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In mathematics, Lawvere's fixed-point theorem is an important result in category theory. [1] It is a broad abstract generalization of many diagonal arguments in mathematics and logic, such as Cantor's diagonal argument, Cantor's theorem, Russell's paradox, Gödel's first incompleteness theorem, Turing's solution to the Entscheidungsproblem, and Tarski's undefinability theorem.
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 Schauder fixed-point theorem is an extension of the Brouwer fixed-point theorem to topological vector spaces, which may be of infinite dimension.It asserts that if is a nonempty convex closed subset of a Hausdorff topological vector space and is a continuous mapping of into itself such that () is contained in a compact subset of , then has a fixed point.
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.
In mathematics, the Caristi fixed-point theorem (also known as the Caristi–Kirk fixed-point theorem) generalizes the Banach fixed-point theorem for maps of a complete metric space into itself. Caristi's fixed-point theorem modifies the ε {\displaystyle \varepsilon } - variational principle of Ekeland (1974, 1979).
In functional analysis, a branch of mathematics, the Ryll-Nardzewski fixed-point theorem states that if is a normed vector space and is a nonempty convex subset of that is compact under the weak topology, then every group (or equivalently: every semigroup) of affine isometries of has at least one fixed point. (Here, a fixed point of a set of ...
The Browder fixed-point theorem is a refinement of the Banach fixed-point theorem for uniformly convex Banach spaces.It asserts that if is a nonempty convex closed bounded set in uniformly convex Banach space and is a mapping of into itself such that ‖ () ‖ ‖ ‖ (i.e. is non-expansive), then has a fixed point.