Boundary Value Problems
Volume 2011 (2011), Article ID 591219, 11 pages
Research Article

Multiple Positive Solutions for m-Point Boundary Value Problem on Time Scales

1Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu, 730050, China
2School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu, 730000, China

Received 29 May 2010; Accepted 6 August 2010

Academic Editor: Feliz Manuel Minhós

Copyright © 2011 Jie Liu and Hong-Rui Sun. 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.


The purpose of this article is to establish the existence of multiple positive solutions of the dynamic equation on time scales (ϕ(uΔ(t)))+h(t)f(t,u(t),uΔ(t))=0,  t(0,T)T, subject to the multi-point boundary condition uΔ(0)=0,  u(T)=i=1m2aiu(ξi), where ϕ: is an increasing homeomorphism and satisfies the relation ϕ(xy)=ϕ(x)ϕ(y) for x,y, which generalizes the usually p-Laplacian operator. An example applying the result is also presented. The main tool of this paper is a generalization of Leggett-Williams fixed point theorem, and the interesting points are that the nonlinearity f contains the first-order derivative explicitly and the operator ϕ is not necessarily odd.