In this paper, we are concerned with the three-point boundary value problem for second-order differential equations
where , , and ; and , satisfies for . The existence of the continuum of a positive solution is established by utilizing the Leray-Schauder global continuation principle. Furthermore, the interval of α about the nonexistence of a positive solution is also given.
MSC: 34B10, 34B18, 34G20.
Keywords:positive solution; global continuous theorem; continuum; differential equation
In this paper, we consider the following three-point boundary value problem for second-order differential equations:
The existence and multiplicity of positive solutions for multi-point boundary value problems have been studied by several authors and many nice results have been obtained; see, for example, [1-6] and the references therein for more information on this problem. The multi-point 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. studied the three-point boundary value problem
where is a parameter, , , and . Based on Krein-Rutmann theorems and the fixed point index theory, they not only established the criteria of the existence and multiplicity of a positive solution, but also obtained the parameter μ in relation with the nonlinear term f and the first eigenvalue of the linear operator.
On the other hand, we note that the nice results in  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 second-order differential equations has been extensively studied in the literature, see [4,7,8].
Motivated by this, in this paper, we consider the three-point boundary value problem for second-order differential equations (1.1) and make use of the Leray-Schauder global continuation theorem in the frame of techniques nicely employed by Ma and Thompson  and convex analysis technique. We consider two cases , and , , and establish the existence of continuum of positive solutions, where and . Moreover, the interval of parameter α about the nonexistence of positive solutions is also given. Our main results extend and improve the corresponding results [1,3,4]. In contrast to [, Theorem 3.1 and Theorem 3.2], we obtain the global structure and behavior of positive solutions, where the parameter α is regarded as a variable.
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 , , and give the existence of the continuum of positive solutions and the interval of parameter α about the nonexistence of positive solutions. In Section 4, we study the case , , and give the existence of global continuum of positive solutions.
2 Preliminaries and lemmas
Define a set by
then P is a cone.
We assume that
Lemma 2.1 (see [, Lemma 2.1])
has a unique solution
For the sake of convenience, we list the following hypotheses:
If , then we have from [, Lemma 2] that the results hold.
then we find from the boundary conditions in (2.2) that
This together with the concavity of u leads to
This is a contradiction.
Consequently, we get from Case I and Case II that the conclusion holds. □
Proof If , then from [, Lemma 3.3] the conclusion holds.
This completes the proof. □
Denote by the closure of the set
Then the set
3 The superlinear case
From (H0)-(H2) and Lemma 2.2, we get that
From (H0)-(H2) and Lemma 2.2, we have that
Lemma 3.1 [, Theorem 3.2]
This together with (3.4) yields
Applying the Newton-Leibniz formula, we find
Consequently, combining (3.5) and (3.6), we conclude that
This together with (3.7) and the Lebesgue dominated convergence theorem, we get
This together with (3.9) and (3.10) yields
On the other hand, multiplying (1.1) by ψ and integrating by parts, we find
Then we obtain
This is a contradiction. Consequently, conclusion (3.8) holds.
Proof We divide the proof into four steps.
Step 1. We construct a continuum.
It follows from Lemma 3.1 and the excision property of the fixed point index that
Let be the closure of the set
Then we know, from Lemma 3.1 and the excision property of the fixed point index, that
Step 3. Let ζ be a continuum satisfying (3.14). We claim that
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
This is a contradiction.
Consequently, the conclusion in (3.17) holds. □
Remark 3.1 In contrast to [, Theorem 3.2], we obtain the global structure and behavior of positive solutions, where the parameter α is regarded as a variable.
Then problem (3.19) has no positive solutions.
then the concavity of u implies that
From the boundary condition of (3.19), it follows that
This is a contradiction.
4 The sublinear case
Lemma 4.1 [, Theorem 3.1]
From conditions (H0)-(H2) and Lemma 2.2, it follows that
Adopting the same proof as in Lemma 3.2, we get a contradiction. Hence, conclusion (4.1) holds.
Next, we only show that
Using the same proof as in Lemma 3.2, we get a contradiction.
Hence, the conclusion holds. □
Remark 4.1 In contrast to [, Theorem 3.1], we obtain the global structure and behavior of positive solutions, where the parameter α is regarded as a variable.
The authors declare that they have no competing interests.
All authors contributed significantly in writing this paper. All authors read and approved the final manuscript.
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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