffusion generated by nonlinear sources, chemically reacting systems as well as concentration in chemical or biological problems. In all these problems, positive solutions are very meaningful.
In [1,2], Webb and Infante considered the existence of positive solutions of nonlinear boundary value problem:
where hg are nonnegative functions, subjected to the nonlocal boundary conditions
Here
involving a Stieltjes integral with a signed measure, that is, A has bounded variation. They dealt with many boundary conditions given in the literature in a unified way by utilizing the fixed point index theory in cones.
Recently, many researchers were interested in the global structure of positive solutions for the nonlinear boundary value problem (see, e.g., [3,6,7]). In 2009, Ma and An [3] considered the problem (1.1). Assume that
(A0)
(A1)
(A2)
(A3)
They obtained the following main result:
Theorem 1.1 ([3], Theorem 4.1])
Assume that (A0)(A3) hold. Then there exists a component
for some
Here, ∑ is the closure of the set of positive solutions of (1.1) on
A natural problem arises: How can we consider the global structure of positive solutions
for the case
In this paper, we first obtain the global structure of positive solutions by the use
of global continuous theorem, and some a priori estimates. Applying the analysis technique,
we construct the lower and upper solutions. Those combined with the fixed point index
theory, the existence, multiplicity and nonexistence of positive solutions to (1.1)
in the case
This paper is arranged as follows. We will give some hypotheses and lemmas in Section 2. In Section 3, new criteria of the existence, multiplicity and nonexistence of a positive solution are obtained. Moreover, an example is given to illustrate our result.
2 Preliminaries and lemmas
Let X denote the Banach space
Define
Denote
Throughout this paper, we suppose that the following conditions hold:
(H0)
(H1)
(H2)
(H3)
Remark 2.1 It is easy to see from (H0) that
Define an operator
Assume that the conditions (H0)(H2) hold, then it is easy to verify that
Lemma 2.1 ([9] Global continuation theorem)
LetXbe a Banach space and letKbe an order cone inX. Consider the equation
where
Remark 2.2
(1) We note that u is a positive solution of the problem (1.1) if and only if
(2) If
Lemma 2.2 ([10])
LetXbe a Banach space, Kan order cone inXand
3 Main results
Lemma 3.1Let (H0)(H3) hold and let
Proof Suppose on the contrary that there exist a sequence
Denote
Choose
Since
This, together with (3.1) and (3.2), implies
Put
On the other hand, multiplying (1.1) by ψ and integrating by parts, we obtain that
leads to
i.e.,
This is a contradiction. □
Lemma 3.2Assume that the hypotheses (H0)(H3) hold. Then there exists
Proof Let u be a positive solution of (1.1) corresponding to μ. From the hypotheses (H2) and (H3), it follows that there exists
Let
and let
implies
This completes the proof. □
Lemma 3.3Assume that (H0)(H3) hold. Then we have
Proof Suppose this fails, that is, there exists
where
where
Hence, we find that
Thus, it implies that
Theorem 3.1Assume that the conditions (H0)(H3) hold and
Proof Define
From Lemma 2.1 and Remark 2.2, we can find that there exists an unbounded continuum
Next, we only show that the problem (1.1) has no positive solution for any
For the sake of obtaining the contradiction, we divide the proof into four steps.
Step 1. Constructing a modified boundary value problem.
Choose arbitrarily a constant
where
Denote
and
Indeed, using (3.6), we have
Define a set
where
Step 2. We will show that if u is a positive solution of (3.9), then
Let u be a positive solution of (3.9), then we claim that
Suppose this fails, that is,
Case I. From (3.7), it implies that there exists a constant
Since f is uniformly continuous on
where
and
This together with (3.11) and (3.12) leads to
This is a contradiction.
Case II. Let
Obviously, we have
We obtain a contradiction. In particular, if
This contradicts (3.13).
Case III. Since
Therefore, we conclude that the claim (3.10) holds.
Step 3.
Using (2.2), we define an operator
Then
where
Step 4. We conclude that the problem (1.1) has at least two positive solutions corresponding to μ.
Since the problem (1.1) is equivalent to the problem (3.9) on
On the other hand, from Lemma 3.2, we choose
Then it is easy to verify that
Hence, by the additivity property and (3.16), we have
Then we conclude that the problem (1.1) has at least two positive solutions corresponding to μ. □
Remark 3.1
(i) From the hypotheses (H2) and (H3), it implies that there exists
Let f attain its maximum at the point
(ii) If u is a positive solution of the equation (1.1) corresponding to μ, then we have
i.e., from (3.17),
Therefore, we get that
Corollary 3.1Assume that (H1)(H3) hold. Consider the followingmpoint boundary value problem
whereμis a positive parameter,
Proof In the boundary condition of (1.1), if we let
where
Then the boundary condition of (1.1) reduces to the mpoint boundary condition of (3.18). Applying the method of Theorem 3.1, we get the conclusion. □
Example 3.1 We consider the following singular boundary value problem:
where
Computing yields
We find that for any
Thus, it implies that (2.3) holds. It is easy to verify that the conditions (H2)
and (H3) hold. Therefore, by Theorem 3.1, we obtain that there exists a constant
Competing interests
The author declares that they have no competing interests.
Author’s contributions
The author typed, read and approved the final manuscript.
Acknowledgement
The author would like to thank the anonymous referees very much for helpful comments and suggestions which led to the improvement of presentation and quality of the work. The work was supported partly by NSCF of Tianyuan Youth Foundation (No. 11126125), K. C. Wong Magna Fund of Ningbo University, Subject Foundation of Ningbo University (No. xkl11044) and Hulan’s Excellent Doctor Foundation of Ningbo University.
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