Fixed point theory ends in masses of packages in mathematics, computer technological, engineering, game concept, fuzzy principle, image processing and so on. In metric areas, this theory starts with the Banach fixed-point theorem which gives a optimistic technique of locating constant factors and an crucial device for solution of some issues in mathematics and engineering and consequently has been generalized in lots of methods. A foremost shift inside the area of fixed point idea got here in 1976, when Jungck 7, 8, 9, defined the idea of commutative and compatible maps and proved the common fixed point results for such maps. Later on, Sessa 15 gave the idea of weaklycompatible, and proved results for set valued maps. Certain altercations of commutativity and compatibility can also be found in 5 7 15 16 . In this paper we establish a unique common fixed point theorem satisfying the pairs of weaklycompatiblemappings in the context of digital metric space. An example is given in the support of our main result.
The first important result on fixed-point for contractive-type mappings was the well-known Banach fixed point theorem, published for the first time in 1922. In 1998, Jungck and Rhoades  introduced the notion of weaklycompatiblemappings and showed that compatible maps are weaklycompatible but not conversely. The concept of fuzzy set was introduced by Zadeh  and after his work there has been a great endeavor to obtain fuzzy analogues of classical theories. In 1994, George and Veeramani  introduced the notion of fuzzy metric space and
We establish a common fixed point theorem for weaklycompatiblemappings generalizing a result of Khan and Kubiaczyk 1988. Also, an example is given to support our generalization. We also prove common fixed point theorems for weaklycompatiblemappings in metric and compact metric spaces.
In this paper, we prove a common ﬁxed point theorem for weaklycompatiblemappings 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,
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 fixed point theorems for weaklycompatiblemappings in intuitionistic fuzzy metric spaces.
results by proving the existence and uniqueness of the solution for a periodic boundary value problem. Later these results were extended and generalized by Sedghi et al. , Fang  and Xin-Qi Hu  etc. Fixed point theorems, involving four self-maps, began with the assumption that they are commuted. Sessa  weakened the condition of commutativity to that of pairwise weakly commuting. Jungck generalized the notion of weak commutativity to that of pairwise compatible  and then pairwise weaklycompatible maps . Jungck and Rhoades  introduced the concept of occasionally weaklycompatible maps. In this paper we introduce some coupled fixed point theorems for occasionally weaklycompatiblemappings in fuzzy metric space.
The fixed point theory has become a part of non-linear functional analysis since 1960. It serves as an essential tool for various branches of mathematical analysis and its applications. Polish mathematician Banach published his contraction Principle in1922. In 1928, Menger introduced semi-metric space as a generalization of metric space. In 1976, Cicchese  introduced the notion of a contractive mapping in semi-metric space and proved the first fixed point theorem for this class of spaces. In 1986, Jungck  introduced the notion of compatiblemappings. In 1997, Hicks and Rhoades generalized Banach contraction principle in semi-metric space. In 1998, Jungck and Rhoades  introduced the notion of weaklycompatiblemappings and showed that compatiblemappings are weaklycompatible but not conversely. Recently in 2006,Jungck and Rhoades  introduced occasionally weaklycompatiblemappings which is more general among the commutativity concepts. Jungck and Rhoades obtained several common fixed point theorems using the idea of occasionally weaklycompatiblemappings. Several interesting and elegant results have been obtained by various authors in this direction. There have been interesting generalized and formulated results in semi- metric space initiated by Frechet , Menger  and Wilson. Also, in this paper, we prove a common fixed point theorem for three pairs of self-mappings using occasionally weaklycompatiblemappings.
Fuzzy set was defined by Zadeh . Kramosil and Michalek  introduced fuzzy metric space, George and Veermani  modified the notion of fuzzy metric spaces with the help of continuous t-norms. Many researchers have obtained common fixed point theorem for mappings satisfying different types of commutativity conditions. Vasuki  proved fixed point theorems for R-weakly commuting mappings. Pant [12, 13, 14] introduced the new concept of reciprocally continuous mappings and established some common fixed point theorems. Balasubramaniam et al.  have shown that Rhoades  open problem on the existence of contractive definition which generates a fixed point but does not force the mappings to be continuous at the fixed point, posses an affirmative answer. Pant and Jha  obtained some anologous results proved by Balasubramanium et al.. Recent literature on fixed point in fuzzy metric space can be viewed in [1, 2, 3, 4,5, 8, 11, 16].
In 1976, Jungck proved some common fixed point theorems for commuting maps which generalize the Banach contraction principle. Further this result was generalized and extended in various ways by several authors. On the other hand Sessa  introduced the concept of weak commutativity and proved a common fixed point theorem for weakly commuting maps. In 1986, G.Jungck introduced the concept of compatible maps which is more general than that of weakly commuting maps. Afterwards Jungck and Rhoades  introduced the notion of weaklycompatible and proved that compatible maps are weaklycompatible but not conversely.
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].
Remark 1.2. The purpose of this paper is to prove some general common fixed point theorem for occasionally weaklycompatiblemappings satisfying implicit relations in intuitionistic fuzzy metric space which generalizes several results from the literature. Our results generalize several fixed point theorems in following respects.
Abstract. In this paper, we prove the existence of fixed points for two set-valued mappings and two single-valued mappings satisfying generalized contractive conditions by using the concept of weaklycompatiblemappings with control functions and implicit relations in complete metric spaces. Our results extend and generalize the corre- sponding result in Mehta and Joshi .
The above example reveals that occasionally weaklycompatiblemappings are not weakly com- patible. Since it has two coincidence points 1/2 and 1 (AP,S) and (BQ,T) are not commuting at x=1/2. We observed that the self mappings (A, P) and (B, Q) are commuting and the mappings A, B, S, T, P and Q have unique common fixed point.
Abstract: The notion of modular metric spaces being a natural generalization of classical modulars over linear spaces Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, and Calderon-Lozanovskii spaces was recently introduced. Chistyakov [4, 6] introduced and studied the concept of modular metric spaces and proved fixed point theorems for contractive map in Modular spaces. It is related to contracting rather “generalized average velocities” than metric distances, and the successive approximations of fixed points converge to the fixed points in a weaker sense as compared to metric convergence. In this paper, we prove some unique common fixed point theorems for generalized contraction type mappings for six self occasionally weaklycompatiblemappings in modular metric spaces.
 Abbas M., Rhoades B.E.,: Common Fixed Point Theorems for Hybrid pairs of Occasionally WeaklyCompatibleMappings Satisfying Generalized Contractive Condition of Intregal type . Fixed Point Theory and Applications , Article ID 54101,9 pages, 2007  George A. and . Veeramani P., On some results in fuzzy metric spaces, Fuzzy sets and Systems vol. 64, pp. 395-399, 1994.  Kramosil O. and Michalek J., Fuzzy metrics and statistical metric spaces. Kybernetica vol.11, pp.326-334, 1975.
Abstract. In this paper, we present a common tripled fixed point theorem for weaklycompatiblemappings under ϕ -contractive condition in M-fuzzy metric spaces. The result generalizes, extends and improves several classical and very recent related results of Sedghi, Altun and Shobe.
Motivated by the above result, we address the same question on -metric space for weaklycompatiblemappings satisfying a Generalized Contraction Principle condition given by (1), we establish a fixed point results in the third part of the paper. Our results are the following.