Math 6420 • PDE

Consider the non-local differential operator
\[
  Lu=Du''-u(g(x))
\]
with boundary conditions \(u(0)=u(1)=0\) and where \(g(x)\) is a monotone increasing function
with \(g(0)=0\), \(g(1)=1\), and \(g(x)<x\) for \(0<x<1\).
\begin{enumerate}
\item[(a)] Find the adjoint of the operator \(L\).
\item[(b)] Prove that the operator \(L\) is invertible.
\item[(c)] Devise an iteration scheme that could be used to solve \(Lu=f(x)\). Under what
conditions on \(D\) is this iteration scheme a contraction mapping?
\end{enumerate}