COMMON FIXED POINT THEOREMS IN KM AND GV-FUZZY METRIC SPACES

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, we prove some common fixed point theorems using (CLR) property in KM and GV-fuzzy metric spaces. Our results extend and unify several fixed point theorems present in the literature.


INTRODUCTION
Zadeh [4] introduced the concept of fuzzy sets in 1965. Fixed point theory is one of the most famous mathematical theories with application in several branches of science, especially in nonlinear programming, economics, theory of differential equations. Fixed point theory in fuzzy metric spaces has been developed starting with the work of Heilpern [6]. He proved some fixed point theorems for fuzzy contraction mappings in metric linear space, which is a fuzzy extension of the Banach's contraction principle. The concept of fuzzy metric space was introduced by Kramosil and Michalek [3] in 1975, which opened an avenue for further development of analysis in such spaces. Further, George and Veeramani [1] modified the concept of fuzzy metric space introduced by Kramosil and Michalek [3]. Sintunavarat and Kumam [8] defined the notion of "common limit in the range" property (or (CLR) property) in fuzzy metric spaces.
The aim of this work is to prove some common fixed point theorems for generalized contractive mappings in fuzzy metric spaces both in the sense of Kramosil and Michalek and in the sense of George and Veeramani by using (CLR) property. Our results does not require condition of closeness of range and so our theorems generalize, unify, and extend many results in literature.

PRELIMINARIES
Definition 2.1. [8] Let (X, M, * ) be a fuzzy metric space and P, Q, R, S be self maps on X. The pairs (P, Q) and (R, S) are said to satisfy the joint common limit in the range of mappings (JCLR) property if there exists sequences {x n } and {y n } in X such that lim n→∞ Px n = lim n→∞ Qx n = lim n→∞ Ry n = lim n→∞ Sy n = Qv = Rv, for some v ∈ X. (ii) * is continuous ; In 1975, Kramosil and Michalek [3] gave a notion of fuzzy metric space which could be considered as a reformulation, in the fuzzy system. (ii) M(x, y, z,t) = 1 if atleast two of x,y,z of X are equal; George and Veeramani [1] introduced and studied a notion of fuzzy metric space which constitutes a modification of the one due to Kramosil and Michalek.
is a nonempty set, * is a continuous t-norm and M is a fuzzy set on X 3 × [0, 1] and the following conditions are satisfied for all x, y ∈ X and t, s > 0: and M and N be fuzzy sets on X 3 × [0, ∞) defined as follows: for all h, n ∈ R + . Then(X, M, * , ⊕) is called a fuzzy metric space.
Following notion will be used in the sequel to prove our theorems: Let ℑ be class of all mappings f : [0, 1] → [0, 1] satisfying the following properties: (1) f is continuous and non decreasing on [0, 1]; (2) f x > x for all x ∈ (0, 1). where t > 0 and 1 ≤ k < 2. If P and Q satisfy (CLR Q ) property, then P and Q have a unique common fixed point in X.

COMMON FIXED POINT IN KM AND GV-FUZZY METRIC SPACES
Proof. Since P and Q satisfy the CLR Q property, there exists a sequence {x n } in X such that Let Pv = Qv. It is not difficult to prove that there exists t 0 > 0 such that If equation (3) does not hold, then we have for all t > 0. Therefore, Qv = Pv = z. Now, we claim that z is common fixed point of Q and P.
Since P and Q are weakly compatible, we have Pz = Qz.
Let us suppose that Pz = z.
Substituting Px = z and y = z in the given condition, we get for all ε ∈ (0, 2t k ). When ε −→ 0, we get which is a contradiction.
So, Pz = Qz = z. Hence, z is a common fixed point of P and Q. Uniqueness of z follows from (1).
Theorem 3.2. Let (X, M, * ) be a GV-fuzzy metric space satisfying the following property: ∀x, y ∈ X, x = y, ∃t > 0 : 0 < M(x, y,t) < 1 and let P, Q be weakly compatible self-mappings of where t > 0 and 1 ≤ k < 2. If P and Q satisfy (CLR Q ) property,then P and Q have a unique common fixed point in X.
Proof. Since P and Q satisfy the CLR Q property,there exists a sequence {x n } in X such that lim n→∞ Px n = lim n→∞ Qx n = Qx Let Pv = Qv. It is not difficult to prove that there exists t 0 > 0 such that If equation (6) does not hold,then we have for all t > 0.
By repeating this process, we obtain . When ε −→ 0,we obtain M(Qv, Pv,t 0 ) ≥ M(Qv, Pv, 2t 0 k ) which is a contradiction Therefore, Qv = Pv = z Now, we claim that z is common fixed point of Q and P.
Since P and Q are weakly compatible,we have Pz = Qz.
Let us suppose that Pz = z.
Substituting Px = z and y = z in the given condition, we get for all ε ∈ (0, 2t k ). When ε −→ 0,we get which is a contradiction.
So,Pz = Qz = z. Hence, z is a common fixed point of P and Q.
Uniqueness of z follows form (1). Corollary 3.3. Let P and Q be self-mappings of a fuzzy metric space (X, M, * ) satisfying the following conditions: (i) P n (X) ⊆ Q m (X), P n Q = QP n and PQ m = Q m P (ii) M(P n x, P n y,t) ≥ f (min{M(Q m x, Q m x,t), sup t 1 +t 2 = 2t k min{M(Q m x, P n x,t 1 ), M(Q m y, P n y,t 2 )}, sup t 3 +t 4 = 2t k max{M(Q m x, P n y,t 3 ), M(Q m y, P n x,t 4 )}}) for all x, y ∈ X,for some n, m = 2, 3, . . . ,t > 0 and for some 1 ≤ k < 2. Suppose that the pair (P n , Q m ) satisfies the property JCLR and (P n , Q m ) is weakly compatible. Then P and Q have a unique common fixed point in X.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.