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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.
In mathematics, the Markov–Kakutani fixed-point theorem, named after Andrey Markov and Shizuo Kakutani, states that a commuting family of continuous affine self-mappings of a compact convex subset in a locally convex topological vector space has a common fixed point. This theorem is a key tool in one of the quickest proofs of amenability of ...
The Kakutani fixed-point theorem is a generalization of Brouwer's fixed-point theorem, holding for generalized correspondences instead of functions.Its most important uses are in proving the existence of Nash equilibria in game theory, and the Arrow–Debreu–McKenzie model of general equilibrium theory in microeconomics.
the Kakutani fixed-point theorem, a fixed-point theorem for set-valued functions; Kakutani's theorem (geometry): the result that every convex body in 3-dimensional space has a circumscribed cube; Kakutani's theorem (measure theory): a result on the mutual equivalence or singularity of infinite product measures
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 ...
Set-valued functions arise in optimal control theory, especially differential inclusions and related subjects as game theory, where the Kakutani fixed-point theorem for set-valued functions has been applied to prove existence of Nash equilibria. This among many other properties loosely associated with approximability of upper hemicontinuous ...
The point is, stick to it, be consistent, and you may eventually like working out. nd3000/istockphoto. Final Thoughts on How to Make Exercise Fun and Empowering.
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 ...