A common fixed point theorem for a sequence of fuzzy mappings

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 , A α : = { x : A ( x ) ≥ α , if α ∈ ( 0 , 1 ] } , and A 0 : = { x : A ( x ) > 0 } ¯ . 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 . For A , B ∈ W ( X ) , α ∈ [ 0 , 1 ] , define P α ( A , B ) = inf x ∈ A α , y ∈ B α d ( x , y ) , P ( A , B ) = sup α P α ( A , B ) , D ( A , B ) = sup α d H ( A α , B α ) , where d H is the Hausdorff metric induced by the metric d . We notc that P α 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 x 0 ∈ X . Then there exists an x 1 ∈ X such that { x 1 } ⊂ F ( x 0 ) .