A Mathematical Theory of System Fluctuations Using Fuzzy Mapping

Kazuo HORIUCHI  Yasunori ENDO  

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E76-A   No.5   pp.678-682
Publication Date: 1993/05/25
Online ISSN: 
DOI: 
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Neural Nets,Chaos and Numerics)
Category: Mathematical Theory
Keyword: 
nondeterministic,  fluctuation problem,  fuzzy mapping,  metric between fuzzy sets,  fixed point theorem,  

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Summary: 
In the direct product space of a complete metric linear space X and its related space Y, a fuzzy mapping G is introduced as an operator by which we can define a projective fuzzy set G(x,y) for any xX and yY. An original system is represented by a completely continuous operator f(x)Y, e.g., in the form x=λ(f(x)), (λ is a linear operator), and a nondeterministic or fuzzy fluctuation induced into the original system is represented by a generalized form of system equation xβG(x,f(x)). By establishing a new fixed point theorem for the fuzzy mapping G, the existence and evaluation problems of solution are discussed for this generalized equation. The analysis developed here for the fluctuation problem goes beyond the scope of the perturbation theory.