#### Abstract

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.

#### 1. Introduction and Preliminaries

The notion of fuzzy sets was introduced by Zadeh  in 1965. Since that time a substantial literature has developed on this subject; see, for example, . 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  are some examples from this line of research.

Fixed point theory in fuzzy metric spaces has been developed starting with the work of Heilpern . 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  have studied existence of fixed points of fuzzy mappings. Butnariu  also proved some useful fixed point results for fuzzy mappings. Badshah and Joshi  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.

Definition 1.1 (Schweizer and Sklar ). A continuous -norm is a binary operation on satisfying the following conditions: (i) is commutative and associative; (ii) for all ; (iii) whenever and ; (iv) the mapping is continuous.

Example 1.2. The following examples are classical examples of a continuous -norms. (TL)(the Lukasiewicz -norm). A mapping which defined through (TP)(the product -norm). A mapping which defined through (TM)(the minimum -norm). A mapping which defined through

In 1975, Kramosil and Michalek  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 .

Definition 1.3 (Kramosil and Michalek ). A fuzzy metric space is a triple where is a nonempty set, is a continuous -norm and is a fuzzy set on such that the following axioms hold: (KM-1) for all ; (KM-2) for all where ; (KM-3) for all ; (KM-4) is left continuous for all ; (KM-5) for all and for all .

We will refer to these spaces as KM-fuzzy metric spaces.

Lemma 1.4 (Grabiec ). For every , the mapping is nondecreasing on .

George and Veeramani [2, 25] introduced and studied a notion of fuzzy metric space which constitutes a modification of the one due to Kramosil and Michalek.

Definition 1.5 (George and Veeramani [2, 25]). A fuzzy metric space is a triple where is a nonempty set, is a continuous -norm and is a fuzzy set on and the following conditions are satisfied for all and : (GV-1); (GV-2); (GV-3); (GV-4) is continuous; (GV-5).

From (GV-1) and (GV-2), it follows that if , then for all . 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 .

Example 1.6. Let be a metric space, and, for all and , Then is a GV-fuzzy metric space, called standard fuzzy metric space induced by .

Definition 1.7. Let be a (KM- or GV-) fuzzy metric space. A sequence in is said to be convergent to if for all .

Definition 1.8. Let be a (KM- or GV-) fuzzy metric space. A sequence in is said to be -Cauchy sequence if for all and .

Definition 1.9. A fuzzy metric space is called -complete if every -Cauchy sequence converges to a point in .

Lemma 1.10 (Schweizer and Sklar ). If is a KM-fuzzy metric space and , are sequences in such that then for every continuity point of .

Definition 1.11 (Jungck and Rhoades ). Let be a nonempty set. Two mappings are said to be weakly compatible if for all which .

In 1995, Subrahmanyam  gave a generalization of Jungck's  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, -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, -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 .

Definition 1.12 (Aamri and El Moutawakil ). Let and be self-mapping of a metric space . We say that and satisfy E.A. property if there exists a sequence in such that for some .

The class of E.A. mappings contains the class of noncompatible mappings.

In a similar mode, it is said that two self-mappings of and of a fuzzy metric space satisfy E.A. property, if there exists a sequence in such that and converge to for some 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  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 be class of all mappings satisfying the following properties:(φ1)φ is continuous and nondecreasing on ; (φ2) for all .

Theorem 1.13 (see [34, Theorem  2.1]). Let be a KM-fuzzy metric space satisfying the following property: and let be weakly compatible self-mappings of such that, for some , for all where . If and satisfy E.A. property and the range of is a closed subspace of , then and have a unique common fixed point.

Theorem 1.14 (see [34, Theorem  3.1]). Let be a GV-fuzzy metric space and weakly compatible self-mappings of such that, for some and some , for all . If and satisfy E.A. property and the range of is a closed subspace of , then and have a unique common fixed point.

We obtain that Theorems 1.13 and 1.14 require special condition, that is, the range of is a closed subspace of . Sometimes, the range of maybe is not a closed subspace of . 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 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.

#### 2. Common Fixed Point in KM and GV-Fuzzy Metric Spaces

We first introduce the concept of new property.

Definition 2.1. Suppose that is a metric space and . Two mappings and are said to satisfy the common limit in the range of g property if for some .

In what follows, the common limit in the range of property will be denoted by the (CLRg) property.

Next, we show examples of mappings and which are satisfying the (CLRg) property.

Example 2.2. Let be the usual metric space. Define by and for all . We consider the sequence . Since therefore and satisfy the (CLRg) property.

Example 2.3. Let be the usual metric space. Define by and for all . Consider the sequence . Since therefore and satisfy the (CLRg) property.

In a similar mode, two self-mappings and of a fuzzy metric space satisfy the (CLRg) property, if there exists a sequence in such that and converge to for some in the sense of Definition 1.7.

