The existence and the uniqueness of symmetric positive solutions for a fourth-order boundary value problem

The purpose of this paper is to investigate the existence and the uniqueness of symmetric positive solutions for a class of fourth-order boundary value problem: {y(4)(t)=f(t,y(t)),t∈[0,1],y(0)=y(1)=y′(0)=y′(1)=0. By using the fixed point index method, we establish the existence of at least one or at least two symmetric positive solutions for the above boundary value problem. Further, by using a fixed point theorem of general αα-concave operators, we also present criteria which guarantee the existence and uniqueness of symmetric positive solutions for the above boundary value problem.