An Existence and Uniqueness Result for a Bending Beam Equation without Growth Restriction

Abstract

We discuss the solvability of the fourth-order boundary value problem $u^{\left(4\right)}=f(t,u,u^{\mathrm{\prime \prime }})$, $0\le t\le 1$, $u\left(0\right)=u\left(1\right)=u^{\mathrm{\prime \prime }}\left(0\right)=u^{\mathrm{\prime \prime }}\left(1\right)=0$, which models a statically bending elastic beam whose two ends are simply supported, where $f:\left[\mathrm{0,1}\right]\times {\mathbb{R}}^2\to \mathbb{R}$ is continuous. Under a condition allowing that $f(t,u,v)$ is superlinear in $u$ and $v$, we obtain an existence and uniqueness result. Our discussion is based on the Leray-Schauder fixed point theorem.