Multiple positive solutions for a third-order three-point BVP with sign-changing Green's function

This article concerns the third-order three-point boundary-value problem $$displaylines{ u'''(t)=f(t,u(t)),quad tin [0,1], cr u'(0)=u(1)=u''(eta)=0. }$$ Although the corresponding Green's function is sign-changing, we still obtain the existence of at least 2m-1 positive solutions for arbitrary positive integer m under suitable conditions on f.