A COMMON FIXED POINT THEOREM FOR A SEQUENCE OF FUZZY MAPPINGS

B. E. RHOADES

《International Journal of Mathematics and Mathematical Sciences》 1995卷 (1995/12) Pp. 447-449

https://doi.org/10.1155/S0161171295000561

摘要

We obtain a common fixed point theorem for a sequence of fuzzy mappings, satisfying a contractive definition more general than that of Lee, Lee, Cho and Kim. Let (X, d) be a complete linear metric space. A fuzzy set A in X is a function from X into [0, 1]. If x ∈ X, the function value A(x) is called the grade of membership of X in A. The α-level set of A, A_α :={x: A(x)≥α, if α∈(0, 1]}, and A_0:= (The equation is abbreviated). W(X) denotes the collection of all the fuzzy sets A in X such that A_α is compact and convex for each α∈[0,1] and sup_(x∈X) A(x)=1. For A,B∈W(X), A⊂B means A(x)≤B(x) for each x∈X. ForA, B ∈ W(X),α∈[0,1], define(The equation is abbreviated) where d_H is the Hausdorff metric induced by the metric d. We note that P_α is a nondecreasing function of α and D is a metric on W(X). Let X be an arbitrary set, Y any linear metric space. F is called a fuzzy mapping if F is a mapping from the set X into W(Y). In earlier papers the author and Bruce Watson, [3] and [4], proved some fixed point theorems for some mappings satisfying a very general contractive condition. In this paper we prove a fixed point theorem for a sequence of fuzzy mappings satisfying a special case of this general contractive condition. We shall first prove the theorem, and then demonstrate that our definition is more general than that appearing in [2]. Let D denote the closure of the range of d. We shall be concerned with a function Q, defined on d and satisfying the following conditions:(a)  0＜Q(s)＜s for each  s ∈ D\{0} and Q(0)=0 (b)  Q is nondecreasing on D, and (c) g(s):=s/(s−Q(s)) is nonincreasing on D\{0}LEMMA 1. [1] Let (X, d) be a complete linear metric space, F a fuzzy mapping from X into W(X) and x_0∈X. Then there exists an x_1∈X such that {x1}⊂F(x_0).

關鍵字

Common fixed point ； fuzzy mappings