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: $\{\begin{array}{c}{y}^{\left(4\right)}\left(t\right)=f(t,y\left(t\right))\text{,}t\in [0,1]\text{,}\hfill \\ y\left(0\right)=y\left(1\right)=y^{\text{'}}\left(0\right)=y^{\text{'}}\left(1\right)=0\text{.}\hfill \end{array}$ 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 $\alpha$ ‐concave operators, we also present criteria which guarantee the existence and uniqueness of symmetric positive solutions for the above boundary value problem.