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.
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).