Unbounded Solutions of Second-Order Multipoint Boundary Value Problem on the Half-Line
© Lishan Liu et al. 2010
Received: 14 May 2010
Accepted: 11 October 2010
Published: 18 October 2010
This paper investigates the second-order multipoint boundary value problem on the half-line , , , , , 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 Leray-Schauder 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.
where , and is continuous, in which .
Yan et al.  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 multi-point 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 multi-point 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 Leray-Schauder 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
then is a Banach space with the norm (see ).
The Arzela-Ascoli 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 ).
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.
in which , and for .
By using arguments similar to those used to prove Lemma 2.2 in , we conclude that (2.7) holds. This completes the proof.
Let us first give the following result of completely continuous operator.
Supposing that and hold, then is completely continuous.
First, we show that is well defined.
So, for any .
We show that is continuous.
We show that is relatively compact.
which implies that is uniformly bounded.
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 .
3. Existence Result
In this section, we present the existence of an unbounded solution for BVP (1.1) by using the Leray-Schauder nonlinear alternative.
Suppose that conditions hold. Then BVP (1.1) has at least one unbounded solution.
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 .
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).
Suppose that and hold. Then, is completely continuous.
Suppose that conditions and hold and the following condition holds:
Then, BVP (1.1) has a unique unbounded positive solution.
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).
So, is indeed a contraction. The Banach contraction mapping principle yields the uniqueness of positive solution to BVP (1.1).
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.
Then, . By Theorem 4.2, BVP (5.4) has a unique unbounded positive solution.
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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