Solvability of singular second order m-point boundary value problems

Let f : [0, 1] × R2 → R be a function satisfying Carathéodory’s conditions and (1 − t)e(t) ∈ L1(0, 1). Let ξi ∈ (0, 1), ai ∈ R, i = 1, . . . , m − 2, 0 < ξ1 < ξ2 < · · · < ξm−2 < 1 be given. This paper is concerned with the problem of existence of a C1[0, 1) solution for the m-point boundary value problem x = f t,x(t),x (t ) + e(t ), 0 < t <1, x (0) = 0, x(1) = m −2 i=1 aix(ξi ). The proof of our main result is based upon the Leray–Schauder continuation theorem.  2004 Elsevier Inc. All rights reserved.