Fixed Point

A fixed point of a function $f:X\to X$ is an element $x^\ast\in X$ satisfying $f(x^\ast)=x^\ast$. Thus, applying the function leaves that element unchanged. The underlying object need not be a geometric point: it may be a number, vector, function, or set. Fixed-point theory studies conditions guaranteeing the existence or uniqueness of such elements and methods for finding them. Different branches of the theory use metric, topological, or order-theoretic structure. (math.ucdavis.edu)

Definition and elementary examples

For a function of one real variable, fixed points occur where its graph intersects the diagonal $y=x$. Equivalently, they are solutions of the equation $f(x)-x=0$. A function may have no fixed points, exactly one, or several; existence depends on both its formula and its domain. (maria-titova.com)

For example, direct substitution shows that $f(x)=x^2$ on $\mathbb R$ has fixed points $0$ and $1$, whereas $f(x)=x+1$ has none. The identity function fixes every element of its domain. The map $f(x)=(x+2)/3$ has the unique fixed point $1$. These examples also distinguish a fixed point from a zero of a function: the former satisfies $f(x)=x$, while the latter satisfies $f(x)=0$.

Principal existence theorems

The Banach fixed-point theorem, also called the contraction mapping theorem, applies to a nonempty complete metric space $(X,d)$. If $f:X\to X$ is a contraction mapping, meaning that some constant $0\leq q<1$ satisfies

for every $x,y\in X$, then $f$ has exactly one fixed point. Starting from any $x_0\in X$, repeated application of $f$ converges to it. Completeness ensures that the resulting Cauchy sequence has a limit within the space. (math.ucdavis.edu)

The Brouwer fixed-point theorem instead uses topological conditions. Every continuous map from a nonempty compact, convex subset of finite-dimensional Euclidean space into itself has a fixed point. Unlike Banach’s theorem, it does not guarantee uniqueness or convergence of ordinary iteration. The Schauder fixed-point theorem extends this existence result to a nonempty compact, convex subset of a Banach space. (arxiv.org)

Order-theoretic fixed points need neither distances nor geometric continuity. The Knaster–Tarski theorem states that an order-preserving self-map of a complete lattice has fixed points, including a least and a greatest one. Here, a complete lattice is a partially ordered set in which every subset has a supremum and an infimum. Moreover, the fixed points themselves form a complete lattice under the inherited order. (cs.utexas.edu)

Iteration and computation

In numerical analysis, fixed-point iteration constructs successive approximations by

If $f$ is continuous and this sequence converges to $x^\ast$, passing to the limit gives $x^\ast=f(x^\ast)$. Under Banach’s hypotheses, convergence is guaranteed, with the error estimate

This provides a quantitative stopping criterion once a contraction constant is known. (eml.berkeley.edu)

Existence alone does not ensure that iteration succeeds. As an elementary example, $f(x)=1-x$ maps $[0,1]$ continuously into itself and has the unique fixed point $1/2$. Nevertheless, every other starting value alternates between $x_0$ and $1-x_0$, rather than converging. Thus, an existence theorem and a convergent computational procedure are distinct kinds of result.

Local dynamics and stability

Fixed points are stationary states of discrete-time dynamical systems described by iteration. An attracting fixed point draws sufficiently nearby trajectories toward itself; a repelling fixed point pushes nearby trajectories away. For a continuously differentiable scalar map, the magnitude of the derivative supplies a local test: $|f'(x^\ast)|<1$ implies attraction, while $|f'(x^\ast)|>1$ implies repulsion. If the magnitude equals $1$, this test alone does not decide the behavior. (pi.math.cornell.edu)

The distinction concerns behavior near a fixed point, not merely whether the point exists. It also explains why two algebraically equivalent fixed-point reformulations can lead to different numerical behavior: their iteration maps may have different derivatives at the same solution.

Applications in analysis and computing

A major application is proving existence and uniqueness for differential equations. An initial-value problem $u'(t)=F(t,u(t))$, $u(t_0)=u_0$, can be rewritten as the fixed-point equation

The right-hand side defines an operator on a space of continuous functions. With continuity and suitable Lipschitz assumptions, this operator becomes a contraction on an appropriate space over a sufficiently short interval. Its fixed point is the local solution. This construction underlies the Picard–Lindelöf theorem. (web.mit.edu)

In theoretical computing, fixed points give mathematical meaning to recursive definitions. The semantics of a loop can be expressed as a fixed point of an operator describing one step followed by further execution. A least fixed point, in a suitable ordering of partial computations, captures behavior built from successive approximations, including possible nontermination. Cornell’s treatment of while-loop semantics uses such an operator and order-theoretic fixed-point results to establish its meaning. (courses.cs.cornell.edu)