Multiple Positive Solutions for m-Point Boundary Value Problem on Time Scales. (10th August 2010) - Liu, Jie; Sun, Hong-Rui
Abstract : The purpose of this article is to establish the existence of multiple positive solutions of the dynamic equation on time scales( ϕ ( u Δ ( t ) ) ) ∇ + h ( t ) f ( t, u ( t ), u Δ ( t ) ) = 0,    t ∈ ( 0, T ) 𝕋, subject to the multi-point boundary conditionu Δ ( 0 ) = 0,    u ( T ) = ∑ i = 1 m − 2 a i u ( ξ i ), whereϕ : ℝ → ℝ is an increasing homeomorphism and satisfies the relationϕ ( x y ) = ϕ ( x ) ϕ ( y ) forx, 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 operatorϕ is not necessarily odd.