© Y. Li and Z.Wei. 2010
Received: 22 January 2010
Accepted: 3 June 2010
Published: 20 June 2010
We study the existence of multiple positive solutions for th-order multipoint boundary value problem. , , , , where , , . We obtained the existence of multiple positive solutions by applying the fixed point theorems of cone expansion and compression of norm type and Leggett-Williams fixed-point theorem. The results obtained in this paper are different from those in the literature.
By using the fixed point theorems of cone expansion and compression of norm type, Yang and Wei in  also obtained the existence of at least one positive solutions for the BVP (1.1) if . This work is motivated by Ma (see ). This method is simpler than that of . In addition, Eloe and Ahmad in  had solved successfully the existence of positive solutions to the BVP (1.1) if = 1. Hao et al. in  had discussed the existence of at least two positive solutions for the BVP (1.1) by applying the Krasonse'skii-Guo fixed point theorem on cone expansion and compression if = 1. However, there are few papers dealing with the existence of multiple positive solutions for th-order multipoint boundary value problem.
In this paper, we study the existence of at least two positive solutions associated with the BVP (1.1) by applying the fixed point theorems of cone expansion and compression of norm type if and the existence of at least three positive solutions for BVP (1.1) by using Leggett-Williams fixed-point theorem. The results obtained in this paper are different from those in the literature and essentially improve and generalize some well-known results (see [1–8]).
The rest of the paper is organized as follows. In Section 2, we present several lemmas. In Section 3, we give some preliminaries and the fixed point theorems of cone expansion and compression of norm type. The existence of at least two positive solutions for the BVP (1.1) is formulated and proved in Section 4. In Section 5, we give Leggett-Williams fixed-point theorem and obtain the existence of at least three positive solutions for the BVP (1.1).
2. Several Lemmas
Lemma 2.2 (see ).
Lemma 2.3 (see ).
We omit the proof Lemma 2.3 here and you can see the detail of Theorem in .
Lemma 2.4 (see ).
For convenience, we make the following assumptions:
From Lemmas 2.2–2.4, we have the following result.
Lemma 3.1 (see ).
The aim of the following section is to establish some simple criteria for the existence of multiple positive solutions of the BVP (1.1), which gives a positive answer to the questions stated above. The key tool in our approach is the following fixed point theorem, which is a useful method to prove the existence of positive solutions for differential equations, for example [2–5, 9].
4. The Existence of Two Positive Solutions
The proof is complete.
The proof is complete.
The proof of Corollary 4.4 is similar to that of Corollary 4.2; so we omit it.
5. The Existence of Three Positive Solutions
Lemma 5.1 (see ).
Now, we establish the existence conditions of three positive solutions for the BVP (1.1).
Then the boundary value problem (1.1) has at least three positive solutions.
The proof is complete.
The authors are grateful to the referee's valuable comments and suggestions. The project is supported by the Natural Science Foundation of Anhui Province (KJ2010B226), The Excellent Youth Foundation of Anhui Province Office of Education (2009SQRZ169), and the Natural Science Foundation of Suzhou University (2009yzk17)
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