Two topics in nonlinear system analysis through fixed point theorems

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 introduced 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.