Non-spurious solutions to discrete boundary value problems through variational methods

Abstract

Using direct variational method we consider the existence of non-spurious solutions to the following Dirichlet problem $\ddot{x}\left(t\right)=f\left(t,x\left(t\right)\right)$, $x\left(0\right)=x\left(1\right)=0$ where $f:\left[0,1\right]\times \mathbb{R}\to \mathbb{R}$ is a jointly continuous function convex in $x$ which does not need to satisfy any further growth conditions.