Solvability of an m-Point Boundary Value Problem for Second Order Ordinary Differential Equations

Let ƒ: [0, 1] × R2 → R be a function satisfying Caratheodory′s conditions and e(t) ∈ L1 [0, 1]. Let ξi ∈ (0, 1), ai ∈ R, all of the ai′s having the same sign, i = 1, 2, ..., m − 2, 0 < ξ1 < ξ2 < ... < ξm−2 < 1 be given. This paper is concerned with the problem of existence of a solution for the m-point boundary value problem x"(t) = ƒ(t, x(t), x′(t)) + e(t), t ∈ (0, 1), (E)[formula] Conditions for the existence of a solution for the above boundary value problem are given using the Leray-Schauder continuation theorem.