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∈C[0,1]×R ³,e（t）∈L ¹[0,1],α≥0, is considered, all the β _j's have not the same sign, 0 <ξ<1,0<η _{1 <η 2<… <η m-2 <1. By using the coincidence degree theory, some existence theorems for the problems at resonance are obtained.}