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Unbounded Solutions of SecondOrder Multipoint Boundary Value Problem on the HalfLine
Boundary Value Problems volumeÂ 2010, ArticleÂ number:Â 236560 (2010)
Abstract
This paper investigates the secondorder multipoint boundary value problem on the halfline ,, , , , where , , , , and is continuous. We establish sufficient conditions to guarantee the existence of unbounded solution in a special function space by using nonlinear alternative of LeraySchauder type. Under the condition that is nonnegative, the existence and uniqueness of unbounded positive solution are obtained based upon the fixed point index theory and Banach contraction mapping principle. Examples are also given to illustrate the main results.
1. Introduction
In this paper, we consider the following secondorder multipoint boundary value problem on the halfline
where , and is continuous, in which .
The study of multipoint boundary value problems (BVPs) for secondorder differential equations was initiated by Bicadze and SamarskÄ [1] and later continued by II'in and Moiseev [2, 3] and Gupta [4]. Since then, great efforts have been devoted to nonlinear multipoint BVPs due to their theoretical challenge and great application potential. Many results on the existence of (positive) solutions for multipoint BVPs have been obtained, and for more details the reader is referred to [5â€“10] and the references therein. The BVPs on the halfline arise naturally in the study of radial solutions of nonlinear elliptic equations and models of gas pressure in a semiinfinite porous medium [11â€“13] and have been also widely studied [14â€“27]. When , BVP (1.1) reduces to the following threepoint BVP on the halfline:
where . Lian and Ge [16] only studied the solvability of BVP (1.2) by the LeraySchauder continuation theorem. When , , and nonlinearity is variable separable, BVP (1.1) reduces to the second order twopoint BVP on the halfline
Yan et al. [17] established the results of existence and multiplicity of positive solutions to the BVP (1.3) by using lower and upper solutions technique.
Motivated by the above works, we will study the existence results of unbounded (positive) solution for second order multipoint BVP (1.1). Our main features are as follows. Firstly, BVP (1.1) depends on derivative, and the boundary conditions are more general. Secondly, we will study multipoint BVP on infinite intervals. Thirdly, we will obtain the unbounded (positive) solution to BVP (1.1). Obviously, with the boundary condition in (1.1), if the solution exists, it is unbounded. Hence, we extend and generalize the results of [16, 17] to some degree. The main tools used in this paper are LeraySchauder nonlinear alternative and the fixed point index theory.
The rest of the paper is organized as follows. In Section 2, we give some preliminaries and lemmas. In Section 3, the existence of unbounded solution is established. In Section 4, the existence and uniqueness of positive solution are obtained. Finally, we formulate two examples to illustrate the main results.
2. Preliminaries and Lemmas
Denote , where . Let
For any , define
then is a Banach space with the norm (see [17]).
The ArzelaAscoli theorem fails to work in the Banach space due to the fact that the infinite interval is noncompact. The following compactness criterion will help us to resolve this problem.
Lemma 2.1 (see [17]).
Let . Then, is relatively compact in if the following conditions hold:
(a) is bounded in ;
(b) the functions belonging to and are locally equicontinuous on ;
(c) the functions from and are equiconvergent, at .
Throughout the paper we assume the following.
Suppose that , and there exist nonnegative functions with such that
, where
Denote
Lemma 2.2.
Supposing that with , then BVP
has a unique solution
where
in which , and for .
Proof.
Integrating the differential equation from to , one has
Then, integrating the above integral equation from to , noticing that and , we have
Since , it holds that
By using arguments similar to those used to prove Lemma 2.2 in [9], we conclude that (2.7) holds. This completes the proof.
Now, BVP (1.1) is equivalent to
Letting , (2.12) becomes
For , define operator by
Then,
Set
Remark 2.3.
is the Green function for the following associated homogeneous BVP on the halfline:
It is not difficult to testify that
Let us first give the following result of completely continuous operator.
Lemma 2.4.
Supposing that and hold, then is completely continuous.
Proof.

