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) = ∑m−2 i=1 αiu(ηi), where 0 < η1 < η2 < ··· < ηm−2 ≤ 1/2, αi > 0 for i= 1,2, . . . ,m− 2 withm−2 i=1 αi < 1,m≥ 3. a : (0,1)→ [0,∞) is continuous, symmetric on the interval (0,1), and maybe singular at t = 0 and t = 1, f : [0,1]× [0,∞)→ [0,∞) is continuous, and f (·,x) is symmetric on the interval [0,1] for all x ∈ [0,∞) 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.