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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.
In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem or Banach–Caccioppoli theorem) 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.
In numerical analysis, fixed-point iteration is a method of computing fixed points of a function.. More specifically, given a function defined on the real numbers with real values and given a point in the domain of , the fixed-point iteration is + = (), =,,, … which gives rise to the sequence,,, … of iterated function applications , (), (()), … which is hoped to converge to a point .
A contraction mapping has at most one fixed point. Moreover, the Banach fixed-point theorem states that every contraction mapping on a non-empty complete metric space has a unique fixed point, and that for any x in M the iterated function sequence x, f (x), f (f (x)), f (f (f (x))), ... converges to the fixed point
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.
During the 20th century numerous fixed-point theorems were developed, and even a branch of mathematics called fixed-point theory. [38] Brouwer's theorem is probably the most important. [ 39 ] It is also among the foundational theorems on the topology of topological manifolds and is often used to prove other important results such as the Jordan ...
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 ...