Fixed points of fuzzy functions. (English) Zbl 0946.54036

Janiš, Vladimír

Tatra Mt. Math. Publ. 12, 13-19 (1997).

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

Keywords:

Hausdorff metric; fuzzy function; fuzzy numbers; Banach-type theorem

Citations:

Zbl 0828.26015