KnowraFixed-point theoremFixed-point theoremA fixed-point theorem guarantees that a function or mapping has at least one point that it leaves unchanged.BriefConnectFixed point (mathematics): A point that a function maps to itself. It is the object whose existence every fixed-point theorem concerns.Banach fixed-point theorem: A theorem stating that a contraction on a nonempty complete metric space has a unique fixed point. It guarantees existence and uniqueness by repeatedly applying a distance-shrinking map.Picard–Lindelöf theorem: A theorem giving local existence and uniqueness for ordinary differential equations with a suitably Lipschitz right-hand side. Its standard proof converts an initial-value problem into a contraction fixed-point problem.Intermediate value theorem: A theorem stating that a continuous real function takes every intermediate value between two values it attains. It guarantees a zero in one dimension, while fixed-point results concern invariant points of maps.Function (mathematics): A rule assigning each input in a domain exactly one output. Fixed-point theorems impose conditions on functions or more general maps.Brouwer fixed-point theorem: A theorem stating that every continuous map from a compact convex subset of Euclidean space to itself has a fixed point. It derives existence from continuity and the geometry of a finite-dimensional domain.Integral equation: An equation in which an unknown function appears inside an integral. Many existence proofs recast differential equations as fixed-point problems for integral operators.Contraction mapping theorem: A theorem asserting that a contraction on a complete metric space has a unique fixed point. It is a narrower, constructive result within the broader family of fixed-point theorems.Metric space: A set equipped with a distance function satisfying the metric axioms. Many fixed-point results use distances to state their assumptions.Schauder fixed-point theorem: A theorem extending Brouwer's fixed-point result to compact continuous maps on suitable convex subsets of Banach spaces. It carries finite-dimensional existence arguments into infinite-dimensional settings.Show all 23Linked from 24 pagesExistence theoremBroader topic: It guarantees a solution by proving that a suitable function has a fixed point.ContinuityRelated: Continuity is a key hypothesis in classical fixed-point results such as Brouwer’s theorem.John NashNarrower topic: Nash used a fixed-point argument to prove equilibrium existence.Transfinite recursionCompared with: Fixed-point methods characterize self-consistent values, whereas recursion specifies values from earlier stages.Connected spaceRelated: Connectedness is among the hypotheses that make several classical fixed-point results possible.Supremum normNarrower topic: Contraction-based fixed-point proofs use normed function spaces and often the supremum norm.Mathematical economicsRelated: Existence proofs for economic equilibria often rely on fixed-point results.L. E. J. BrouwerNarrower topic: Brouwer’s theorem became a defining example in a broad family of mathematical results.Show all 24