Discrete Dynamics in Nature and Society
Volume 2009 (2009), Article ID 123283, 15 pages
doi:10.1155/2009/123283
Research Article

Triple Positive Solutions for Third-Order m-Point Boundary Value Problems on Time Scales

Jian Liu1 and Fuyi Xu2

1School of Statistics and Mathematics Science, Shandong Economics University, Jinan, Shandong 250014, China
2School of Science, Shandong University of Technology, Zibo, Shandong 255049, China

Received 25 February 2009; Accepted 25 May 2009

Academic Editor: Leonid Shaikhet

Copyright © 2009 Jian Liu and Fuyi Xu. 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

We study the following third-order m-point boundary value problems on time scales (φ(uΔ∇))∇+a(t)f(u(t))=0, t∈[0,T]T, u(0)=∑i=1m−2biu(ξi), uΔ(T)=0, φ(uΔ∇(0))=∑i=1m−2ciφ(uΔ∇(ξi)), where φ:R→R is an increasing homeomorphism and homomorphism and φ(0)=0, 0<ξ1<⋯<ξm−2<ρ(T). We obtain the existence of three positive solutions by using fixed-point theorem in cones. The conclusions in this paper essentially extend and improve the known results.