where are given. The authors examine and discuss these solutions.
Top1. Introduction
In this paper, we concern with the existence of multiple solutions for higher order boundary value problem
where
is a positive integer,
are given and
is the p-Laplacian operator, that is,
for
. Clearly,
is invertible with inverse
. Here
In recent years, the existence of positive solutions for nonlinear boundary value problems with p-Laplacian operator received wide attention. As we know, two point boundary value problems are used to describe a number of physical, biological and chemical phenomena. Recently, some authors have obtained some existence results of positive solutions of multi-points boundary value problems for second order ordinary differential equations (Wang & Ge, 2007; Yu, Wong, Yeh, & Lin, 2007; Zhao, Wang, & Ge, 2007; Zhou, & Su, 2007). In this paper, we establish the existence of positive solutions of general multi-points boundary value problem (BVP) and related results (Bai, Gui, & Ge, 2004; Guo & Lakshmikantham, 1988; Guo, Lakshmikantham, & Liu, 1996; He & Ge, 2004; Lian & Wong, 2000; Liu, 2002; Ma, 1999; Ma & Cataneda, 2001; Sun, Ge, & Zhao, 2007; Wang, 1997).
In order to abbreviate our discussion, throughout this paper, we assume
are both nondecreasing continuous and odd functions defined on 
and at least one of them satisfies the condition that there exists 
such that 
for all 
Top2. Preliminaries And Lemmas
Let
Then, B is a Banach space with norm 
And let
Obviously, K is a cone in B.
In order to discuss our results, we need the following some lemmas: