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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)].

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