Fixed points for fuzzy mappings |
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Authors: | Dan Butnariu |
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Affiliation: | Dept. of Mathematics, Polytechnic Institute of Iassy, Iassy 6600, Romania |
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Abstract: | In this paper the problem of the existence and computation of fixed points for fuzzy mappings is approached. A fuzzy mapping R over a set X is defined to be a function attaching to each x in X a fuzzy subset Rχ of X. An element x of X is called fixed point of R iff its membership degree to Rχ is at least equal to the membership degree to Rχ of any y?X, i.e. Rχ(χ)? Rχ(y)(?y?X). Two existence theorems for fixed points of a fuzzy mapping are proved and an algorithm for computing approximations of such a fixed point is described. The convergence theorem of our algorithm is proved under the restrictive assumption that for any x in X, the membership function of Rχ has a ‘complementary function’. Examples of fuzzy mappings having this property are given, but the problem of proving general criteria for a function to have a complementary remain open. |
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Keywords: | Fuzzy sets Fuzzy mapping Fixed point Triangulation Linear function relative to a triangulation Kuhn's fixed point algorithm Eaves' fixed point algorithm |
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