Existence of positive solutions for a nonlinear third order boundary value problem

This work concerned withthe following third-order three point boundaryvalue problem (BVP). Our main objective is to investigatethe existence, uniqueness and existence of positive solutions forthe boundary value problem (P1), by using Banach contractionprinciple, Leray Schauder nonlinear alternative, properties of theGreen function and Guo-Krasnosel'skii fixed point theorem in cone,in the case where the nonlinearity $f$ is either superlinear orsublinear.