A FLUCTUATION THEORY OF SYSTEMS BY FUZZY MAPPING CONCEPT AND ITS APPLICATIONS

This paper proposes a methodology for fine evaluation of the uncertain behaviors of systems affected by any fluctuation of internal structur es and internal parameters, by the use of a new concept on the fuzzy m apping. For a uniformly convex real Banach space X and Y, a fuzzy mapp ing G is introduced as the operator by which we can define a bounded c losed compact fuzzy set G(x,y) for any (x,y) is-an-element-of X x Y. A n original system is represented by a completely continuous operator f defined on X, for instance, in a form x = lambda(f(x) by a continuous operator lambda : Y --> X. The nondeterministic fluctuations induced into the original system are represented by a generalized form of the fuzzy mapping equation x is-an-element-of G(beta)(x, f(x)) DELTA {xi i s-an-element-of X \ mu(G)(x,f(x))(xi)) greater-than-or-equal-to beta}, in order to give a fine evaluation of the solutions with respect to a n arbitrarily-specified beta-level. By establishing a useful fixed poi nt theorem, the existence and evaluation problems of the ''beta-level- likely'' solutions are discussed for this fuzzy mapping equation. The theory developed here for the fluctuation problems is applied to the f ine estimation of not only the uncertain behaviors of system-fluctuati ons but also the validity of system-models and -simulations with uncer tain properties.