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

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.