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 RR onto RR with ϕ(0)=0ϕ(0)=0, θ(0)=0θ(0)=0. Let f:[0,1]×R×R↦R be a function satisfying Carathéodory's conditions, and for each i  , i=1,2,…,m−2i=1,2,…,m−2, let ai:R↦R, be a continuous function, with ∑i=1m−2ai(0)=1, ξi∈(0,1)ξi∈(0,1), 0<ξ1<ξ2<⋯<ξm−2<10<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))=∑i=1m−2θ(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))=∑i=1m−2θ(u(ξi))ai(u′(ξi)) has the non-trivial solution u(t)=ρu(t)=ρ, where ρ∈Rρ∈R is an arbitrary constant vector, in view of the assumption ∑i=1m−2ai(0)=1.