Positive solutions for some second-order four-point boundary value problems

In this paper, the second-order four-point boundary value problem x (t) +λh(t)f t,x(t) = 0, 0 < t <1, x(0) = ax(ξ ), x(1) = bx(η), is studied, where 0 < ξ <η<1, 0 a, b < 1, and h: [0, 1] → [0,∞), f : [0, 1]×[0,∞)→[0,∞) are nonnegative continuous functions. By the use of the property of the corresponding Green’s function, fixed-point index theory, Leray–Schauder degree and upper and lower solution method, some existence, nonexistence, and multiplicity results of positive solutions are acquired. © 2006 Elsevier Inc. All rights reserved