© Changlong Yu et al. 2009
Received: 5 April 2009
Accepted: 6 July 2009
Published: 17 August 2009
Multipoint boundary value problems (BVPs) for second-order differential equations in a finite interval have been studied extensively and many results for the existence of solutions, positive solutions, multiple solutions are obtained by use of the Leray-Schauder continuation theorem, Guo-Krasnosel'skii fixed point theorem, and so on; for details see [1–4] and the references therein.
where have the same signal, and are given. We first present the Green function for second-order multipoint BVPs on the half-line and then give the existence results for (1.2) using the properties of this Green function and the Leray-Schauder continuation theorem.
2. Preliminary Results
In this section, we present some definitions and lemmas, which will be needed in the proof of the main results.
Definition 2.1 (see ).
Therefore, we get the result.
Theorem 2.6 (see ).
3. Main Results
The main result of this paper is following.
Combining (3.9) (3.13), we can see that is continuous. Let be a bounded subset; it is easy to prove that is uniformly bounded. In the same way, we can prove (3.5),(3.6), and (3.12), we can also show that is equicontinuous and equiconvergent. Thus, by Theorem 2.6, is completely continuous. The proof is completed.
Proof of Theorem 3.1.
The Natural Science Foundation of Hebei Province (A2009000664) and the Foundation of Hebei University of Science and Technology (XL200759) are acknowledged.
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