Positive solutions of m-point boundary value problems for higher order ordinary differential equations

In this paper, we study m-point boundary value problems for higher order ordinary differential equation {y(2n)(t)=f(t,y(t),y″(t),…,y(2(n−1))(t)),0≤t≤1,y(2i)(0)=0,y(2i)(1)=∑j=1m−2kijy(2i)(ξj),0≤i≤n−1, where f is allowed to change sign, and 0=ξ0<ξ1<ξ2<⋯<ξm−2<ξm−1=1. We show sufficient conditions for the existence of at least two positive solutions by applying a new fixed point theorem in cones and the associated Green's function. In particular, the second positive solutions for the above problem is not concave.