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 **weakly** **compatible**, 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 **weakly** **compatible** **mappings** in the context of digital metric space. An example is given in the support of our main result.

Show more
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 [10] introduced the notion of **weakly** **compatible** **mappings** and showed that **compatible** maps are **weakly** **compatible** but not conversely. The concept of fuzzy set was introduced by Zadeh [20] and after his work there has been a great endeavor to obtain fuzzy analogues of classical theories. In 1994, George and Veeramani [5] introduced the notion of fuzzy metric space and

14 Read more

We establish a common fixed point theorem for **weakly** **compatible** **mappings** 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 **weakly** **compatible** **mappings** in metric and compact metric spaces.

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,

11 Read more

In this paper, we prove a common fixed point theorem for occasionally weakly compatible mappings in fuzzy.. metric spaces using the property (E.A.).[r]

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.

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. [9], Fang [2] and Xin-Qi Hu [10] etc. Fixed point theorems, involving four self-maps, began with the assumption that they are commuted. Sessa [8] weakened the condition of commutativity to that of pairwise **weakly** commuting. Jungck generalized the notion of weak commutativity to that of pairwise **compatible** [4] and then pairwise **weakly** **compatible** maps [5]. Jungck and Rhoades [6] introduced the concept of occasionally **weakly** **compatible** maps. In this paper we introduce some coupled fixed point theorems for occasionally **weakly** **compatible** **mappings** in fuzzy metric space.

Show more
19 Read more

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[17] introduced semi-metric space as a generalization of metric space. In 1976, Cicchese [6] 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 [13] introduced the notion of **compatible** **mappings**. In 1997, Hicks and Rhoades[8] generalized Banach contraction principle in semi-metric space. In 1998, Jungck and Rhoades [14] introduced the notion of **weakly** **compatible** **mappings** and showed that **compatible** **mappings** are **weakly** **compatible** but not conversely. Recently in 2006,Jungck and Rhoades [15] introduced occasionally **weakly** **compatible** **mappings** which is more general among the commutativity concepts. Jungck and Rhoades[15] obtained several common fixed point theorems using the idea of occasionally **weakly** **compatible** **mappings**. 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 [7], Menger [17] and Wilson[20]. Also, in this paper, we prove a common fixed point theorem for three pairs of self-**mappings** using occasionally **weakly** **compatible** **mappings**.

Show more
Fuzzy set was defined by Zadeh [18]. Kramosil and Michalek [10] introduced fuzzy metric space, George and Veermani [7] 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 [17] 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. [6] have shown that Rhoades [15] 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 [14] obtained some anologous results proved by Balasubramanium et al.[6]. Recent literature on fixed point in fuzzy metric space can be viewed in [1, 2, 3, 4,5, 8, 11, 16].

Show more
Now, we prove a coupled fixed point theorem for a pair of weakly compatible maps satisfying -contractive conditions in Menger PM-space with a continuous t -norm of H -type.. At the end[r]

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 [5] introduced the concept of weak commutativity and proved a common fixed point theorem for **weakly** commuting maps. In 1986, G.Jungck[1] introduced the concept of **compatible** maps which is more general than that of **weakly** commuting maps. Afterwards Jungck and Rhoades [4] introduced the notion of **weakly** **compatible** and proved that **compatible** maps are **weakly** **compatible** but not conversely.

Show more
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].

Show more
11 Read more

Remark 1.2. The purpose of this paper is to prove some general common fixed point theorem for occasionally **weakly** **compatible** **mappings** 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 **weakly** **compatible** **mappings** with control functions and implicit relations in complete metric spaces. Our results extend and generalize the corre- sponding result in Mehta and Joshi [21].

13 Read more

The above example reveals that occasionally **weakly** **compatible** **mappings** 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.

“Common fixed point theorems of Gregus type for weakly compatible mappings satisfying contractive conditions of integral type.” Journal of Mathematical Analysis and Applicatio[r]

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 **weakly** **compatible** **mappings** in modular metric spaces.

Show more
10 Read more

[4] Abbas M., Rhoades B.E.,: Common Fixed Point Theorems for Hybrid pairs of Occasionally **Weakly** **Compatible** **Mappings** Satisfying Generalized Contractive Condition of Intregal type . Fixed Point Theory and Applications , Article ID 54101,9 pages, 2007 [5] George A. and . Veeramani P., On some results in fuzzy metric spaces, Fuzzy sets and Systems vol. 64, pp. 395-399, 1994. [6] 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 **weakly** **compatible** **mappings** 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.

10 Read more

Motivated by the above result, we address the same question on -metric space for **weakly** **compatible** **mappings** 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.

12 Read more