Let be a compact continuous self-map of a Banach space. If the set of solutions of for is bounded, then has a fixed point. The theorem converts a priori estimates for a homotopy of equations into existence.
Every monotone self-map of a complete lattice has a complete lattice of fixed points. Its least fixed point is the meet of its prefixed points, and its greatest fixed point is the join of its postfixed points.
If is a homeomorphism and has the fixed-point property, then a continuous induces . A fixed point of the conjugate map is carried by to a fixed point of .
For continuous , the function is nonnegative at zero and nonpositive at one, so the intermediate value theorem gives a fixed point.
A contraction of a nonempty complete metric space has one fixed point, and every orbit converges to it geometrically.
For contraction mappings , compare their fixed points using on both arguments, then bound the change from to by the uniform metric. The resulting inequality gives the displayed estimate. Continuity at only needs its own contraction constant , even if nearby maps have constants approaching one.
A metric space is complete if and only if every contraction mapping of every nonempty closed subset into itself has a fixed point. For the converse, a nonconvergent Cauchy sequence contains a rapidly approaching subsequence whose point set is closed and whose shift is a contraction without a fixed point. Requiring the property only on the whole space is a different, weaker condition.
Suppose are contractions of one complete metric space with a common constant , and depends continuously on for each fixed . Their fixed points depend continuously on , because for a fixed parameter ,
If some iterate is a contraction on a complete metric space, then itself has exactly one fixed point: uniqueness for forces to fix its fixed point.
For the Newton map near a simple root ,Bounds and make on a sufficiently small closed interval around . The contraction mapping theorem then gives the unique local fixed point.
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A fixed-point theorem is a fundamental result in various branches of mathematics, particularly in analysis and topology, that asserts the existence of fixed points under certain conditions. A fixed point of a function is a point that is mapped to itself by the function. Formally, if \( f: X \rightarrow X \) is a function on a set \( X \), then a point \( x \in X \) is a fixed point if \( f(x) = x \).