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.

Abstract

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=1m−2aiu(ξi), where ϕ:ℝ→ℝ is an increasing homeomorphism and satisfies the relation ϕ(xy)=ϕ(x)ϕ(y) for x,y∈ℝ, 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.