Math 6420 • PDE

Let \(K:L^2[0,1]\to L^2[0,1]\) be the linear operator defined by
\[
  (Kx)(t)=\int_0^1 k(t,\tau)x(\tau)\,d\tau
\]
where \(k:[0,1]\times[0,1]\to\mathbb{R}\) is continuous.
\begin{enumerate}
\item[(a)] Show that \(K\) is bounded and give an upper bound for \(\lVert K\rVert\) (in terms
of \(k\)).

\item[(b)] Show that there is a constant \(C\) such that if \(\lvert\mu\rvert<C\), the operator
\(I+\mu K\) is invertible.

\item[(c)] Assuming \(\lvert\mu\rvert<C\), where \(C\) is as in part (b), express
\((I+\mu K)^{-1}\) as a series. Specify whether the series converges and in what sense.
\end{enumerate}