Let φ and θ be two odd increasing homeomorphism from R onto R with φ(0)=0,θ(0)=0, and let f:[0,1]×R×R→R be a function satisfying Caratheodory's conditions. Let a i∈ℝ,ξi(0,1),i=1,2,...,m-2,0<ξ 1<ξ2<⋯<ξm-2<1 be given. We are interested in the problem of existence of solutions for the m-point boundary value problem:(φ(x′))′=f(t,x,x′),t∈(0,1),x′(0)=0, θ(x′(1))=∑i=1m-2aiθ(x′(ξ i)). We note that this non-linear m-point boundary value problem is always at resonance since the associated m-point boundary value problem (φ(x′))′=0,t∈(0,1),x′(0)=0,θ(x′(1))= ∑i=1m-2aiθ(x′(ξi)) has non-trivial solutions x(t)=ρ,ρ∈R (an arbitrary constant). Our results are obtained by a suitable homotopy, Leray-Schauder degree properties, and a priori bounds. © 2003 Elsevier Ltd. All rights reserved.
García-Huidobro, M., Gupta, C. P., & Manásevich, R. (2004). An m-point boundary value problem of Neumann type for a p-Laplacian like operator. Nonlinear Analysis, Theory, Methods and Applications, 56(7), 1071–1089. https://doi.org/10.1016/j.na.2003.11.003