Banach Fixed-point Theorem - Generalizations

Generalizations

There are a number of generalizations as immediate corollaries, which are of some interest for the sake of applications. Let be a map on a complete non-empty metric space.

• Assume that some iterate of T is a contraction. Then T has a unique fixed point.
• Assume that for all and in, Then T has a unique fixed point.

However, in most applications the existence and unicity of a fixed point can be shown directly with the standard Banach fixed point theorem, by a suitable choice of the metric that makes the map T a contraction. Indeed, the above result by Bessaga strongly suggests to look for such a metric. See also the article on fixed point theorems in infinite-dimensional spaces for generalizations.

A different class of generalizations arise from suitable generalizations of the notion of metric space, e.g. by weakening the defining axioms for the notion of metric. Some of these have applications, e.g., in the theory of programming semantics in theoretical computer science.