Math 6420 • PDE

Let \(X\) be a real Banach space, \(f:X\to\mathbb{R}\) be a bounded linear functional and
\(T:X\to X\) a bounded linear operator. In the following \(y\in X\) is fixed and
\(\alpha\in\mathbb{R}\).
\begin{enumerate}
\item[(a)] Prove that there exists a constant \(C>0\) such that if \(\lvert\alpha\rvert<C\),
then the non-linear equation
\[
  x+\alpha f(x)Tx=y,
  \tag{1}
\]
has a unique solution \(x\) in the ball
\(B=\{x\in X\mid \lVert x-y\rVert\le1\}\).

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