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

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 $\mathbb{R}$ \rightarrow $\mathbb{R}$$ is a jointly continuous function convex in $x$ which does not need to satisfy any further growth conditions.