International Journal of Mathematics and Mathematical Sciences
Volume 18 (1995), Issue 3, Pages 447-449
doi:10.1155/S0161171295000561

A common fixed point theorem for a sequence of fuzzy mappings

Abstract

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 [2].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, Aα:={x:A(x)≥α, if α∈(0,1]}, and A0:={x:A(x)>0}¯. W(X) denotes the collection of all the fuzzy sets A in X such that Aα is compact and convex for each α∈[0,1] and supx∈XA(x)=1. For A,B∈W(X), A⊂B means A(x)≤B(x) for each x∈X. For A,B∈W(X), α∈[0,1], define Pα(A,B)=infx∈Aα,y∈Bαd(x,y), P(A,B)=supαPα(A,B), D(A,B)=supαdH(Aα,Bα), where dH is the Hausdorff metric induced by the metric d. We notc that Pα is a nondecrcasing 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 x0∈X. Then there exists an x1∈X such that {x1}⊂F(x0).