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The Banach fixed-point theorem (1922) gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point. [2]By contrast, the Brouwer fixed-point theorem (1911) is a non-constructive result: it says that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point, [3] but it doesn ...
In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem or Banach–Caccioppoli theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces and provides a constructive method to find those fixed points.
The Brouwer fixed-point theorem (1911) says that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point, but it doesn't describe how to find the fixed point. The Lefschetz fixed-point theorem (and the Nielsen fixed-point theorem) from algebraic topology give a way to count fixed points.
We focus on functions :, where the domain X is a nonempty subset of the Euclidean space .ch(X) denotes the convex hull of X.Iimura-Murota-Tamura theorem: [2] If X is a finite integrally-convex subset of , and : is a hypercubic direction-preserving (HDP) function, then f has a fixed-point.
In numerical analysis, fixed-point iteration is a method of computing fixed points of a function.. More specifically, given a function defined on the real numbers with real values and given a point in the domain of , the fixed-point iteration is + = (), =,,, … which gives rise to the sequence,,, … of iterated function applications , (), (()), … which is hoped to converge to a point .
Schauder fixed-point theorem: Let C be a nonempty closed convex subset of a Banach space V. If f : C → C is continuous with a compact image, then f has a fixed point. Tikhonov (Tychonoff) fixed-point theorem: Let V be a locally convex topological vector space. For any nonempty compact convex set X in V, any continuous function f : X → X has ...
The Kakutani fixed point theorem generalizes the Brouwer fixed-point theorem in a different direction: it stays in R n, but considers upper hemi-continuous set-valued functions (functions that assign to each point of the set a subset of the set). It also requires compactness and convexity of the set.
The Kakutani fixed point theorem is a generalization of the Brouwer fixed point theorem. The Brouwer fixed point theorem is a fundamental result in topology which proves the existence of fixed points for continuous functions defined on compact, convex subsets of Euclidean spaces. Kakutani's theorem extends this to set-valued functions.