Banach fixed-point theorem: Difference between revisions

Revision as of 20:49, 4 May 2024

Revision as of 19:41, 22 May 2024

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# Ω′ := (''I'' + ''g'')(Ω) is an open subset of ''E'': precisely, for any ''x'' in Ω such that {{nobr|''B''(''x'', ''r'') ⊂ Ω}} one has {{nobr|''B''((''I'' + ''g'')(''x''), ''r''(1 − ''k'')) ⊂ Ω′;}}

# ''I'' + ''g'' : Ω → Ω′ is a bi-Lipschitz homeomorphism;

: precisely, (''I'' + ''g'')⁻¹ is still of the form {{nobr|''I'' + ''h'' : Ω → Ω′}} with ''h'' a Lipschitz map of constant ''k''/(1 − ''k''). A direct consequence of this result yields the proof of the [[inverse function theorem]].

* It can be used to give sufficient conditions under which Newton's method of successive approximations is guaranteed to work, and similarly for Chebyshev's third-order method.

* It can be used to prove existence and uniqueness of solutions to integral equations.

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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 |first1=Pascal |last1=Hitzler | author-link1=Pascal Hitzler|first2=Anthony |last2=Seda |title=Mathematical Aspects of Logic Programming Semantics |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 |first1=Anthony K. |last1=Seda |first2=Pascal |last2=Hitzler | author-link2=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>

==Example of numerical application – calculating high accuracy [[Pi|{{pi}}]]==

Banach theorem allows for example fast and accurate calculation of the {{pi}} number using the trigonometric

functions which numerically are the power [[Taylor_series|Taylor series]].

Because <math>\sin(\pi)=0</math> and the {{pi}} is the fixed point of for example the function

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: <math>f(\pi)=\pi</math>

and also the function <math>f</math> is around {{pi}} the contraction mapping from the obvious reasons because its derivative in {{pi}} vanishes therefore {{pi}} can be obtained from the infinite superposition for example for the argument value 3:

: <math>\pi=f(f(f(\cdots f(3)\cdots))))</math>