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

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 multipoint boundary condition uΔ 0 0, u T ∑m−2 i 1 aiu 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.