Math 6710 • Applied Linear Operators and Spectral Methods

Let \(X\) be a Banach space, \(f : X \to \mathbb{R}\) be a bounded linear functional, \(y \in X\) be fixed,
and \(\alpha \in \mathbb{R}\).
\begin{enumerate}

\item
Prove that there exists a constant \(C > 0\) such that if \(|\alpha| < C\), then the non-linear
equation

\begin{equation}
x + \alpha f(x)x = y
\tag{1}
\end{equation}


has a unique solution \(x\) in the ball \(B = \{x \in X : \|x - y\| \le 1\}\).


\item
Suggest an interative procedure for approximating the unique solution \(x\) to equation
(1).

\end{enumerate}