Some multipoint boundary value problems of Neumann–Dirichlet type involving a multipoint p-Laplace like operator

Let φ and θ be two increasing homeomorphisms from R onto R with φ(0) = 0, θ(0) = 0. Let f : [0, 1]× R×R →R be a function satisfying Carathéodory’s conditions, and for each i, i = 1, 2, . . . , m − 2, let ai :R →R, be a continuous function, with m−2 i=1 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: φ(u ) = f (t,u,u ), t ∈ (0, 1), u (0) = 0, θ u(1) = m−2 i=1 θ u(ξi ) ai u (ξi ) . We note that this problem is at resonance, in the sense that the associated m-point boundary value problem φ u (t) = 0, t∈ (0, 1), u (0) = 0, θ u(1) = m−2 i=1 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 m−2 i=1 ai (0) = 1. © 2006 Elsevier Inc. All rights reserved.