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The function f(x) = x 2 − 4 has two fixed points, shown as the intersection with the blue line; its least one is at 1/2 − √ 17 /2.. In order theory, a branch of mathematics, the least fixed point (lfp or LFP, sometimes also smallest fixed point) of a function from a partially ordered set ("poset" for short) to itself is the fixed point which is less than each other fixed point, according ...
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 ...
A fixed-point theorem is a result saying that at least one fixed point exists, under some general condition. [1] For example, the Banach fixed-point theorem (1922) gives a general criterion guaranteeing that, if it is satisfied, fixed-point iteration will always converge to a fixed point.
Since complete lattices cannot be empty (they must contain a supremum and infimum of the empty set), the theorem in particular guarantees the existence of at least one fixed point of f, and even the existence of a least fixed point (or greatest fixed point). In many practical cases, this is the most important implication of the theorem.
Computation of the least fixpoint of f(x) = 1 / 10 x 2 +atan(x)+1 using Kleene's theorem in the real interval [0,7] with the usual order. In the mathematical areas of order and lattice theory, the Kleene fixed-point theorem, named after American mathematician Stephen Cole Kleene, states the following: Kleene Fixed-Point Theorem.
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.
In mathematics, the Lefschetz fixed-point theorem [1] is a formula that counts the fixed points of a continuous mapping from a compact topological space to itself by means of traces of the induced mappings on the homology groups of .
The notation . (and its dual) are inspired from the lambda calculus; the intent is to denote the least (and respectively greatest) fixed point of the expression where the "minimization" (and respectively "maximization") are in the variable , much like in lambda calculus . is a function with formula in bound variable; [6] see the denotational ...