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

TL;DR

This paper employs variational methods to establish the existence of non-spurious solutions for a Dirichlet boundary value problem with a convex, continuous nonlinearity, without requiring growth conditions.

Contribution

It demonstrates the existence of solutions to a discrete boundary value problem using direct variational techniques under minimal assumptions.

Findings

Existence of solutions proven without growth restrictions

Solutions are non-spurious and valid for convex nonlinearities

Method applicable to a class of Dirichlet problems

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 $f:\left[ 0,1\right] \times \mathbb{R} \rightarrow \mathbb{R}$ is a jointly continuous function convex in $x$ which does not need to satisfy any further growth conditions.