Abstract

Let φ{symbol} and θ be two increasing homeomorphisms from R onto R with φ{symbol} (0) = 0, θ (0) = 0. Let f : [0, 1] × R × R {mapping} R be a function satisfying Carathéodory's conditions, and for each i, i = 1, 2, ..., m - 2, let ai : R {mapping} R, be a continuous function, with ∑i = 1 m - 2 ai (0) = 1, ξi ∈ (0, 1), 0 < ξ1 < ξ2 < ⋯ < ξm - 2 < 1. In this paper we first prove a suitable continuation lemma of Leray-Schauder type which we use to obtain several existence results for the m-point boundary value problem:{Mathematical expression}. We note that this problem is at resonance, in the sense that the associated m-point boundary value problem(φ{symbol} (u′ (t)))′ = 0, t ∈ (0, 1), u′ (0) = 0, θ (u (1)) = underover(∑, i = 1, m - 2) θ (u (ξi)) ai (u′ (ξi)) has the non-trivial solution u (t) = ρ, where ρ ∈ R is an arbitrary constant vector, in view of the assumption ∑i = 1 m - 2 ai (0) = 1. © 2006 Elsevier Inc. All rights reserved.