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The fixed point iteration x n+1 = cos x n with initial value x 1 = −1.. An attracting fixed point of a function f is a fixed point x fix of f with a neighborhood U of "close enough" points around x fix such that for any value of x in U, the fixed-point iteration sequence , (), (()), ((())), … is contained in U and converges to x fix.
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.
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 ...
Applying the Banach fixed-point theorem shows that the fixed point π is the unique fixed point on the interval, allowing for fixed-point iteration to be used. For example, the value 3 may be chosen to start the fixed-point iteration, as 3 π / 4 ≤ 3 ≤ 5 π / 4 {\displaystyle 3\pi /4\leq 3\leq 5\pi /4} .
Upon iteration, one may find that there are sets that shrink and converge towards a single point. In such a case, the point that is converged to is known as an attractive fixed point. Conversely, iteration may give the appearance of points diverging away from a single point; this would be the case for an unstable fixed point. [11]
Given a function :, consider the problem of finding a fixed point of , which is a solution to the equation () =.A classical approach to the problem is to employ a fixed-point iteration scheme; [2] that is, given an initial guess for the solution, to compute the sequence + = until some convergence criterion is met.
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. Suppose ( L , ⊑ ) {\displaystyle (L,\sqsubseteq )} is a directed-complete partial order (dcpo) with a least element, and let f : L → L {\displaystyle ...
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.