Theorem 2.4. Let be a KM-fuzzy metric space satisfying the following property: and let be weakly compatible self-mappings of such that, for some , for all , where . If and satisfy the (CLRg) property, then and have a unique common fixed point.

Proof. Since and satisfy the (CLRg) property, there exists a sequence in such that for some . Let be a continuity point of . Then for all . By making , we have for every . We claim that . If not, then It follows from the condition of that , which is a contradiction. Therefore .
Next, we let . Since and are weakly compatible mappings, which implies that We claim that . Assume not, then by (2.4), it implies that for some . By condition of , we have . Using condition (2.5) again, we get for all , which is a contradiction. Hence , that is, . Therefore is a common fixed point of and .
For the uniqueness of a common fixed point, we suppose that is another common fixed point in which . It follows from condition (2.4) that there exists such that . Since , we have by virtue of (). From (2.5), we have for all , which is a contradiction. Therefore, it must be the case that which implies that and have a unique a common fixed point. This finishes the proof.

Next, we will give example which cannot be used [34, Theorem  2.1]. However, we can apply Theorem 2.4 for this case.

Example 2.5. Let and, for each and , It is well known (see ) that is a GV-fuzzy metric space. If the mappings are defined on through and , then the range of is which is not a closed subspace of . So Theorem  2.1 of Mihet in  cannot be used for this case. It is easy to see that the mappings and satisfy the (CLRg) property with a sequence . Therefore all hypothesis of the above theorem holds, with for . Their common fixed point is .

Corollary 2.6 ([34, Theorem  2.1]). Let be a KM-fuzzy metric space satisfying the following property: and let be weakly compatible self-mappings of such that, for some , for all , where . If and satisfy E.A. property and the range of is a closed subspace of , then and have a unique common fixed point.

Proof. Since and satisfy E.A. property, there exists a sequence in such that for some . It follows from being a closed subspace of that for some and then and satisfy the (CLRg) property. By Theorem 2.4, we get that and have a unique common fixed point.

Corollary 2.7. Let be a KM-fuzzy metric space satisfying the following property: and let be weakly compatible self-mappings of such that, for some , for all , where and If and satisfy the (CLRg) property, then and have a unique common fixed point.

Proof. As is nondecreasing and for , we have So inequality (2.18) implies that By Theorem 2.4, we get and have a unique common fixed point.

If is a fuzzy metric space in the sense of George and Veeramani, then some of the hypotheses in the preceding theorem can be relaxed.

Theorem 2.8. Let be a GV-fuzzy metric space and weakly compatible self-mappings of such that, for some , for all , where . If and satisfy the (CLRg) property, then and have a unique common fixed point.

Proof. It follows from and satisfying the (CLRg) property that we can find a sequence in such that for some .
Let be a continuity point of . Then for all . By taking the limit as tends to infinity in (2.25), we have for every . Now, we show that . If , then from (GV-1) and (GV-2), for all . From condition of , which is a contradiction. Hence .
Similarly in the proof of Theorem 2.4, by denoting a point by . Since and are weakly compatible mappings, which implies that .
Next, we will show that . We will suppose that . By (GV-1) and (GV-2), it implies that for all . By , we know that . It follows from condition (2.23) that for all , which is contradicting the above inequality. Therefore , and then . Consequently, and have a common fixed point that is .
Finally, we will prove that a common fixed point of and is unique. Let us suppose that is a common fixed point of and in which . It follows from condition of (GV-1) and (GV-2) that for every , we have which implies that . On the other hand, we know that for all , which is contradiction. Hence we conclude that . It finishes the proof of this theorem.

Corollary 2.9 ([34, Theorem  3.1]). Let be a GV-fuzzy metric space and weakly compatible self-mappings of such that, for some , for all , where . If and satisfy E.A. property and the range of is a closed subspace of , then and have a unique common fixed point.

Proof. Since and satisfy E.A. property, there exists a sequence in satisfies for some . It follows from being a closed subspace of that there exists in which . Therefore and satisfy the (CLRg) property. It follows from Theorem 2.8 that there exists a unique common fixed point of and .

Corollary 2.10. Let be a GV-fuzzy metric space and weakly compatible self-mappings of such that, for some , for all , where and If and satisfy the (CLRg) property, then and have a unique common fixed point.

Proof. Since is nondecreasing and where we get Now, we know that inequality (2.32) implies that It follows from Theorem 2.8 that and have a unique common fixed point.

#### Acknowledgments

The authors would like to thank the reviewer, who have made a number of valuable comments and suggestions which have improved the paper greatly. The first author would like to thank the Research Professional Development Project under the Science Achievement Scholarship of Thailand (SAST) and the Faculty of Science, KMUTT for financial support during the preparation of this paper for the Ph.D. Program at KMUTT. Moreover, they also would like to thank the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission for financial support (NRU-CSEC Project no. 54000267). This work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission.