Fixed point (mathematics)

Formally, c is a fixed point of a function f if c belongs to both the domain and the codomain of f, and f(c) = c. In particular, f cannot have any fixed point if its domain is disjoint from its codomain.If f is defined on the real numbers, it corresponds, in graphical terms, to a curve in the Euclidean plane, and each fixed-point c corresponds to an intersection of the curve with the line y = x, cf.In numerical analysis, fixed-point iteration is a method of computing fixed points of a function.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 Brouwer fixed-point theorem (1911) says that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point, but it doesn't describe how to find the fixed point.is said to have the fixed point property (FPP) if for any continuous function there existsAccording to the Brouwer fixed-point theorem, every compact and convex subset of a Euclidean space has the FPP.Compactness alone does not imply the FPP, and convexity is not even a topological property, so it makes sense to ask how to topologically characterize the FPP.In 1932 Borsuk asked whether compactness together with contractibility could be a necessary and sufficient condition for the FPP to hold.The problem was open for 20 years until the conjecture was disproved by Kinoshita, who found an example of a compact contractible space without the FPP.[2] In domain theory, the notion and terminology of fixed points is generalized to a partial order.[4] Malkis justifies the definition presented here as follows: "since f is before the inequality sign in the term f(x) ≤ x, such x is called a prefix point.Prefixpoints and postfixpoints have applications in theoretical computer science.One way to express the Knaster–Tarski theorem is to say that a monotone function on a complete lattice has a least fixed point that coincides with its least prefixpoint (and similarly its greatest fixed point coincides with its greatest postfixpoint).[7] In combinatory logic for computer science, a fixed-point combinator is a higher-order functionthat returns a fixed point of its argument function, if one exists.Their development has been motivated by descriptive complexity theory and their relationship to database query languages, in particular to Datalog.In many fields, equilibria or stability are fundamental concepts that can be described in terms of fixed points.
The function (shown in red) has the fixed points 0, 1, and 2.
other uses of "fixed point"stationary pointsmathematicstransformationfunctionsinvariant setdomaincodomainreal numbersEuclidean planeFixed-point iterationnumerical analysissequenceiterated functionconvergeperiodic pointsFixed-point theoremsBanach fixed-point theoremBrouwer fixed-point theoremcontinuous functionunit ballEuclidean spaceLefschetz fixed-point theoremNielsen fixed-point theoremalgebraic topologyalgebragroup actionfixed-point subgroupautomorphismsubgroupfixed-point subringsubringGalois theoryfield automorphismsfixed fieldFixed-point propertytopological spacefixed point propertytopological invarianthomeomorphismretractioncompactconvexsubsetBorsukcontractibilitydomain theorypartial ordertheoretical computer scienceLeast fixed pointorder theoryfunctionpartially ordered setKnaster–Tarski theoremmonotone functioncomplete latticeFixed point combinatorcombinatory logiccomputer sciencehigher-order functionFixed-point logicmathematical logicdescriptive complexity theorydatabase query languagesDatalogstabilityprojective geometryprojectivityeconomicsNash equilibriumbest response correspondenceJohn NashKakutani fixed-point theoremphysicstheory of phase transitionsWilsonrenormalization groupcritical phenomenonProgramming languagecompilersdata-flow analysisoptimizationabstract interpretationtype theoryfixed-point combinatoruntyped lambda calculusPageRanklinear transformationWorld Wide WebMarkov chainiterated functionsCycles and fixed pointsEigenvectorEquilibriumIdempotenceInfinite compositions of analytic functionsInvariant (mathematics)Fund. Math.Pacific Journal of MathematicsWayback MachineCoxeter, H. S. M.University of Toronto PressG. B. HalstedBibcode