Uniqueness of Positive Solutions for a Class of Fourth-Order Boundary ValueProblems

Abstract

The purpose of this paper is to investigate the existence and uniqueness of positive solutions for the following fourth-order boundary value problem: $y^{\left(4\right)}\left(t\right)=f(t,y(t\left)\right)$, $t\in [0,1]$, $y\left(0\right)=y\left(1\right)=y^{\prime }\left(0\right)=y^{\prime }\left(1\right)=0$. Moreover, under certain assumptions, we will prove that the above boundary value problem has a unique symmetric positive solution. Finally, we present some examples and we compare our results with the ones obtained in recent papers. Our analysis relies on a fixed point theorem in partially ordered metric spaces.