Existence and multiplicity of symmetric positive solutions for three-point boundary value problem

In this paper, we are concerned with the existence and multiplicity of symmetric positive solutions for the following second-order three-point boundary value problem u (t)+ a(t)f t,u(t) = 0, 0 < t <1, u(t) = u(1−t), u (0)−u (1) = u(1/2), where a : (0, 1) → [0,∞) is symmetric on (0, 1) and maybe singular at t = 0 and t = 1, f : [0, 1] × [0,∞) → [0,∞) is continuous and f (·,u) is symmetric on [0, 1] for all u ∈ [0,∞). Growth conditions are imposed on f which yield the existence of at least one or at least two positive solutions. Our proof based on Krasnoselskii’s fixed point theorem in a cone. © 2006 Elsevier Inc. All rights reserved