Existence of Solutions for -point Boundary Value Problems on a Half-Line
© Changlong Yu et al. 2009
Received: 5 April 2009
Accepted: 6 July 2009
Published: 17 August 2009
By using the Leray-Schauder continuation theorem, we establish the existence of solutions for -point boundary value problems on a half-line , where and are given.
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.
We use the space exists, exists with the norm , where is supremum norm on the half-line, and is absolutely integrable on with the norm .
and we suppose are the same signal in this paper and we always assume
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 ).
for each is measurable on
for almost every is continuous on ,
for each , there exists with on such that implies , for a.e. .
Therefore, the unique solution of (2.1) is which completes the proof.
Remark of Lemma 2.2 . Obviously satisfies the properties of a Green function, so we call the Green function of the corresponding homogeneous multipoint BVP of (2.1) on the half-line.
Therefore, we get the result.
Let are positive, and note , then for every , we have so that is, Because is continuous on the interval , there exists satisfying , where .
Theorem 2.6 (see ).
is uniformly bounded in ;
the functions from are equicontinuous on any compact interval of ;
the functions from are equiconvergent, that is, for any given , there exists a such that , for any .
3. Main Results
The main result of this paper is following.
Let be an S-Carathéodory function. Then, for each is completely continuous in .
where . Thus, .
so we can get .
We claim that is completely continuous in , that is, for each , is continuous in and maps a bounded subset of into a relatively compact set.
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.
According to Lemma 2.5, we know that for any , there exists satisfying . Hence, there are three cases as follow.
Case 1 ( ).
Set , which is independent of .
Case 2 ( ).
Set , which is independent of and is what we need.
Case 3 ( ).
and so for all .
Set and which is we need. So (1.2) has at least one solution.
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.