Lecture15

# Lecture15 - \documentclass[12pt,letterpaper]cfw_article

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\documentclass[12pt,letterpaper]{article} \usepackage{amsmath} \usepackage{amssymb} \usepackage{latexsym} \usepackage{array} \usepackage{enumerate} \usepackage{amsthm} \usepackage{amscd} \newtheorem{lemma}{Lemma} \newtheorem{theorem}{Theorem} \newtheorem{definition}{Definition} \newcommand{\bC}{{\mathbb C}} \newcommand{\bH}{{\mathbb H}} \newcommand{\bR}{{\mathbb R}} \newcommand{\ra}{\rightarrow} \newcommand{\Ra}{\Rightarrow} \newcommand{\norm}[1]{\| #1 \|} \newcommand{\skipline}{\vspace{\baselineskip}} \newcommand{\ivp}{initial value problem} \newcommand{\closure}[1]{\overline{#1}} \date{} \title{\bf Lecture 15} \author{} \begin{document} \maketitle \begin{definition} Let $M$ be a metric space. A {\bf contraction} on $M$ is a function $\phi: M \ra M$ such that there exists a positive constant $k < 1$ with $$a, b \in M \implies d(\phi(a), \phi(b)) \leq kd(a,b).$$ \end{definition} \begin{theorem} (Contraction Principle''): Each contraction on a complete metric space $M$ has a unique fixed point: there exists a unique $z \in M$ such that $\phi(z) = z$. \end{theorem} \begin{proof} Uniqueness is plain (without completeness). Existence is constructive. Choose any $z_0 \in M$ and for $n \in \mathbb{N}$, define $z_n = \phi^n (z_0)$. (That is, inductively $z_{n+1} = \phi(z_n)$.)

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## Lecture15 - \documentclass[12pt,letterpaper]cfw_article

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