# A Refined Fixed Point Theorem for Recurrent System of Fuzzy-Set-Valued Nonlinear Mapping Equations and Its Application to Ring Nonlinear Network Systems(Nonlinear Theory and its Applications)

## 概要

Let us introduce n (≥ 2) nonlinear mappings f_I(I = 1,2,・・・,n) defined on complete linear metric spaces (X_<I-1>,p)(I = 1,2,・・・,n), respectively, and let fi : X_<I--1> → X_I be completely continuous on boundedconvex closed subsets X^<(0)>_<I-1> ⊂ X_<I-1>,(I = 1,2,・・・,n ≡ 0), such that f_I(X^<(0)>_<I-1>) ⊂ X^<(0)>_I . Moreover, let us introduce n fuzzy-set-valued nonlinear mappings F_I : X_<I-1> × X_I → {a family of all non-empty closed compact fuzzy subsets of X_I}. Here, by introducing arbitrary constant β, ∈ (0,1], for every integer I(I = 1,2,・・・,n ≡ 0), separately, we have a fixed point theorem on the recurrent system of β_I-level fuzzy-set-valued mapping equations: x_I ∈ F_<iβ_I>(x_<I-1>,f_I(x_<I-1>), (I = 1,2,・・・,n = 0), where the fuzzy set F_I is characterized by a membership function μ_<F_I>(x_I) : X_I→ [0,1], and the β_I-level set F_<iβ_I> of the fuzzy set F_I is defined as F_<iβ_I> ≜ {ξ_I ∈ X_I | μ_<F_I>(ξ_I) ≥ β_I}, for any constant β_I ∈ (0,1]. This theorem can be applied immediately to discussion for characteristics of ring nonlinear network systems disturbed by undesirable uncertain fluctuations and to extremely fine estimation of available behaviors of those disturbed systems. In this paper, its mathematical situation and proof are discussed, in detail.