Multiple positive solutions for second-order three-point boundary-value problems with sign changing nonlinearities

In this article, we study the second-order three-point boundary-value problem $$displaylines{ u''(t)+a(t)u'(t)+f(t,u)=0,quad 0 leq t leq 1, cr u'(0)=0,quad u(1)=alpha u(eta), }$$ where $0