Common Fixed Point Theorems for a Pair of Weakly Compatible Mappings in Fuzzy Metric Spaces

We prove some common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces both in the sense of Kramosil and Michalek and in the sense of George and Veeramani by using the new property and give some examples. Our results improve and generalize the main results of Mihet in (Mihet, 2010) and many fixed point theorems in fuzzy metric spaces.


Introduction and Preliminaries
The notion of fuzzy sets was introduced by Zadeh 1 in 1965. Since that time a substantial literature has developed on this subject; see, for example, 2-4 . Fixed point theory is one of the most famous mathematical theories with application in several branches of science, especially in chaos theory, game theory, nonlinear programming, economics, theory of differential equations, and so forth. The works noted in 5-10 are some examples from this line of research.
Fixed point theory in fuzzy metric spaces has been developed starting with the work of Heilpern 11 . He introduced the concept of fuzzy mappings and proved some fixed point theorems for fuzzy contraction mappings in metric linear space, which is a fuzzy extension of the Banach's contraction principle. Subsequently several authors 12-20 have studied existence of fixed points of fuzzy mappings. Butnariu 21 also proved some useful fixed point results for fuzzy mappings. Badshah and Joshi 22 studied and proved a common fixed point theorem for six mappings on fuzzy metric spaces by using notion of semicompatibility and reciprocal continuity of mappings satisfying an implicit relation.
For the reader's convenience we recall some terminologies from the theory of fuzzy metric spaces, which will be used in what follows. ii a * 1 a for all a ∈ 0, 1 ; iv the mapping * : 0, 1 × 0, 1 → 0, 1 is continuous. In 1975, Kramosil and Michalek 4 gave a notion of fuzzy metric space which could be considered as a reformulation, in the fuzzy context, of the notion of probabilistic metric space due to Menger 24 .
From GV-1 and GV-2 , it follows that if x / y, then 0 < M x, y, t < 1 for all t > 0. In what follows, fuzzy metric spaces in the sense of George and Veeramani will be called GV-fuzzy metric spaces.
From now on, by fuzzy metric we mean a fuzzy metric in the sense of George and Veeramani. Several authors have contributed to the development of this theory, for instance 26-29 .
Then X, M, * is a GV-fuzzy metric space, called standard fuzzy metric space induced by X, d .
for all t > 0.
for all t > 0 and m ∈ N.  In 1995, Subrahmanyam 31 gave a generalization of Jungck's 32 common fixed point theorem for commuting mappings in the setting of fuzzy metric spaces. Even if in the recent literature weaker conditions of commutativity, as weakly commuting mappings, compatible mappings, R-weakly commuting mappings, weakly compatible mappings and several authors have been utilizing, the existence of a common fixed point requires some conditions on continuity of the maps, G-completeness of the space, or containment of ranges.
The concept of E.A. property in metric spaces has been recently introduced by Aamri and El Moutawakil 33 . for some t ∈ X.
The class of E.A. mappings contains the class of noncompatible mappings. In a similar mode, it is said that two self-mappings of f and T of a fuzzy metric space X, M, * satisfy E.A. property, if there exists a sequence {x n } in X such that fx n and gx n converge to t for some t ∈ X in the sense of Definition 1.7.
The concept of E.A. property allows to replace the completeness requirement of the space with a more natural condition of closeness of the range.
Recently, Mihet 34 proved two common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces both in the sense of Kramosil and Michalek and in the sense of George and Veeramani by using E.A. property.
Let Φ be class of all mappings ϕ : 0, 1 → 0, 1 satisfying the following properties: ϕ1 ϕ is continuous and nondecreasing on 0, 1 ; ϕ2 ϕ x > x for all x ∈ 0, 1 . for all x, y ∈ X. If f and g satisfy E.A. property and the range of g is a closed subspace of X, then f and g have a unique common fixed point.

Journal of Applied
We obtain that Theorems 1.13 and 1.14 require special condition, that is, the range of g is a closed subspace of X. Sometimes, the range of g maybe is not a closed subspace of X. Therefore Theorems 1.13 and 1.14 cannot be used for this case.
The aim of this work is to introduce the new property which is so called "common limit in the range" for two self-mappings f, g and give some examples of mappings which satisfy this property. Moreover, we establish some new existence of a common fixed point theorem 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 new property and give some examples. Ours results does not require condition of closeness of range and so our theorems generalize, unify, and extend many results in literature.

Common Fixed Point in KM and GV-Fuzzy Metric Spaces
We first introduce the concept of new property. for some x ∈ X.

Journal of Applied Mathematics
In what follows, the common limit in the range of g property will be denoted by the CLRg property.
Next, we show examples of mappings f and g which are satisfying the CLRg property. In a similar mode, two self-mappings f and g of a fuzzy metric space X, M, * satisfy the CLRg property, if there exists a sequence {x n } in X such that fx n and gx n converge to gx for some x ∈ X in the sense of Definition 1.7. It follows from the condition of ϕ2 that ϕ M gx, fx, t 0 > M gx, fx, t 0 , which is a contradiction. Therefore gx fx.
Next, we let z : fx gx. Since f and g are weakly compatible mappings, fgx gfx which implies that fz fgx gfx gz.

2.10
We claim that fz z. Assume not, then by 2.4 , it implies that 0 < M fz, z, t 1 < 1 for some t 1 > 0. By condition of ϕ2 , we have ϕ M fz, z, t 1 > M fz, z, t 1  for all t > 0, which is a contradiction. Hence fz z, that is, z fz gz. Therefore z is a common fixed point of f and g.
For the uniqueness of a common fixed point, we suppose that w is another common fixed point in which w / z. It follows from condition 2.4 that there exists t 2 > 0 such that