Existence of positive solutions for nonlinear third-order three-point boundary value problems

This paper is concerned with the following nonlinear third-order three-point boundary value problem: u‴(t)+a(t)f(u(t))=0,t∈(0,1),u(0)=u′(0)=0,u′(1)=αu′(η), where 0<η<10<η<1 and 1<α<1η. First, the Green’s function for the associated linear boundary value problem is constructed, and then, some useful properties of the Green’s function are obtained by a new method. Finally, existence results for at least one positive solution for the above problem are established when ff is superlinear or sublinear.