Boundary Value Problems
Volume 2009 (2009), Article ID 584145, 19 pages
doi:10.1155/2009/584145
Research Article

Several Existence Theorems of Multiple Positive Solutions of Nonlinear m-Point BVP for an Increasing Homeomorphism and Homomorphism on Time Scales

Wei Han1 and Shugui Kang2

1Department of Mathematics, North University of China, Taiyuan, Shanxi 030051, China
2Institute of Applied Mathematics, Shanxi Datong University, Datong, Shanxi 037009, China

Received 24 July 2009; Accepted 29 November 2009

Academic Editor: Kanishka Perera

Copyright © 2009 Wei Han and Shugui Kang. 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.

Abstract

By using fixed point theorems in cones, the existence of multiple positive solutions is considered for nonlinear m-point boundary value problem for the following second-order boundary value problem on time scales (ϕ(uΔ))∇+a(t)f(t,u(t))=0, t∈(0,T), ϕ(uΔ(0))=∑i=1m−2aiϕ(uΔ(ξi)), u(T)=∑i=1m−2biu(ξi), where ϕ:R→R is an increasing homeomorphism and homomorphism and ϕ(0)=0. Some new results are obtained for the existence of twin or an arbitrary odd number of positive solutions of the above problem by applying Avery-Henderson and Leggett-Williams fixed point theorems, respectively. In particular, our criteria generalize and improve some known results by Ma and Castaneda (2001). We must point out for readers that there is only the p-Laplacian case for increasing homeomorphism and homomorphism. As an application, one example to demonstrate our results is given.