Some Fixed Point Theorem for Successively Recurrent System of Set-Valued Mapping Equations

摘要

Let us introduce n (≥ 2) mappings f_i (i = 1, 2, . . ., n) defined on complete linear metric spaces (X_(i-1), ρ) (i = 1,2, ..., n), respectively, and let f_i : X_(i-1) → X_i be completely continuous on bounded convex closed subsets X_(i-1)~(0) is contained in X_(i-1), (i = 1, 2, ..., n ≡ 0), such that f_i(X_(i-1~(0))) is contained in X_i~(0). Moreover, let us introduce n set-valued mappings F_i : X_(i-1) X X_i → F(X_i)(the family of all non-empty closed compact subsets of X_i), (i = 1, 2. ..., n ≡ 0). Here, we have a fixed point theorem on the successively recurrent system of set-valued mapping equations: x_i ∈ F_i(x_(x-i), f_i(x_(i-1))), (i = 1, 2, ..., n ≡ 0). This theorem can be applied immediately to analysis of the availability of system of circular networks of channels undergone by uncertain fluctuations and to evaluation of the tolerability of behaviors of those systems. In this paper, mathematical situation and detailed proof are discussed, about this theorem.