- Research Article
- Open Access
© Changlong Yu et al. 2009
- Received: 5 April 2009
- Accepted: 6 July 2009
- Published: 17 August 2009
- Differential Equation
- Unique Solution
- Ordinary Differential Equation
- Functional Equation
- Green Function
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.
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 ).
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.
- Gupta CP: A note on a second order three-point boundary value problem. Journal of Mathematical Analysis and Applications 1994,186(1):277-281. 10.1006/jmaa.1994.1299MathSciNetView ArticleMATHGoogle Scholar
- Gupta CP, Trofimchuk SI: A sharper condition for the solvability of a three-point second order boundary value problem. Journal of Mathematical Analysis and Applications 1997,205(2):586-597. 10.1006/jmaa.1997.5252MathSciNetView ArticleMATHGoogle Scholar
- Ma R: Positive solutions for second-order three-point boundary value problems. Applied Mathematics Letters 2001,14(1):1-5. 10.1016/S0893-9659(00)00102-6MathSciNetView ArticleGoogle Scholar
- Guo Y, Ge W: Positive solutions for three-point boundary value problems with dependence on the first order derivative. Journal of Mathematical Analysis and Applications 2004,290(1):291-301. 10.1016/j.jmaa.2003.09.061MathSciNetView ArticleMATHGoogle Scholar
- O'Regan D: Theory of Singular Boundary Value Problems. World Scientific, River Edge, NJ, USA; 1994:xii+154.View ArticleMATHGoogle Scholar
- Agarwal RP, O'Regan D: Infinite Interval Problems for Differential, Difference and Integral Equations. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2001:x+341.View ArticleMATHGoogle Scholar
- Baxley JV: Existence and uniqueness for nonlinear boundary value problems on infinite intervals. Journal of Mathematical Analysis and Applications 1990,147(1):122-133. 10.1016/0022-247X(90)90388-VMathSciNetView ArticleMATHGoogle Scholar
- Jiang D, Agarwal RP:A uniqueness and existence theorem for a singular third-order boundary value problem on . Applied Mathematics Letters 2002,15(4):445-451. 10.1016/S0893-9659(01)00157-4MathSciNetView ArticleMATHGoogle Scholar
- Ma R: Existence of positive solution for second-order boundary value problems on infinite intervals. Applied Mathematics Letters 2003, 16: 33-39. 10.1016/S0893-9659(02)00141-6MathSciNetView ArticleMATHGoogle Scholar
- Bai C, Fang J: On positive solutions of boundary value problems for second-order functional differential equations on infinite intervals. Journal of Mathematical Analysis and Applications 2003,282(2):711-731. 10.1016/S0022-247X(03)00246-4MathSciNetView ArticleMATHGoogle Scholar
- Yan B, Liu Y: Unbounded solutions of the singular boundary value problems for second order differential equations on the half-line. Applied Mathematics and Computation 2004,147(3):629-644. 10.1016/S0096-3003(02)00801-9MathSciNetView ArticleMATHGoogle Scholar
- Tian Y, Ge W: Positive solutions for multi-point boundary value problem on the half-line. Journal of Mathematical Analysis and Applications 2007,325(2):1339-1349. 10.1016/j.jmaa.2006.02.075MathSciNetView ArticleMATHGoogle Scholar
- Tian Y, Ge W, Shan W: Positive solutions for three-point boundary value problem on the half-line. Computers & Mathematics with Applications 2007,53(7):1029-1039.MathSciNetView ArticleMATHGoogle Scholar
- Zima M: On positive solutions of boundary value problems on the half-line. Journal of Mathematical Analysis and Applications 2001,259(1):127-136. 10.1006/jmaa.2000.7399MathSciNetView ArticleMATHGoogle Scholar
- Lian H, Ge W: Solvability for second-order three-point boundary value problems on a half-line. Applied Mathematics Letters 2006,19(10):1000-1006. 10.1016/j.aml.2005.10.018MathSciNetView ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.