Discrete Dynamics in Nature and Society
Volume 2008 (2008), Article ID 143040, 16 pages
doi:10.1155/2008/143040
Research Article
Positive Solutions for Third-Order Nonlinear p-Laplacian m-Point Boundary Value Problems on Time Scales
School of Mathematics and Information Science, Shandong University of Technology, Zibo, Shandong 255049, China
Received 7 August 2008; Revised 8 October 2008; Accepted 9 November 2008
Academic Editor: Leonid Shaikhet
Copyright © 2008 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 p-Laplacian m-point boundary value problems
on time scales: (ϕp(uΔ∇))∇+a(t)f(t,u(t))=0, t∈[0,T]T, βu(0)−γuΔ(0)=0, u(T)=∑i=1m−2aiu(ξi), ϕp(uΔ∇(0))=∑i=1m−2biϕp(uΔ∇(ξi)), where ϕp(s) is p-Laplacian operator, that is, ϕp(s)=|s|p−2s, p>1, ϕp−1=ϕq, 1/p+1/q=1, 0<ξ1<⋯<ξm−2<ρ(T). We obtain the existence of positive solutions by using fixed-point theorem in
cones. The conclusions in this paper essentially extend and improve the known results.