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 .
Let be a complete linear metric space. A fuzzy set in is a function from into
. If , the function value is called the grade of membership of in . The -level
set of , , and . denotes the
collection of all the fuzzy sets in such that is compact and convex for each
and . For , means for each . For
, , define
where is the Hausdorff metric induced by the metric . We notc that is a nondecrcasing
function of and is a metric on .
Let be an arbitrary set, any linear metric space. is called a fuzzy mapping if is a
mapping from the set into .
In earlier papers the author and Bruce Watson,  and , 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 .
Let denote the closure of the range of . We shall be concerned with a function , defined
on and satisfying the following conditions:
LEMMA 1.  Let be a complete linear metric space, a fuzzy mapping from
into and . Then there exists an such that .