Existence Results of Third Order Multi-Point Boundary Value Problem at Resonance

A kind of third order multi-point boundary value problems, x（t）=f（t,x（t）,x′（t）,x″（t））+ e（t）, t∈（0,1）,x（0）=αx（ξ）,x′（0）=0,x（1）=∑m-2j=1β_jx（η_j）, f∈C0,1×R³,e（t）∈L¹0,1,α≥0, is considered, all the β_j's have not the same sign, 0<ξ<1,0<η₁<η₂<…<η m-2 <1. By using the coincidence degree theory, some existence theorems for the problems at resonance are obtained.