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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Contraction mapping ： ウィキペディア英語版 Contraction mapping In mathematics, a contraction mapping, or contraction or contractor, on a metric space ''(M,d)'' is a function ''f'' from ''M'' to itself, with the property that there is some nonnegative real number such that for all ''x'' and ''y'' in ''M'', :$d(f(x),f(y))\backslash leq\; k\backslash ,d(x,y).$ The smallest such value of ''k'' is called the Lipschitz constant of ''f''. Contractive maps are sometimes called Lipschitzian maps. If the above condition is instead satisfied for ''k'' ≤ 1, then the mapping is said to be a non-expansive map. More generally, the idea of a contractive mapping can be defined for maps between metric spaces. Thus, if (''M'',''d'') and (''N'',''d) are two metric spaces, and $f:M\; \backslash rightarrow\; N$, then there is a constant such that :$d\text{'}(f(x),f(y))\backslash leq\; k\backslash ,d(x,y)$ for all ''x'' and ''y'' in ''M''. Every contraction mapping is Lipschitz continuous and hence uniformly continuous (for a Lipschitz continuous function, the constant ''k'' is no longer necessarily less than 1). A contraction mapping has at most one fixed point. Moreover, the Banach fixed point theorem states that every contraction mapping on a nonempty complete metric space has a unique fixed point, and that for any ''x'' in ''M'' the iterated function sequence ''x'', ''f'' (''x''), ''f'' (''f'' (''x'')), ''f'' (''f'' (''f'' (''x''))), ... converges to the fixed point. This concept is very useful for iterated function systems where contraction mappings are often used. Banach's fixed point theorem is also applied in proving the existence of solutions of ordinary differential equations, and is used in one proof of the inverse function theorem.〔Theodore Shifrin, ''Multivariable Mathematics'', Wiley, 2005, ISBN 0-471-52638-X, pp. 244–260.〕 ==Firmly non-expansive mapping== A non-expansive mapping with $k=1$ can be strengthened to a firmly non-expansive mapping in a Hilbert space ''H'' if the following holds for all ''x'' and ''y'' in ''H'': :$\backslash |f(x)-f(y)\; \backslash |^2\; \backslash leq\; \backslash ,\; \backslash langle\; x-y,\; f(x)\; -\; f(y)\; \backslash rangle.$ where :$d(x,y)\; =\; \backslash |x-y\backslash |$ This is a special case of $\backslash alpha$ averaged nonexpansive operators with $\backslash alpha\; =\; 1/2$.〔''Solving monotone inclusions via compositions of nonexpansive averaged operators'', Patrick L. Combettes, 2004〕 A firmly non-expansive mapping is always non-expansive, via the Cauchy–Schwarz inequality. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Contraction mapping」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.