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 . 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 urn:x-wiley:01611712:media:ijmm465769:ijmm465769-math-0002 , 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: urn:x-wiley:01611712:media:ijmm465769:ijmm465769-math-0003 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).