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In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contraction mapping principle) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides a constructive method to find those fixed points. The theorem is named after Stefan Banach (1892–1945), and was first stated by him in 1922.〔http://www.emis.de/journals/BJMA/tex_v1_n1_a1.pdf〕 ==Statement== ''Definition.'' Let (''X'', ''d'') be a metric space. Then a map ''T'' : ''X'' → ''X'' is called a contraction mapping on ''X'' if there exists ''q'' ∈ [0, 1) such that : for all ''x'', ''y'' in ''X''. Banach Fixed Point Theorem. Let (''X'', ''d'') be a non-empty complete metric space with a contraction mapping ''T'' : ''X'' → ''X''. Then ''T'' admits a unique fixed-point ''x ''Remark 1.'' The following inequalities are equivalent and describe the speed of convergence: : Any such value of ''q'' is called a ''Lipschitz constant'' for ''T'', and the smallest one is sometimes called "the best Lipschitz constant" of ''T''. ''Remark 2.'' ''d''(''T''(''x''), ''T''(''y'')) < ''d''(''x'', ''y'') for all ''x'' ≠ ''y'' is in general not enough to ensure the existence of a fixed point, as is shown by the map ''T'' : [1, ∞) → [1, ∞), ''T''(''x'') = ''x'' + 1/''x'', which lacks a fixed point. However, if ''X'' is compact, then this weaker assumption does imply the existence and uniqueness of a fixed point, that can be easily found as a minimizer of ''d''(''x'', ''T''(''x'')), indeed, a minimizer exists by compactness, and has to be a fixed point of ''T''. It then easily follows that the fixed point is the limit of any sequence of iterations of ''T''. ''Remark 3.'' When using the theorem in practice, the most difficult part is typically to define ''X'' properly so that ''T''(''X'') ⊆ ''X''. 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「Banach fixed-point theorem」の詳細全文を読む スポンサード リンク
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