## Fixed points of fuzzy monotone maps.(English)Zbl 1047.03044

This short paper presents two theorems proving sufficient conditions for the existence of fixed points of a mapping that is fuzzy monotone with respect to a fuzzy order. From the various existing definitions of a fuzzy order, the author uses the following one, relying mainly on the ideas proposed in 1986 by C. Z. Luo and in 1987 by A. Billot [see, e.g., A. Billot, Economic theory of fuzzy equilibria. An axiomatic analysis. 2nd ed. (Springer, Berlin) (1995; Zbl 0861.90017)]: A fuzzy order on a crisp set $$X$$ is a normalized fuzzy binary relation $$R$$ on $$X$$ with a membership function $$r$$ fulfilling the conditions $$(\forall x,y \in X)r(x,y)\star _{\L }r(y,x)>0 \Rightarrow x=y$$ (antisymmetry), and $$(\forall x,y,z \in X)r(x,y)\geq r(y,x)\& r(y,z)\geq r(z,y)\Rightarrow r(x,z)\geq r(z,x)$$ (f-transitivity), where $$\star _{\L }$$ denotes the Łukasiewicz t-norm. In both theorems, a basic part of the sufficient condition is the requirement $$(\exists x\in X)r(x,f(x))\geq r(f(x),x)$$. In addition, the first theorem requires every fuzzy chain in $$X$$ to have a supremum, whereas the second one requires only every countable fuzzy chain in $$\{y:y\in X\& r(x,y)\geq r(y,x)\}$$ to have a supremum; instead the mapping is required to be not only fuzzy monotone, but also fuzzy order-continuous. A substantial difference between both theorems consists in the way how the existence of a fixed point of the considered mapping $$f$$ is established: In the first theorem, its existence follows from Fuzzy Zorn’s Lemma, whereas in the second theorem it is $$\sup _n f^n(x)$$ that is proved to be a fixed point of $$f$$, and the existence of that supremum is already assumed in the sufficient condition.
The theorems are proved in the setting of fuzzy sets with $$[0,1]$$-valued membership functions. In the reviewer’s opinion, they can be proved also with the more general membership functions assuming values in a linearly ordered lattice, and also the t-norm in the definition of a fuzzy order can be quite general.

### MSC:

 03E72 Theory of fuzzy sets, etc. 06A06 Partial orders, general

Zbl 0861.90017
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