(1)
First, we show that is well defined.
For any , there exists such that . Then,
so
Similarly,
Further,
On the other hand, for any and , by Remark 2.3, we have
Hence, by , the Lebesgue dominated convergence theorem, and the continuity of , for any , we have
So, for any .
We can show that . In fact, by (2.23) and (2.24), we obtain
Hence, is well defined.

(2)
We show that is continuous.
Suppose , and . Then, as , and there exists such that . The continuity of implies that
as . Moreover, since
we have from the Lebesgue dominated convergence theorem that
Thus, is continuous.

(3)
We show that is relatively compact.
(a) Let be a bounded subset. Then, there exists such that for all . By the similar proof of (2.20) and (2.22), if , one has
which implies that is uniformly bounded.
(b) For any , if , we have
Thus, for any there exists such that if , then
Since is arbitrary, then and are locally equicontinuous on .
(c) For , from (2.27), we have
which means that and are equiconvergent at . By Lemma 2.1, is relatively compact.
Therefore, is completely continuous. The proof is complete.
Let be Banach space, be a bounded open subset of , and be a completely continuous operator. Then either there exist such that , or there exists a fixed point .
Let be a bounded open set in real Banach space , let be a cone of , and let be completely continuous. Suppose that
Then,
3. Existence Result
In this section, we present the existence of an unbounded solution for BVP (1.1) by using the LeraySchauder nonlinear alternative.
Theorem 3.1.
Suppose that conditions hold. Then BVP (1.1) has at least one unbounded solution.
Proof.
Since , by , we have , a.e. , which implies that . Set
From Lemmas 2.2 and 2.4, BVP (1.1) has a solution if and only if is a fixed point of in . So, we only need to seek a fixed point of in .
Suppose such that . Then
Therefore,
which contradicts . By Lemma 2.5, has a fixed point . Letting , boundary conditions imply that is an unbounded solution of BVP (1.1).
4. Existence and Uniqueness of Positive Solution
In this section, we restrict the nonlinearity and discuss the existence and uniqueness of positive solution for BVP (1.1).
Define the cone as follows:
Lemma 4.1.
Suppose that and hold. Then, is completely continuous.
Proof.
Lemma 2.4 shows that is completely continuous, so we only need to prove . Since , and from Remark 2.3, we have
Then,
Therefore, .
Theorem 4.2.
Suppose that conditions and hold and the following condition holds:
suppose that and there exist nonnegative functions with such that
Then, BVP (1.1) has a unique unbounded positive solution.
Proof.
We first show that implies . By (4.4), we have
By Lemma 4.1, is completely continuous. Let . Then, . Set
For any , by (4.5), we have
Therefore, for all , that is, for any . Then, Lemma 2.6 yields , which implies that has a fixed point . Let . Then, is an unbounded positive solution of BVP (1.1).
Next, we show the uniqueness of positive solution for BVP (1.1). We will show that is a contraction. In fact, by (4.4), we have
So, is indeed a contraction. The Banach contraction mapping principle yields the uniqueness of positive solution to BVP (1.1).
5. Examples
Example 5.1.
Consider the following BVP:
We have
Let
Then, , and it is easy to prove that is satisfied. By direct calculations, we can obtain that . By Theorem 3.1, BVP (5.1) has an unbounded solution.
Example 5.2.
Consider the following BVP:
In this case, we have
Let
Then, . By Theorem 4.2, BVP (5.4) has a unique unbounded positive solution.
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Acknowledgments
The authors are grateful to the referees for valuable suggestions and comments. The first and second authors were supported financially by the National Natural Science Foundation of China (11071141, 10771117) and the Natural Science Foundation of Shandong Province of China (Y2007A23, Y2008A24). The third author was supported financially by the Australia Research Council through an ARC Discovery Project Grant.
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Liu, L., Hao, X. & Wu, Y. Unbounded Solutions of SecondOrder Multipoint Boundary Value Problem on the HalfLine. Bound Value Probl 2010, 236560 (2010). https://doi.org/10.1155/2010/236560
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DOI: https://doi.org/10.1155/2010/236560