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Fixed-point theorems in infinite-dimensional spaces

Index Fixed-point theorems in infinite-dimensional spaces

In mathematics, a number of fixed-point theorems in infinite-dimensional spaces generalise the Brouwer fixed-point theorem. [1]

28 relations: Algebraic topology, Banach fixed-point theorem, Banach space, Brouwer fixed-point theorem, Closed set, Compact space, Continuous function, Contraction mapping, Convex set, Earle–Hamilton fixed-point theorem, Empty set, Existence theorem, Fixed point (mathematics), James Dugundji, Jean Leray, Juliusz Schauder, Kakutani fixed-point theorem, Locally convex topological vector space, Markov–Kakutani fixed-point theorem, Mathematics, Metric space, Partial differential equation, Ryll-Nardzewski fixed-point theorem, Schauder fixed-point theorem, Sheaf (mathematics), Simplicial complex, Topological degree theory, Uniformly convex space.

Algebraic topology

Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces.

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Banach fixed-point theorem

In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contraction mapping principle) 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.

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Banach space

In mathematics, more specifically in functional analysis, a Banach space (pronounced) is a complete normed vector space.

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Brouwer fixed-point theorem

Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer.

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Closed set

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.

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Compact space

In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).

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Continuous function

In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.

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Contraction mapping

In mathematics, a contraction mapping, or contraction or contractor, on a metric space (M,d) is a function f from M to itself, with the property that there is some nonnegative real number 0\leq k such that for all x and y in M, The smallest such value of k is called the Lipschitz constant of f. Contractive maps are sometimes called Lipschitzian maps.

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Convex set

In convex geometry, a convex set is a subset of an affine space that is closed under convex combinations.

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Earle–Hamilton fixed-point theorem

In mathematics, the Earle–Hamilton fixed point theorem is a result in geometric function theory giving sufficient conditions for a holomorphic mapping of an open domain in a complex Banach space into itself to have a fixed point.

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Empty set

In mathematics, and more specifically set theory, the empty set or null set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.

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Existence theorem

In mathematics, an existence theorem is a theorem with a statement beginning 'there exist(s)..', or more generally 'for all,,...

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Fixed point (mathematics)

In mathematics, a fixed point (sometimes shortened to fixpoint, also known as an invariant point) of a function is an element of the function's domain that is mapped to itself by the function.

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James Dugundji

James Dugundji (August 30, 1919 – January, 1985) was an American mathematician, a professor of mathematics at the University of Southern California.

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Jean Leray

Jean Leray (7 November 1906 – 10 November 1998) was a French mathematician, who worked on both partial differential equations and algebraic topology.

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Juliusz Schauder

Juliusz Paweł Schauder (21 September 1899, Lwów, Austria-Hungary – September 1943, Lwów, Occupied Poland) was a Polish mathematician of Jewish origin, known for his work in functional analysis, partial differential equations and mathematical physics.

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Kakutani fixed-point theorem

In mathematical analysis, the Kakutani fixed-point theorem is a fixed-point theorem for set-valued functions.

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Locally convex topological vector space

In functional analysis and related areas of mathematics, locally convex topological vector spaces or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces.

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Markov–Kakutani fixed-point theorem

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.

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Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

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Metric space

In mathematics, a metric space is a set for which distances between all members of the set are defined.

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Partial differential equation

In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives.

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Ryll-Nardzewski fixed-point theorem

In functional analysis, a branch of mathematics, the Ryll-Nardzewski fixed-point theorem states that if E is a normed vector space and K is a nonempty convex subset of E that is compact under the weak topology, then every group (or equivalently: every semigroup) of affine isometries of K has at least one fixed point.

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Schauder fixed-point theorem

The Schauder fixed point theorem is an extension of the Brouwer fixed point theorem to topological vector spaces, which may be of infinite dimension.

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Sheaf (mathematics)

In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space.

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Simplicial complex

In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration).

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Topological degree theory

In mathematics, topological degree theory is a generalization of the winding number of a curve in the complex plane.

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Uniformly convex space

In mathematics, uniformly convex spaces (or uniformly rotund spaces) are common examples of reflexive Banach spaces.

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Redirects here:

Fixed point theorems in infinite-dimensional spaces, Tikhonov fixed point theorem, Tikhonov's fixed point theorem, Tychonoff fixed point theorem, Tychonoff fixed-point theorem.

References

[1] https://en.wikipedia.org/wiki/Fixed-point_theorems_in_infinite-dimensional_spaces

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