International Journal of Innovations in Engineering and Technology (IJIET) http://dx.doi.org/10.21172/ijiet.131.14 Volume 13 Issue 1 April 2019 106 ISSN: 2319-1058 Fuzzy Fixed Point Mappings on Metric Space J. Ravinder 1 , S. Srinivasan 2 1,2 Department of Mathematics and Actuarial Science, B.S.Abdur Rahman Crescent Institute of Science and Technology, Chennai-48, India. Abstract:- M.A. Ahamed [1] gave the refined and generalized the common fixed point theorem, which have been proved by S.C. Arora and C. Sharma [2]. In this paper, we shall improve the theorem of M.A. Ahamed [1]. Also, we establish the error estimation as well as the rate of convergence of generalized common fixed point theorem. Key Words:- Fuzzy mappings, Fuzzy fixed point mappings, Fixed point theorem, Metric spaces. I. INTRODUCTION The concept of Fuzzy sets was investigated by L.A. Zadeh [16] in 1965. Fuzzy metric space was introduced by Kramosil and Michalek [10] in 1975. Then in 1994, the notion of fuzzy metric spaces was modified by George and Veera-mani [6]. Many researchers have been obtained the common fixed point theorems for self mappings with different types of contraction and commu-tativity conditions. Sessa [14] was initiated the weakly commuting maps on metric spaces to improve commutativity in fixed point theorems, later on, this method was enlarged to compatible maps by Jungck [9]. Then Tas et.al [15] was extended the Jungck’s compatibility conditions to four self mappings on complete and compact metric spaces. Recently, Ahamed [1] generalized the improved results of S.C. Arora and V. Sharma [2]. This paper widely inspired by Tas et al. [15] and Ahamed [1]. We give different approach of Ahamed’s results and we establish the error estimation as well as the rate of convergence of common fuzzy fixed point mappings on metric spaces. II. PRELIMINARY NOTES Let X be any metric space with the metric d and I = [0, 1] be unit in-terval. A fuzzy set A in a metric space X is said to be an approximate quantity if and only if for each α ∈ I the α-level set of A is non empty compact convex set in X and supx∈ X A(x) = 1. W (X) is the family of all approximate quantities in X. That is, f or any α ∈ I, W (X) is given by {Aα ∈ IX : Aα is non empty compact convex set with supx∈ X A(x) = 1}, where IX is collection of fuzzy subsets of X. Note that, a set A is more accurate than the set B in W (X), denoted by A⊂B, if and only if A(x) ≤ B(x) for each x ∈ X, where A(x), B(x) denotes the membership values of x in X. For x ∈ X, we write {x} the characteristic function of the ordinary subset {x} of X. We denote W 0(X) = {{x} : x ∈ X}. For some α ∈ I and A, B ∈ W (X), pα(A, B) = inf d(x, y); Dα(A, B) = H(Aα, Bα); x∈ Aα,y∈ Bα and D(A, B) = sup Dα(A, B), p(A, B) = sup pα(A, B); α∈ I α∈ I where H is the Hausdorff metric induced by the metric d, pα is a non-decreasing function of α and D is a metric on W (X). Definition 1: [5] Let Y be an arbitrary set, X be a metric linear space. A mapping T : Y → W(X) is said to be a fuzzy mapping, if for each y ∈ Y, T y ∈ W(X). Thus if we characterize a fuzzy set Ty in a metric linear space X by a member ship function Ty, then Ty(x) is the grade of member ship of x in Ty. Note that, a fuzzy mapping T is a fuzzy subset on X×Y with membership function Tx(y). Definition 2: [14] Self-mappings f and g on a metric space (X, d) are said to weakly commute if and only if d(fgx, gfx) < d(fx, gx) ∀ x ∈ X. Definition 3: [9] Self-mappings f and g on a metric space (X, d) are said to be compatible if and only if whenever xn is a sequence in X such that limn→∞ fxn = limn→∞ gxn = t for some t ∈ X, then limn→∞ d(fgxn, gfxn) = 0. The following proposition and lemmas are needed in the sequel. Proposition 1: [9] Let A, B be compatible self mappings on a complete metric space (X, d). If for some t ∈ X, At = Bt, then ABt = BAt. Suppose that limn→∞ Axn = t = limn→∞ Bxn, for some t ∈ X.