Banach fixed-point theorem: Difference between revisions

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There are a number of generalizations (some of which are immediate [[Corollary|corollaries]]).<ref name=Latif2014>{{cite book |first=Abdul |last=Latif |title=Topics in Fixed Point Theory |pages=33–64 |chapter=Banach Contraction Principle and its Generalizations |publisher=Springer |year=2014 |doi=10.1007/978-3-319-01586-6_2 |isbn=978-3-319-01585-9 }}</ref>

Let <math>''T\,\colon'' : ''X\to'' → ''X</math>'' be a map on a complete non-empty metric space. Then, for example, some generalizations of the Banach fixed-point theorem are:

*Assume that some iterate ''T<mathsup>T^n</mathsup>'' of <math>''T</math>'' is a contraction. Then <math>''T</math>'' has a unique fixed point.

*Assume that for each <math>''n\in\mathbb N</math>'', there exists aexist ''c<mathsub>c_nn</mathsub>'' such that <math>''d(T^ⁿ(x), T^ⁿ(y))\leq c_nd≤ c_nd(x, y)</math>'' for all <math>''x</math>'' and <math>''y</math>'', and that

::<math>\sum_nsum\nolimits_n c_n <\infty.</math>

:Then <math>''T</math>'' has a unique fixed point.

In applications, the existence and unicity of a fixed point often can be shown directly with the standard Banach fixed point theorem, by a suitable choice of the metric that makes the map <math>''T</math>'' 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.<ref>{{cite book |first=Pascal |last=Hitzler | authorlink1=Pascal Hitzler|first2=Anthony |last2=Seda |title=Mathematical Aspects of Logic Programming Semantics |___location= |publisher=Chapman and Hall/CRC |year=2010 |isbn=978-1-4398-2961-5 }}</ref> Some of these have applications, e.g., in the theory of programming semantics in theoretical computer science.<ref>{{cite journal |first=Anthony K. |last=Seda |first2=Pascal |last2=Hitzler | authorlink2=Pascal Hitzler|title=Generalized Distance Functions in the Theory of Computation |journal=The Computer Journal |volume=53 |issue=4 |pages=443–464 |year=2010 |doi=10.1093/comjnl/bxm108 }}</ref>