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{\LARGE {Mathematics 118A, Winter 1998}}
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\large Homework Assignment 3, due Friday, January 30
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\bigskip

\begin{enumerate}
\item Let $X$ be a Banach space with norm $\| \cdot \|$.  Recall that
the norm of a linear transformation $T\colon X \to X$ is defined by
\[
  \| T \|_{\mathcal{L}} =
    \sup_{\substack{x \in X \\ x \neq 0}}
    \frac{\| T x \|}{\| x \|}.
\]
Prove that $\| A^k \| \leq \| A \|^k$ for every $k \in \N$.
  

\item Let $f(t) = 5(1-e^t)$.  Use the contraction mapping theorem to
show that the equation $f(t)=t$ has exactly one solution, which is
located in the interval $[4,5]$.

\item
\begin{enumerate}
\item Acquaint yourself with \emph{Mathematica}, for more details see
\begin{verbatim}
  http://www.math.uci.edu/~moliver/math118/math118.html
\end{verbatim}


\item Revisit Problem 3 on Homework Assignment 1.  Let
\emph{Mathematica} solve the equation, and generate the graphs of a
few solutions.

\item Revisit Problems 3 and 4a on Homework Assignment 2.  Repeat the
calculations on \emph{Mathematica}, and graph the vector fields (for
different values of $\lambda$) as well as some phase curves.  Can
\emph{Mathematica} find exact solutions to these equations?

\end{enumerate}

\end{enumerate}
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