This paper proposes a methodology for fine evaluation of the uncertain behaviors of systems affected by any fluctuation of internal structures and internal parameters, by the use of a new concept on the fuzzy mapping. For a uniformly convex real Banach space X and Y, a fuzzy mapping G is introduced as the operator by which we can define a bounded closed compact fuzzy set G(x,y) for any (x,y)∈X×Y. An original system is represented by a completely continuous operator f defined on X, for instance, in a form xλ(f(x)) by a continuous operator λ: YX. The nondeterministic fluctuations induced into the original system are represented by a generalized form of the fuzzy mapping equation xGβ (x,f(x)) {ζX|µG(x,f(x))(ζ)β}, in order to give a fine evaluation of the solutions with respect to an arbitrarily–specified β–level. By establishing a useful fixed point theorem, the existence and evaluation problems of the "β–level-likely" solutions are discussed for this fuzzy mapping equaion. The theory developed here for the fluctuation problems is applied to the fine estimation of not only the uncertain behaviors of system–fluctuations but also the validity of system–models and -simulations with uncertain properties.

This paper reviews two topics of nonlinear system analysis done in Japan. The first half of this paper concerns with nonlinear system analysis through the nondeterministic operator theory. The nondeterministic operator is a set-valued or fuzzy set valued operator by K. Horiuchi. From 1975 Horiuchi has developed fixed point theorems for nondeterministic operators. Using such fixed point theorems, he developed a unique theory for nonlinear system analysis. Horiuchi's theory provides a fundamental view point for analysis of fluctuations in nonlinear systems. In this paper, it is pointed out that Horiuchi's theory can be viewed as an extension of the interval analysis. Next, Urabe's theory for nonlinear boundary value problems is discussed. From 1965 Urabe has developed a method of computer assisted existence proof for solutions of nonlinear boundary value problems. Urabe has presented a convergence theorem for a certain simplified Newton method. Urabe's theorem is essentially based on Banach's contraction mapping theorem. In this paper, reformulation of Urabe's theory using the interval analysis is presented. It is shown that sharp error estimation can be obtained by this reformulation. Both works discussed in this paper have been done independently with the interval analysis. This paper points out that they have deep relationship with the interval analysis. Moreover, it is also pointed out that these two works suggest future directions of the interval analysis.

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.