In 2002, Aamri and El-Moutawakil [2] defined the notion of (E.A) property for self **mappings** which contained the class of non-**compatible** **mappings** in **metric** **spaces**. It was pointed out that (E.A) property allows replacing the com- pleteness requirement of the space with a more natural condition of closedness of the range as well as relaxes the compleness of the whole space, continuity of one or more **mappings** and containment of the range of one mapping into the range of other which is utilized to construct the sequence of joint iterates. Many authors have proved **common** **fixed** **point** **theorems** in **fuzzy** **metric** **spaces** for different contractive conditions. For details, we refer to [5, 10, 11, 15– 17, 20, 24–26, 29, 30, 32–34, 36, 37]. Recently, Sintunavarat and Kumam [35] defined the notion of (CLRg) property in **fuzzy** **metric** **spaces** and improved the results of Mihet¸ [18] without any requirement of the closedness of the subspace. In this paper, we prove a **common** **fixed** **point** theorem for a pair of **weakly** **compatible** **mappings** by using (CLRg) property in **fuzzy** **metric** space. We also present a **common** **fixed** **point** theorem for two finite families of self **mappings** in **fuzzy** **metric** space by using the notion of pairwise commuting due to Imdad, Ali and Tanveer [12]. Our results improve the results of Sedghi, Shobe and Aliouche [31].

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In this paper, we prove a **common** **fixed** **point** theorem for a pair of **weakly** **compatible** **mappings** in **fuzzy** **metric** space using the joint **common** limit in the range property of **mappings** called (JCLR) property. An example is also furnished which demonstrates the validity of main result. We also extend our main result to two finite families of self **mappings**. Our results improve and generalize results of Cho et al. [Y. J. Cho, S. Sedghi and N. Shobe, “Generalized **fixed** **point** **theorems** for **compatible** **mappings** with some types in **fuzzy** **metric** **spaces**,” Chaos, Solitons & Fractals, Vol. 39, No. 5, 2009, pp. 2233-2244.] and several known results existing in the literature.

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11. S.Manro, S.S.Bhatia and S.Kumar, **Common** **fixed** **point** **theorems** for **weakly** **compatible** maps satisfying **common** (E.A) property in intuitionistic **fuzzy** **metric** **spaces** using implicit relation, Journal of Advanced Studies in Topology, 3(2) (2012) 38-44.

the help of continuous t- norm and continuous t- conorm, as a generalization of **fuzzy** **metric** space due to George and Veeramani [6]. Alaca et al.[3] defined the notion of intuitionistic **fuzzy** **metric** **spaces** and proved **common** **fixed** **point** **theorems**. Turkoglu et al. [14] first formulate the definition of **weakly** commuting

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In 2008, Al-Thagafi and Shahzad [Generalized I-nonexpansive selfmaps and invariant approxima- tions, Acta Math. Sinica 24(5) (2008), 867–876] introduced the notion of occasionally **weakly** com- patible **mappings** (shortly owc maps) which is more general than all the commutativity concepts. In the present paper, we prove **common** **fixed** **point** **theorems** for families of owc maps in Menger **spaces**. As applications to our results, we obtain the corresponding **fixed** **point** **theorems** in **fuzzy** **metric** **spaces**. Our results improve and extend the results of Kohli and Vashistha [**Common** **fixed** **point** **theorems** in probabilistic **metric** **spaces**, Acta Math. Hungar. 115(1-2) (2007), 37-47], Vasuki [**Common** **fixed** points for R-**weakly** commuting maps in **fuzzy** **metric** **spaces**, Indian J. Pure Appl. Math. 30 (1999), 419–423], Chugh and Kumar [**Common** **fixed** **point** theorem in **fuzzy** **metric** **spaces**, Bull. Cal. Math. Soc. 94 (2002), 17–22] and Imdad and Ali [Some **common** **fixed** **point** **theorems** in **fuzzy** **metric** **spaces**, Math. Commun. 11(2) (2006), 153-163].

