Copyright © 2009 Fuyi Xu and Zhaowei Meng. 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]Tκ, u(0)=∑i=1m−2biu(ξi), uΔ(T)=0, ϕp(uΔ∇(0))=∑i=1m−2ciϕp(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. In particular, the nonlinear term f(t,u) is allowed to change sign. The conclusions in this paper essentially extend and improve the known results.