Non-spurious solutions to second order O.D.E. by monotonicity methods

Filip Pietrusiak

Published 2016-06-22Version 1

We consider in $H_{0}^{1}\left( 0,1\right) $ the following second order O.D.E Dirichlet problem $\ddot{x}\left( t\right) =f\left(t,\dot{x} \left(t\right) ,x\left( t\right) \right) -h\left( t\right) $, $x\left( 0\right)=x\left( 1\right) =0$ where $f$ is continuous and satisfies some other conditions, $h\in H_{0}^{1}\left( 0,1\right) $ together with its discretization which reads \begin{equation*} -\Delta ^{2}x(k-1)+\frac{1}{n^{2}}f\left(\frac{k}{n},n\Delta x\left(k\right),x\left(k\right)\right)=\frac{1}{n^{2}} h\left(\frac{k}{n} \right)\text{ for } k \in \left[ 1,\ldots,n\right]\text{.} \end{equation*} Using monotonicity methods we obtain the existence of non-spurious solutions to the above problem.