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The evolution of **fuzzy** mathematics commenced with the introduction of the notion of **fuzzy** sets by Zadeh [18] in 1965, as a new way to represent the vague- ness in every day life. In mathematical programming, problems are expressed as optimizing some goal function given certain constraints, and there are real life problems that consider multiple objectives. Generally, it is very difficult to get a feasible solution that brings us to the optimum of all objective functions. A possible method of resolution, that is quite useful, is the one using **fuzzy** sets [17]. The concept of **fuzzy** **metric** space has been introduced and generalized by many ways ( [4], [7] ). George and Veeramani ( [5] ) modified the concept of **fuzzy** **metric** space introduced by Kramosil and Michalek [8]. They also ob- tained a Hausdorff topology for this kind of **fuzzy** **metric** space which has very important applications in quantum particle physics, particularly in connection with both string and ∞ theory (see, [12] and references mentioned therein). Many authors have proved **fixed** **point** and **common** **fixed** **point** **theorems** in **fuzzy** **metric** **spaces** ( [10], [13], [16]). Regan and Abbas [14] obtained some necessary and sufficient conditions for the existence of **common** **fixed** **point** in **fuzzy** **metric** **spaces**. Recently, Cho et al [3] established some **fixed** **point** theo- rems for **mappings** satisfying generalized contractive condition in **fuzzy** **metric** space. The aim of this paper is to obtain **common** **fixed** **point** of **mappings** sat- isfying generalized contractive type conditions without exploiting the notion of continuity in the setting of **fuzzy** **metric** **spaces**. Our results generalize several comparable results in existing literature (see, [3], [2] and references mentioned therein).

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Meanwhile, jungck[16] defined the concept of **compatible** **mappings**, Jungck and Roadhes [17] generalized the last concept to the **weakly** **compatible** **mappings**, which is weaken than the **compatible** ones. Mishra et al [20] generalized the concept of compatibility in the setting of **fuzzy** **metric** **spaces**, he obtained some **common** **fixed** **point** **theorems** for **compatible** **mappings** in such **spaces**. Recently Bouhadjera and Godet Tobie [4] introduced the concept of subsequential continuity and utilized it with the concept of subcompatible **mappings** to establish a **common** **fixed** **point**, later Imdad et al.[13] improved these results and replaced subcompatibility by compatibility and subsequential continuity by reciprocal continuity, more recently, Gopal and Imdad [11] combined subsequential continuous maps with **compatible** maps concept to obtain some results in **fuzzy** **metric** **spaces**. In present work, we will generalize certain definitions to intuitionistic **fuzzy** **metric** **spaces** in order to obtain some **common** **fixed** **point** **theorems** by combining the concept of **weakly** subsequentially continuous **mappings** due to second author [3] with **compatible** of type (E) **mappings** given by Singh et al.[27, 28].

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The aim of present paper is to introduce the notion of t- conorm of H-type analogous to t-norm of H-type introduced by Hadzic [9] and using this notion we prove coupled **fixed** **point** **theorems** for **weakly** **compatible** **mappings** in intuitionistic **fuzzy** **metric** **spaces**.

The notion of **fuzzy** sets was introduced by Zadeh in 1965 which laid the foundations of **fuzzy** set theory and **fuzzy** mathematics. In 1986, Atanassov generalized the notion of **fuzzy** sets by treating membership as a **fuzzy** logical value rather than a single truth value and introduced the notion of intuitionistic **fuzzy** sets. The intuitionistic **fuzzy** **fixed** **point** theory has become an area of interest for specialists in **fixed** **point** theory as intuitionistic **fuzzy** mathematics has covered new possibilities for **fixed** **point** theorists. In this paper, we use the idea of chainable intuitionistic **fuzzy** **metric** **spaces** and prove a **common** **fixed** **point** theorem for four **weakly** **compatible** **mappings** of chainable intuitionistic **fuzzy** **metric** **spaces**.

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Abstract. The aim of this paper is to prove some **common** **fixed** **point** **theorems** for two **weakly** subsequentially continuous and **compatible** of type (E) pairs of self **mappings** satisfying an implicit relation in **fuzzy** **metric** **spaces**. Two examples are given to illustrate our results.

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In the last years, **fixed** **point** **theorems** have been applied to show the existence and uniqueness of the solutions of diﬀerential equations, integral equations and many other branches mathematics see, e.g., 1–3. Some **common** **fixed** **point** **theorems** for **weakly** commuting, **compatible**, δ-**compatible** and **weakly** **compatible** **mappings** under diﬀerent contractive conditions in **metric** **spaces** have appeared in 4–15. Throughout this paper, X, d is a **metric** space.

The concept of **Fuzzy** sets was initially investigated by Zadeh [14]. Subsequently, it was developed by many authors and used in various fields. To use this concept, several researchers have defined **Fuzzy** **metric** space in various ways. In 1986, Jungck [6] introduced the notion of **compatible** maps for a pair of self **mappings**. However, the study of **common** **fixed** points of non-**compatible** maps is also very interesting. Aamri and El Moutawakil [15] generalized the concept of non-compatibility by defining the notion of property (E.A) and in 2005, Liu, Wu and Li [23] defined **common** (E.A) property in **metric** **spaces** and proved **common** **fixed** **point** **theorems** under strict contractive conditions. Jungck and Rhoades [7] initiated the study of **weakly** **compatible** maps in **metric** space and showed that every pair of **compatible** maps is **weakly** **compatible** but reverse is not true. Many results have been proved for contraction maps satisfying property (E.A) in different settings such as probabilistic **metric** **spaces** [11, 21], **fuzzy** **metric** **spaces** [5, 18, and 19]. Atanassov [12] introduced and studied the concept of intuitionistic **fuzzy** sets as a generalization of **fuzzy** sets [14] and later there has been much progress in the study of intuitionistic **fuzzy** sets [4, 9].

