In 2002, Aamri and El-Moutawakil  defined the notion of (E.A) property for self mappings which contained the class of non-compatiblemappings in metricspaces. 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 commonfixedpointtheorems in fuzzymetricspaces 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  defined the notion of (CLRg) property in fuzzymetricspaces and improved the results of Mihet¸  without any requirement of the closedness of the subspace. In this paper, we prove a commonfixedpoint theorem for a pair of weaklycompatiblemappings by using (CLRg) property in fuzzymetric space. We also present a commonfixedpoint theorem for two finite families of self mappings in fuzzymetric space by using the notion of pairwise commuting due to Imdad, Ali and Tanveer . Our results improve the results of Sedghi, Shobe and Aliouche .
In this paper, we prove a commonfixedpoint theorem for a pair of weaklycompatiblemappings in fuzzymetric 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 fixedpointtheorems for compatiblemappings with some types in fuzzymetricspaces,” Chaos, Solitons & Fractals, Vol. 39, No. 5, 2009, pp. 2233-2244.] and several known results existing in the literature.
11. S.Manro, S.S.Bhatia and S.Kumar, Commonfixedpointtheorems for weaklycompatible maps satisfying common (E.A) property in intuitionistic fuzzymetricspaces 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 fuzzymetric space due to George and Veeramani . Alaca et al. defined the notion of intuitionistic fuzzymetricspaces and proved commonfixedpointtheorems. Turkoglu et al.  first formulate the definition of weakly commuting
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 commonfixedpointtheorems for families of owc maps in Menger spaces. As applications to our results, we obtain the corresponding fixedpointtheorems in fuzzymetricspaces. Our results improve and extend the results of Kohli and Vashistha [Commonfixedpointtheorems in probabilistic metricspaces, Acta Math. Hungar. 115(1-2) (2007), 37-47], Vasuki [Commonfixed points for R-weakly commuting maps in fuzzymetricspaces, Indian J. Pure Appl. Math. 30 (1999), 419–423], Chugh and Kumar [Commonfixedpoint theorem in fuzzymetricspaces, Bull. Cal. Math. Soc. 94 (2002), 17–22] and Imdad and Ali [Some commonfixedpointtheorems in fuzzymetricspaces, Math. Commun. 11(2) (2006), 153-163].
The evolution of fuzzy mathematics commenced with the introduction of the notion of fuzzy sets by Zadeh  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 . The concept of fuzzymetric space has been introduced and generalized by many ways ( ,  ). George and Veeramani (  ) modified the concept of fuzzymetric space introduced by Kramosil and Michalek . They also ob- tained a Hausdorff topology for this kind of fuzzymetric space which has very important applications in quantum particle physics, particularly in connection with both string and ∞ theory (see,  and references mentioned therein). Many authors have proved fixedpoint and commonfixedpointtheorems in fuzzymetricspaces ( , , ). Regan and Abbas  obtained some necessary and sufficient conditions for the existence of commonfixedpoint in fuzzymetricspaces. Recently, Cho et al  established some fixedpoint theo- rems for mappings satisfying generalized contractive condition in fuzzymetric space. The aim of this paper is to obtain commonfixedpoint of mappings sat- isfying generalized contractive type conditions without exploiting the notion of continuity in the setting of fuzzymetricspaces. Our results generalize several comparable results in existing literature (see, ,  and references mentioned therein).
Meanwhile, jungck defined the concept of compatiblemappings, Jungck and Roadhes  generalized the last concept to the weaklycompatiblemappings, which is weaken than the compatible ones. Mishra et al  generalized the concept of compatibility in the setting of fuzzymetricspaces, he obtained some commonfixedpointtheorems for compatiblemappings in such spaces. Recently Bouhadjera and Godet Tobie  introduced the concept of subsequential continuity and utilized it with the concept of subcompatible mappings to establish a commonfixedpoint, later Imdad et al. improved these results and replaced subcompatibility by compatibility and subsequential continuity by reciprocal continuity, more recently, Gopal and Imdad  combined subsequential continuous maps with compatible maps concept to obtain some results in fuzzymetricspaces. In present work, we will generalize certain definitions to intuitionistic fuzzymetricspaces in order to obtain some commonfixedpointtheorems by combining the concept of weakly subsequentially continuous mappings due to second author  with compatible of type (E) mappings given by Singh et al.[27, 28].
