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Metric topology of fuzzy numbers and fuzzy analysis. (English) Zbl 0988.26019

Dubois, Didier (ed.) et al., Fundamentals of fuzzy sets. Foreword by Lotfi A. Zadeh. Dordrecht: Kluwer Academic Publishers. Handb. Fuzzy Sets Ser. 7, 583-641 (2000).
The chapter consists of 9 sections: Introduction, Calculus of Compact Convex Subsets in \(R^n\), The Space \(E^n\), Metrics on \(E^n\), Compactness Criteria, Fuzzy Set Valued Mappings of Real Variables, Interpolation and Approximation, Fuzzy Differential Equations and Conclusion. The authors use the methods similar to those of convex analysis to describe the properties of special types of sets and especially fuzzy sets in a finite-dimensional real vector space \(R^n\). Special attention is given to classes of compact and compact convex subsets of \(R^n\). The second section brings an overview of results achieved for multivalued mappings connected to their measurability, de Blasi and Hukuhara differentiability and Auman integrals. The notion of support functions is extensively used as a technical tool. The sections 3 and 4 study normal fuzzy convex upper semicontinuous fuzzy subsets of \(R^n\) with bounded support. The set of such fuzzy subsets is denoted by \(E^n\). Various metrics on this space are presented as well as properties of corresponding metric spaces. The section 5 deals with compactness in these metric spaces. The section 6 is devoted to mappings from the space \(R^k\) to \(E^n\). As a metric in the space of images the function \(d_\infty\) is used, where \(d_\infty (U,V)=\sup \{ d_H ([U]^\alpha\), \([V]^\alpha)\), \(\alpha \in [0,1] \}\), \(d_H\) is the Hausdorff metric in \(R^n\) and \([Z]^\alpha\) is the \(\alpha\)-cut of the fuzzy set \(Z\). Differentiation in the sense of De Blasi and Hukuhara is generalized from multivalued to fuzzy mappings as well as the Aumann integral. The seventh section introduces results on interpolation for a set of (crisp) nodes and fuzzy values. A Lagrange-type interpolation formula is constructed. Bernstein approximation is also shortly discussed. Section 8 brings information on fuzzy differential equations and differential inclusions. The paper contains an extensive collection of references (69 items) and sections 2-8 are followed by bibliographical notes.
For the entire collection see [Zbl 0942.00007].

MSC:

26E50 Fuzzy real analysis
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