International Journal of Mathematics and Mathematical Sciences
Volume 18 (1995), Issue 3, Pages 447-449

A common fixed point theorem for a sequence of fuzzy mappings

B. E. Rhoades

Department of Mathematics, Indiana University, Bloomington 47405, Indiana, USA

Received 29 September 1994

Copyright © 1995 B. E. Rhoades. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 xX, 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 supxXA(x)=1. For A,BW(X), AB means A(x)B(x) for each xX. For A,BW(X), α[0,1], define Pα(A,B)=infxAα,yBαd(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  sD\{0} and Q(0)=0(b)  Q is nondecreasing on D, and(c)  g(s):=s/(sQ(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 x0X. Then there exists an x1X such that {x1}F(x0).