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.