Summary
International Symposium on Nonlinear Theory and its Applications
2009
Session Number:A4L-C
Session:
Number:A4L-C1
A Fixed Point Theorem in Weak Topology for Successively Recurrent System of Fuzzy-Set-Valued Nonlinear Mapping Equations and Its Application
Kazuo HORIUCHI,
pp.-
Publication Date:2009/10/18
Online ISSN:2188-5079
Summary:
Let us introduce n (? 2) nonlinear mappings fi (i = 1, . . . , n ) defined on reflexive real Banach spaces Xi?1 and let fi : Xi?1 → Yi (a Banach space) be completely continuous on bounded convex closed subsets X(0) i?1 ⊂ Xi?1 . Moreover, let us introduce n fuzzy-set-valued nonlinear mappings Fi : Xi?1×Yi →{ a family of all non-empty closed compact fuzzy subsets of Xi} .Here, by introducing arbitrary constant βi ∈ (0, 1], for every integer i (i = 1, . . . , n ≡ 0), separately, we have a fixed point theorem on the recurrent system of βi -level fuzzy-set-valued mapping equations: xi ∈ Fiβi (xi?1, fi (xi?1 )), (i = 1, . . . , n ≡ 0), where the fuzzy set Fi is characterized by a membership function μFi (xi ) : Xi → [0, 1], and the βi -level set Fiβi of the fuzzy set Fi is defined as Fiβi △= {ξi ∈ Xi | μFi (ξi ) ? βi} , for any constant βi ∈ (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, the mathematical situation and proof are discussed, in weak topology.