1.1 Contraction Mapping Theorem
The contraction mapping theorem is a fundamental tool for proving existence and uniqueness of fixed points. This section establishes the basic definitions and the classical theorem.
A metric space \((X, d)\) is complete if every Cauchy sequence converges in \(X\), i.e., if given any \(\varepsilon {\gt} 0\) there exists \(N(\varepsilon )\) such that \(n, m {\gt} N(\varepsilon )\) implies \(d(x_n, x_m) {\lt} \varepsilon \), then there is some \(y \in X\) with \(\lim _{n\rightarrow \infty } x_n = y\).
Let \((X, d)\) be a metric space. A function \(T: X \rightarrow X\) is a contraction if there is a number \(\kappa \in [0, 1)\), called a contraction constant, such that
for all \(x, y \in X\).
Let \((X, d)\) be a complete metric space. If \(T: X \rightarrow X\) is a contraction with contraction constant \(\kappa \), then there exists a unique fixed point \(\tilde{x} \in X\) of \(T\). Furthermore, \(\tilde{x}\) is globally attracting, and for any \(x \in X\),
Choose \(x_0 \in X\) and recursively define \(x_{n+1} := T(x_n)\). By the contraction property, \(d(x_{n+1}, x_n) \leq \kappa \cdot d(x_n, x_{n-1})\), so by induction \(d(x_{n+1}, x_n) \leq \kappa ^n d(x_1, x_0)\).
For \(n {\lt} m\), the triangle inequality and geometric series give
Thus \(\{ x_n\} \) is Cauchy. By completeness, there exists \(\tilde{x} = \lim _{n\rightarrow \infty } x_n\).
By continuity of \(T\), \(\tilde{x} = \lim T(x_{n-1}) = T(\tilde{x})\).
For uniqueness, if \(\tilde{y}\) is another fixed point with \(d(\tilde{y}, \tilde{x}) {\gt} 0\), then \(d(\tilde{y}, \tilde{x}) = d(T(\tilde{y}), T(\tilde{x})) \leq \kappa \cdot d(\tilde{y}, \tilde{x})\), giving \(\kappa \geq 1\), a contradiction.
Given a contraction mapping \(T: X \rightarrow X\), the rate at which points in \(X\) converge to the globally attracting fixed point \(\tilde{x}\) is determined by \(\kappa \). In particular, the smaller \(\kappa \) is, the faster iterates under \(T\) converge to \(\tilde{x}\).