Contraction mapping

A non-expansive mapping with k=1 can be generalized to a firmly non-expansive mapping in a Hilbert space \mathcal{H} if the following holds for all x and y in \mathcal{H}: \|f(x)-f(y) \|^2 \leq \, \langle x-y, f(x) - f(y) \rangle, where d(x,y) = \|x-y\|. This is a special case of \alpha averaged nonexpansive operators with \alpha = 1/2. A firmly non-expansive mapping is always non-expansive, via the Cauchy–Schwarz inequality. The class of firmly non-expansive maps is closed under convex combinations, but not compositions. This class includes proximal mappings of proper, convex, lower-semicontinuous functions, hence it also includes orthogonal projections onto non-empty closed convex sets. The class of firmly nonexpansive operators is equal to the set of resolvents of maximally monotone operators. Surprisingly, while iterating non-expansive maps has no guarantee to find a fixed point (e.g. multiplication by -1), firm non-expansiveness is sufficient to guarantee global convergence to a fixed point, provided a fixed point exists. More precisely, if \operatorname{Fix}f := \{x \in \mathcal{H} \ | \ f(x) = x\} \neq \varnothing, then for any initial point x_0 \in \mathcal{H}, iterating x_{n+1} = f(x_n), \quad \forall n \in \mathbb{N} yields convergence to a fixed point x_n \to z \in \operatorname{Fix} f. This convergence might be weak in an infinite-dimensional setting. ==Subcontraction map==

A subcontraction map or subcontractor is a map f on a metric space (M, d) such that d(f(x), f(y)) \leq d(x,y); d(f(f(x)),f(x)) If the image of a subcontractor f is compact, then f has a fixed point. ==Locally convex spaces==

In a locally convex space (E, P) with topology given by a set P of seminorms, one can define for any p ∈ P a p-contraction as a map f such that there is some kp n+1 = f(xn), and if (E, P) is Hausdorff, then the fixed point is unique. ==See also==