The Existence of Three Symmetric Positive Solutions to A Fourth-Order Two-Point Boundary Value Problem

Applying the generalized Leggett-Williams fixed-point theorem, the existence of positive solutions to the fourth-order boundary value problem is studied: ${{u}^{\left( 4 \right)}}\left( t \right)=f\left( u\left( t \right) \right)\ \ \ \ \ \left( t\in \left[ 0, 1 \right] \right), u\left( 0 \right)=u\left( 1 \right)=0, {u}''\left( 0 \right)={u}''\left( 1 \right)=0$, where $f:\mathbb{R}\to \left[ 0, +\infty \right)$ is continuous. Under some conditions on f, there exist at least three symmetric positive solutions.