Search results
Results from the WOW.Com Content Network
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.
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 ...
Early in his career, Brouwer proved a number of theorems in the emerging field of topology. The most important were his fixed point theorem, the topological invariance of degree, and the topological invariance of dimension. Among mathematicians generally, the best known is the first one, usually referred to now as the Brouwer fixed point theorem.
Download as PDF; Printable version; ... Fixed-point theorems. Sperner's lemma; Brouwer fixed-point theorem; Kakutani fixed-point theorem; Game theory
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 ...
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.
Fixed-point computation refers to the process of computing an exact or approximate fixed point of a given function. [1] In its most common form, the given function satisfies the condition to the Brouwer fixed-point theorem: that is, is continuous and maps the unit d-cube to itself.
Banach fixed-point theorem; Bekić's theorem; Blackwell's contraction mapping theorem; Borel fixed-point theorem; Borsuk–Ulam theorem; Bourbaki–Witt theorem; Brouwer fixed-point theorem; Browder fixed-point theorem; Bruhat–Tits fixed point theorem