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Fixed points of fuzzy functions. (English) Zbl 0946.54036
Summary: By a fuzzy function we understand a function \(f\) defined on a metric space \((X,d)\), whose values are fuzzy numbers. We deal with fuzzy mappings of a complete metric space into itself. An \(\varepsilon\)-fixed point of \(f\) is a point \(x_0\) such that the value of the membership function which is the image of \(x_0\) is greater or equal to \(1-\varepsilon\) in \(x_0\). A Banach-type theorem on existence of such point is proved. It is also shown that the set of all \(\varepsilon\)-fixed points is closed and bounded in \((X,d)\). The second part of the paper deals with the fixed point of a fuzzy continuous function as it is defined in [M. Burgin and A. Šostak, Fuzzy Sets Syst. 62, No. 1, 71-81 (1994; Zbl 0828.26015)].

54H25 Fixed-point and coincidence theorems (topological aspects)
26E50 Fuzzy real analysis
54A40 Fuzzy topology
54E40 Special maps on metric spaces