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In mathematics, a fixed-point theorem is a result saying that a function ''F'' will have at least one fixed point (a point ''x'' for which ''F''(''x'') = ''x''), under some conditions on ''F'' that can be stated in general terms.〔 〕 Results of this kind are amongst the most generally useful in mathematics. == In mathematical analysis == The Banach fixed-point theorem gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point. By contrast, the Brouwer fixed-point theorem is a non-constructive result: it says that any continuous function from the closed unit ball in ''n''-dimensional Euclidean space to itself must have a fixed point,〔 Eberhard Zeidler, ''Applied Functional Analysis: main principles and their applications'', Springer, 1995.〕 but it doesn't describe how to find the fixed point (See also Sperner's lemma). For example, the cosine function is continuous in () and maps it into (1 ), and thus must have a fixed point. This is clear when examining a sketched graph of the cosine function; the fixed point occurs where the cosine curve ''y''=cos(''x'') intersects the line ''y''=''x''. Numerically, the fixed point is approximately ''x''=0.73908513321516 (thus ''x''=cos(''x'') for this value of ''x''). The Lefschetz fixed-point theorem (and the Nielsen fixed-point theorem) from algebraic topology is notable because it gives, in some sense, a way to count fixed points. There are a number of generalisations to Banach fixed-point theorem and further; these are applied in PDE theory. See fixed-point theorems in infinite-dimensional spaces. The collage theorem in fractal compression proves that, for many images, there exists a relatively small description of a function that, when iteratively applied to any starting image, rapidly converges on the desired image. 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「Fixed-point theorem」の詳細全文を読む スポンサード リンク
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