Math 6420 • PDE

Let \(T:L^1[0,1]\to L^2[0,1]\) be a bounded linear operator. Let
\(a\in L^\infty[0,1]\) and \(g\in L^1[0,1]\).
\begin{enumerate}
\item[(a)] Prove that there exists a constant \(C>0\) such that if
\(\lVert a\rVert_{L^\infty}<C\) then the equation
\[
  f+a(Tf)^2=g
  \tag{1}
\]
has a unique solution \(f\) in the set
\[
  B=\{u\in L^1[0,1]:\lVert u-g\rVert_{L^1}\le1\}.
\]
\textbf{Note:} the constant \(C\) may depend on \(T\) and/or \(g\).

\item[(b)] Suggest an iterative procedure for approximating the unique solution \(f\) to
equation (1).
\end{enumerate}