Existence of symmetric positive solutions for an m-point boundary value problem

We study the second-order m-point boundary value problem u ''(t) + a(t) f (t, u(t)) = 0, 0 < t < 1, u(0) = u(1) = Sigma(m-2)(i=1) alpha(i)u(eta(i)), where 0 < eta(1) < eta(2) <center dot center dot center dot < eta(m-2) <= 1/2, alpha(i) > 0 for i = 1, 2,..., m - 2 with Sigma(m-2)(i=1) alpha(i) < 1, m >= 3. a : (0,1).[0, infinity) is continuous, symmetric on the interval (0, 1), and maybe singular at t = 0 and t = 1, f : [0,1] x [0, infinity) -> [0, infinity) is continuous, and f (., x) is symmetric on the interval [0,1] for all x is an element of [0, infinity) and satisfies some appropriate growth conditions. By using Krasnoselskii's fixed point theorem in a cone, we get some existence results of symmetric positive solutions. Copyright (c) 2007 Y. Sun and X. Zhang.