Existence results of a m-point boundary value problem at resonance

AbstractLet ξi∈(0,1), ai∈(0,∞), i=1,…,m−2, be given constants satisfying ∑m−2i=1ai=1 and 0<ξ1<ξ2<⋯<ξm−2<1. We show the existence of solutions for the m-point boundary value problem x″=f(t,x,x′),t∈(0,1),x′(0)=0,x(1)=∑i=1m−2aix(ξi), where f:[0,1]×R2→R is continuous and f(t,r1,0)⩽0, f(t,r2,0)⩾0 for some r1,r2 with r1