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  and using this notion we prove coupled fixedpointtheorems for weaklycompatiblemappings in intuitionistic fuzzymetricspaces.
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 fuzzyfixedpoint theory has become an area of interest for specialists in fixedpoint theory as intuitionistic fuzzy mathematics has covered new possibilities for fixedpoint theorists. In this paper, we use the idea of chainable intuitionistic fuzzymetricspaces and prove a commonfixedpoint theorem for four weaklycompatiblemappings of chainable intuitionistic fuzzymetricspaces.
Abstract. The aim of this paper is to prove some commonfixedpointtheorems for two weakly subsequentially continuous and compatible of type (E) pairs of self mappings satisfying an implicit relation in fuzzymetricspaces. Two examples are given to illustrate our results.
In the last years, fixedpointtheorems 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 commonfixedpointtheorems for weakly commuting, compatible, δ-compatible and weaklycompatiblemappings under diﬀerent contractive conditions in metricspaces have appeared in 4–15. Throughout this paper, X, d is a metric space.
The concept of Fuzzy sets was initially investigated by Zadeh . Subsequently, it was developed by many authors and used in various fields. To use this concept, several researchers have defined Fuzzymetric space in various ways. In 1986, Jungck  introduced the notion of compatible maps for a pair of self mappings. However, the study of commonfixed points of non-compatible maps is also very interesting. Aamri and El Moutawakil  generalized the concept of non-compatibility by defining the notion of property (E.A) and in 2005, Liu, Wu and Li  defined common (E.A) property in metricspaces and proved commonfixedpointtheorems under strict contractive conditions. Jungck and Rhoades  initiated the study of weaklycompatible maps in metric space and showed that every pair of compatible maps is weaklycompatible but reverse is not true. Many results have been proved for contraction maps satisfying property (E.A) in different settings such as probabilistic metricspaces [11, 21], fuzzymetricspaces [5, 18, and 19]. Atanassov  introduced and studied the concept of intuitionistic fuzzy sets as a generalization of fuzzy sets  and later there has been much progress in the study of intuitionistic fuzzy sets [4, 9].
In this paper, we prove a common ﬁxed point theorem for weaklycompatiblemappings under φ -contractive conditions in fuzzymetricspaces. We also give an example to illustrate the theorem. The result is a genuine generalization of the corresponding result of Hu (FixedPoint Theory Appl. 2011:363716, 2011,
This paper mainly aims to employ the common CLR property to obtain commonfixedpoint results for two pair of weaklycompatiblemappings satis- fying contractive condition of integral type on the partial metric space. Definition 1.1. , [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 :
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 commonfixedpoint 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.
and for each W, ≥ \ here D F is standard negator. The sequence . is said to be convergent to ∈ I in the modified intuitionistic fuzzymetric 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 fuzzymetric space is said to be complete if and only if every Cauchy sequence is convergent.
Definition 2.4.  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 fuzzymetric 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:
In this paper, we prove several common ﬁxed pointtheorems for nonlinear mappings with a function φ in fuzzymetricspaces. In these ﬁxed pointtheorems, 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
Erceg , George and Veeramani  have introduced fuzzymetric which was used in topology and analysis. Recently Gregori et. al [10, 12] and Rafi et. al  have studied some property in fuzzymetricspaces. Many authors [1, 6-8, 9, 11, 13, 14, 17, 18, 25, 26, 27-33, 37, 38] have studied fixedpoint theory in fuzzymetricspaces.
Bhaskar and Lakshmikantham 4, Lakshmikantham and ´ Ciri´c 5 discussed the mixed monotone mappings and gave some coupled fixedpointtheorems 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 fixedpoint theorem for contractions in fuzzymetricspaces, and Fang 7 gave some commonfixedpointtheorems under φ-contractions for compatible and weaklycompatiblemappings in Menger probabilistic metricspaces. Many authors 8– 23 have proved fixedpointtheorems in intuitionistic fuzzymetricspaces or probabilistic metricspaces.