Advances in Difference Equations
Volume 2009 (2009), Article ID 169321, 14 pages
Research Article

The Existence of Positive Solutions for Third-Order p-Laplacian m-Point Boundary Value Problems with Sign Changing Nonlinearity on Time Scales

School of Science, Shandong University of Technology, Zibo, Shandong 255049, China

Received 25 February 2009; Revised 10 April 2009; Accepted 2 June 2009

Academic Editor: Alberto Cabada

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.


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=1m2biu(ξi), uΔ(T)=0, ϕp(uΔ(0))=i=1m2ciϕp(uΔ(ξi)), where ϕp(s) is p-Laplacian operator, that is, ϕp(s)=|s|p2s, p>1, ϕp1=ϕq,1/p+1/q=1, 0<ξ1<<ξm2<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.