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In this paper, we prove a **common** ﬁxed **point** theorem for **weakly** **compatible** **mappings** under φ -contractive conditions in **fuzzy** **metric** **spaces**. We also give an example to illustrate the theorem. The result is a genuine generalization of the corresponding result of Hu (**Fixed** **Point** Theory Appl. 2011:363716, 2011,

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This paper mainly aims to employ the **common** CLR property to obtain **common** **fixed** **point** results for two pair of **weakly** **compatible** **mappings** satis- fying contractive condition of integral type on the partial **metric** space. Definition 1.1. [16], [20, Definition 1.1] A partial **metric** space (briefly P M S) is a pair (X, p) where p : X × X → R + is continuous map and R + = [0, ∞) such that for all x, y, z ∈ X :

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Case III. If max = 4γG(Ay, y, y), G(Ay, y, y) ≤ φ (4γ G(Ay, y, y)), G(Ay, y, y) < 4γ G(Ay, y, y), G(Ay, y, y) < G(Ay, y, y) as 3α + 7β + 6γ < 1. This leads to contradiction. Thus G(Ay, y, y) = 0 ⇒ Ay = y. Hence Ay = y and Ry = Ay ⇒ Ay = Ry = y. Hencey is **common** **fixed** **point** of R and A. Since y = Ay ∈ A(X ) ⊆ T (X), there exists v ∈ X such that T v = y. We prove that Bv = y.

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and for each W, ≥ \ here D F is standard negator. The sequence . is said to be convergent to ∈ I in the modified intuitionistic **fuzzy** **metric** space ;I, J M,N , 4= and denoted by . → J M,N if J M,N . , , K → 1 ∗ whenever → ∞ for every K > 0 . A modified intuitionistic **fuzzy** **metric** space is said to be complete if and only if every Cauchy sequence is convergent.

Definition 2.4. [7] Let Ψ be the class of all **mappings** ψ : [0, 1] → [0, 1] such that ψ is continuous, non-increasing and ψ(t) < t, ∀ t ∈ (0, 1). Let Φ be the class of all **mappings** φ : [0, 1] → [0, 1] such that φ is continuous, non-decreasing and φ(t) > t, ∀ t ∈ (0, 1). Let (X, µ, ν, ∗, ) be an intuitionistic **fuzzy** **metric** space and ψ ∈ Ψ and φ ∈ Φ. A mapping f : X → X is called an intuitionistic **fuzzy** ψ-φ-contractive mapping if the following implications hold:

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In this paper, we prove several **common** ﬁxed **point** **theorems** for nonlinear **mappings** with a function φ in **fuzzy** **metric** **spaces**. In these ﬁxed **point** **theorems**, very simple conditions are imposed on the function φ . Our results improve some recent ones in the literature. Finally, an example is presented to illustrate the main result of this paper. MSC: 54E70; 47H25

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Erceg [6], George and Veeramani [8] have introduced **fuzzy** **metric** which was used in topology and analysis. Recently Gregori et. al [10, 12] and Rafi et. al [28] have studied some property in **fuzzy** **metric** **spaces**. Many authors [1, 6-8, 9, 11, 13, 14, 17, 18, 25, 26, 27-33, 37, 38] have studied **fixed** **point** theory in **fuzzy** **metric** **spaces**.

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Bhaskar and Lakshmikantham 4, Lakshmikantham and ´ Ciri´c 5 discussed the mixed monotone **mappings** and gave some coupled **fixed** **point** **theorems** which can be used to discuss the existence and uniqueness of solution for a periodic boundary value problem. Sedghi et al. 6 gave a coupled **fixed** **point** theorem for contractions in **fuzzy** **metric** **spaces**, and Fang 7 gave some **common** **fixed** **point** **theorems** under φ-contractions for **compatible** and **weakly** **compatible** **mappings** in Menger probabilistic **metric** **spaces**. Many authors 8– 23 have proved **fixed** **point** **theorems** in intuitionistic **fuzzy** **metric** **spaces** or probabilistic **metric** **spaces**.

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