- Research Article
- Open Access
© Y. Ji and Y. Guo. 2009
- Received: 5 July 2009
- Accepted: 30 October 2009
- Published: 4 November 2009
We consider the existence of countably many positive solutions for nonlinear th-order three-point boundary value problem , , , , , where , for some and has countably many singularities in . The associated Green's function for the th-order three-point boundary value problem is first given, and growth conditions are imposed on nonlinearity which yield the existence of countably many positive solutions by using the Krasnosel'skii fixed point theorem and Leggett-Williams fixed point theorem for operators on a cone.
- Banach Space
- Unique Solution
- Ordinary Differential Equation
- Functional Equation
- Open Subset
The existence of positive solutions for nonlinear second-order and higher-order multipoint boundary value problems has been studied by several authors, for example, see [1–12] and the references therein. However, there are a few papers dealing with the existence of positive solutions for the th-order multipoint boundary value problems with infinitely many singularities. Hao et al.  discussed the existence and multiplicity of positive solutions for the following th-order nonlinear singular boundary value problems:
where , , may be singular at and/or . Hao et al. established the existence of at least two positive solution for the boundary value problems if is either superlinear or sublinear by applying the Krasnosel'skii-Guo theorem on cone expansion and compression.
In , Kaufmann and Kosmatov showed that there exist countably many positive solutions for the two-point boundary value problems with infinitely many singularities of following form:
In , Ji and Guo proved the existence of countably many positive solutions for the th-order ordinary differential equation
where , , , , , ( , ), , for some and has countably many singularities in . We show that the problem (1.5) has countably many solutions if and satisfy some suitable conditions. Our approach is based on the Krasnosel'skii fixed point theorem and Leggett-Williams fixed point theorem in cones.
Suppose that the following conditions are satisfied.
Assuming that satisfies the conditions - (we cite [15, Example ] to verify existence of ) and imposing growth conditions on the nonlinearity , it will be shown that problem (1.5) has infinitely many solutions.
The paper is organized as follows. In Section 2, we provide some necessary background material such as the Krasnosel'skii fixed-point theorem and Leggett-Williams fixed point theorem in cones. In Section 3, the associated Green's function for the th-order three-point boundary value problem is first given and we also look at some properties of the Green's function associated with problem (1.5). In Section 4, we prove the existence of countably many positive solutions for problem (1.5) under suitable conditions on and . In Section 5, we give two simple examples to illustrate the applications of obtained results.
The following Krasnosel'skii fixed point theorem and Leggett-Williams fixed point theorem play an important role in this paper.
Theorem 2.4 (, Krasnosel'skii fixed point theorem).
is a completely continuous operator such that, either
Theorem 2.5 (, Leggett-Williams fixed point theorem).
Theorem 2.6 (, Holder's inequality).
To prove the main results, we need the following lemmas.
Lemma 3.1 (see ).
Lemma 3.2 (see ).
Lemma 3.3 (see ).
We omit the proof as it is immediate from Lemma 3.4 and (3.4).
Next, we prove that (3.15) holds.
Theorems 2.4 and 2.5 require the operator to be completely continuous and cone preserving. If is continuous and compact, then it is completely continuous. The next lemma shows that for and that is continuous and compact.
For convenience, we denote
For convenience, we denote
In this section, we cite an example (see ) to verify existence of , and two simple examples are presented to illustrate the applications for obtained conclusion of Theorems 4.1 and 4.2.
In [8–12], the existence of solutions for local or nonlocal boundary value problems of higher-order nonlinear ordinary (fractional) differential equations that has been treated did not discuss problems with singularities. In , the singularity only allowed to appear at and/or , the existence and multiplicity of positive solutions were asserted under suitable conditions on . Although, [14, 15] seem to have considered the existence of countably many positive solutions for the second-order and higher-order boundary value problems with infinitely many singularities in . However, in , only the boundary conditions or have been considered. It is clear that the boundary conditions of Examples 5.1 and 5.2 are and . Hence, we generalize second-order and higher-order multipoint boundary value problem.
The project is supported by the Natural Science Foundation of Hebei Province (A2009000664), the Foundation of Hebei Education Department (2008153), the Foundation of Hebei University of Science and Technology (XL2006040), and the National Natural Science Foundation of PR China (10971045).
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