Abstract
In this paper, we are concerned with the threepoint boundary value problem for secondorder differential equations
where
MSC: 34B10, 34B18, 34G20.
Keywords:
positive solution; global continuous theorem; continuum; differential equation1 Introduction
In this paper, we consider the following threepoint boundary value problem for secondorder differential equations:
where
The existence and multiplicity of positive solutions for multipoint boundary value problems have been studied by several authors and many nice results have been obtained; see, for example, [16] and the references therein for more information on this problem. The multipoint boundary conditions of ordinary differential equations arose in different areas of applied mathematics and physics. In addition, they are often used to model many physical phenomena which include gas diffusion through porous media, nonlinear diffusion generated by nonlinear sources, chemically reacting systems, infectious diseases as well as concentration in chemical or biological problems. In all these problems, only positive solutions are very meaningful.
In 2009, Sun et al.[1] studied the threepoint boundary value problem
where
On the other hand, we note that the nice results in [1] only gave the existence and multiplicity of positive solutions, and if the parameter α is regarded as a variable, then an interesting problem as to what happens to the global structure of positive solutions of (1.2) was not considered. However, this relationship is very useful for computing the numerical solution of (1.2) as it can be used to guide the numerical work. For example, the global bifurcation of solutions for secondorder differential equations has been extensively studied in the literature, see [4,7,8].
Motivated by this, in this paper, we consider the threepoint boundary value problem
for secondorder differential equations (1.1) and make use of the LeraySchauder global
continuation theorem in the frame of techniques nicely employed by Ma and Thompson
[4] and convex analysis technique. We consider two cases
The rest of this paper is arranged as follows. In Section 2, we give Green’s function
and some lemmas. In Section 3, we consider the case
2 Preliminaries and lemmas
Let
Define a set by
then P is a cone.
We assume that
(H0)
Lemma 2.1 (see [[1], Lemma 2.1])
Suppose that condition (H0) holds and
has a unique solution
where
For the sake of convenience, we list the following hypotheses:
(H1)
(H2)
(H3)
(H4)
Lemma 2.2Assume that (H0) holds. Let
Then
Proof We only show that if
If
Next, we consider the case
We separate the proof into two cases: Case I:
Case I. If
then we find from the boundary conditions in (2.2) that
This together with the concavity of u leads to
This contradicts the hypothesis
Case II. Consider the case
(1) If
Consequently, we obtain that
(2) If
(3) If
and
where
leads to
This is a contradiction.
Consequently, we get from Case I and Case II that the conclusion holds. □
Remark 2.1 If
Lemma 2.3Let (H0) hold and let
and
Then there exists
Proof If
Next, we consider the case
(1) If
(2) If
and
The assumption
This completes the proof. □
From (2.1), we define an operator
Assume that (H0)(H2) hold, then it is easy to verify that
By a positive solution of (1.1) we mean a solution of (1.1) which is positive on
Denote by the closure of the set
in
Using the LeraySchauder global continuation theorem [[8], Theorem 14.C], Ma and Thompson [[4], Lemma 2.2] obtained the following result.
Lemma 2.4LetPbe a cone in a Banach spaceX. Let
(1) the equation
(2)
Then the set
has a continuum ℒ of solutions in
3 The superlinear case
For a given
From (H0)(H2) and Lemma 2.2, we get that
For any
From (H0)(H2) and Lemma 2.2, we have that
Lemma 3.1 [[1], Theorem 3.2]
Let conditions (H0)(H3) hold. Then there exist two constants
where
Proof Since
Lemma 3.2Assume that (H0)(H3) hold. Let
where
Proof First we claim that if
Suppose this fails, that is, there exists a sequence
We may suppose that
where
From Remark 2.1, we know that the set
This together with (3.4) yields
In light of Lemma 2.3, there exists
Applying the NewtonLeibniz formula, we find
Consequently, combining (3.5) and (3.6), we conclude that
for some constant
On the other hand, from
uniformly holds for all
This together with (3.7) and the Lebesgue dominated convergence theorem, we get
contradicts
Next, we prove that if
Suppose on the contrary that there exists a sequence
We define
and
Take
Since
This together with (3.9) and (3.10) yields
Put
On the other hand, multiplying (1.1) by ψ and integrating by parts, we find
leads to
Then we obtain
This is a contradiction. Consequently, conclusion (3.8) holds.
Combining (3.3) and (3.8), we let
Theorem 3.1Assume that conditions (H0)(H3) hold. Thencontains a continuum which joins
Proof We divide the proof into four steps.
Step 1. We construct a continuum.
For arbitrarily given
It follows from Lemma 3.1 and the excision property of the fixed point index that
From Lemma 3.2, we know that
Let be the closure of the set
and let
Since
Step 2. We show that there exists
If it is not true, then there exists
Taking
Then we know, from Lemma 3.1 and the excision property of the fixed point index, that
From Lemma 3.2,
Step 3. Let ζ be a continuum satisfying (3.14). We claim that
Suppose on the contrary that there exists a sequence
and
Adopting the same proof as in the second step in Lemma 3.2, we can find a contradiction. Hence, the result in (3.16) holds.
Step 4. Let ζ be a continuum satisfying (3.14). Next we show that
If it is not true, then we have
for some
and
From conditions (H0)(H2) and Lemma 2.2, we get that
(1) If
implies
This is a contradiction.
(2) If
Then
From the strict concavity of
a contradiction.
Consequently, the conclusion in (3.17) holds. □
Remark 3.1 In contrast to [[1], Theorem 3.2], we obtain the global structure and behavior of positive solutions, where the parameter α is regarded as a variable.
Theorem 3.2Suppose that
Then problem (3.19) has no positive solutions.
Proof If
Suppose on the contrary that there exists a positive solution
Therefore, we know from equation (3.19) that
(1) Case I.
Clearly, we know from the boundary conditions of (3.19) that
implies
i.e.,
contradicts the condition
(2) Case II.
If
If
then the concavity of u implies that
Thus
For
contradicts (3.21).
For
where
From the boundary condition of (3.19), it follows that
leads to
This is a contradiction.
Therefore, we conclude that if
4 The sublinear case
Lemma 4.1 [[1], Theorem 3.1]
Let conditions (H0)(H2) and (H4) hold. Then there exist two constants
where
Lemma 4.2Assume that (H0)(H2) and (H4) hold. Let
where
Proof First, we claim that if
Suppose this fails, that is, there exists a sequence
From conditions (H0)(H2) and Lemma 2.2, it follows that
The concavity of
Choose
Since
implies
Adopting the same proof as in Lemma 3.2, we get a contradiction. Hence, conclusion (4.1) holds.
Now, we show that if
Define the nondecreasing function
Since
Suppose that conclusion (4.2) fails, that is, there exists a sequence
Define
Since
uniformly holds for
If we let
Theorem 4.1Let (H0)(H2) and (H4) hold. Thencontains a continuum which joins
Proof Applying the method as in Theorem 3.1, we find from Lemma 2.2, Lemma 4.1 and Lemma 4.2
that there exists a continuum
and
Next, we only show that
Suppose on the contrary that there exists a sequence
Define
and the concavity of
Take
Using the same proof as in Lemma 3.2, we get a contradiction.
Hence, the conclusion holds. □
Remark 4.1 In contrast to [[1], Theorem 3.1], we obtain the global structure and behavior of positive solutions, where the parameter α is regarded as a variable.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed significantly in writing this paper. All authors read and approved the final manuscript.
Acknowledgements
The work was supported partly by NSFC (No. 11201248), K.C. Wong Magna Fund of Ningbo University and Ningbo Natural Science Foundation (No. 2012A610031).
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