- Open Access
Global structure of positive solutions for three-point boundary value problems
© Gu et al.; licensee Springer 2013
- Received: 29 April 2013
- Accepted: 8 July 2013
- Published: 25 July 2013
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.
- positive solution
- global continuous theorem
- differential equation
where , , and ; and , satisfies for .
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.
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.
then P is a cone.
We assume that
(H0) , , and .
Lemma 2.1 (see [, Lemma 2.1])
For the sake of convenience, we list the following hypotheses:
(H2) satisfies for .
(H3) , (superlinear).
(H4) , (sublinear).
Then for . Moreover, if for some , then for all .
Proof We only show that if for some , then for all .
If , then we have from [, Lemma 2] that the results hold.
We separate the proof into two cases: Case I: and Case II: .
This contradicts the hypothesis .
- (1)If , then and the concavity of u imply that , . Hence, we get that , and (since , we have that leads to . This contradicts (2.3)). Again, since u is concave, we have
If , then, adopting the same proof as in Case I, we get a contradiction.
- (3)If , then it follows that . In light of and the concavity of u, we get from (2.3) that
This is a contradiction.
Consequently, we get from Case I and Case II that the conclusion holds. □
Remark 2.1 If is positive, then we know from the proof in Lemma 2.2 that may only have zero point at and .
Proof If , then from [, Lemma 3.3] the conclusion holds.
If , then . This together with the concavity of u yields . Since , there exists such that .
- (2)If , then . Consequently, there exists such that
This completes the proof. □
Assume that (H0)-(H2) hold, then it is easy to verify that is well defined and completely continuous. We note that u is a positive solution of problem (1.1) if and only if on P.
By a positive solution of (1.1) we mean a solution of (1.1) which is positive on .
the equation has no solution on ;
has a continuum ℒ of solutions in , which connects the set with the set .
Lemma 3.1 [, Theorem 3.2]
where is defined by (3.2).
for some constant independent of n. Utilizing the Ascoli-Arzela theorem, we have that is a relatively compact set on . Assume, taking a subsequence if necessary, that in . Then and in .
contradicts . Therefore, the claim (3.3) holds.
This is a contradiction. Consequently, conclusion (3.8) holds.
Combining (3.3) and (3.8), we let . Thus, the result holds. □
Theorem 3.1 Assume that conditions (H0)-(H3) hold. Then contains a continuum which joins with .
Proof We divide the proof into four steps.
Step 1. We construct a continuum.
From Lemma 3.2, we know that has no solutions in . Therefore, from Lemma 2.4, there exists a continuum which joins with . Here is defined by (3.2), and and are defined by (3.1).
Since , we know that .
From Lemma 3.2, has no solution in . Again, using Lemma 2.4, we find a continuum which joins with . This contradicts (3.15). Therefore, the conclusion in (3.14) holds.
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.
- (1)If , then we obtain from the boundary condition of (3.18) that and . From the strict concavity of , it follows that
- (2)If , then (since , we know that and the strict concavity of imply on , which is a contradiction). Put
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.
Proof If , then we know that for is a trivial solution of (3.19).
Case I. .
Case II. .
If , then and imply that , . This contradicts (3.20).
This is a contradiction.
Therefore, we conclude that if , then problem (3.19) has no positive solutions. □
Lemma 4.1 [, Theorem 3.1]
where is defined by (3.2).
Adopting the same proof as in Lemma 3.2, we get a contradiction. Hence, conclusion (4.1) holds.
uniformly holds for . Again, applying the proof method as that in Lemma 3.2, we get a contradiction. Consequently, conclusion (4.2) holds.
If we let , then combining (4.1) and (4.2), we have that the result holds. □
Theorem 4.1 Let (H0)-(H2) and (H4) hold. Then contains a continuum which joins with .
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 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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