Common Fixed Point Results for Occasionally Weakly Compatible Maps in G-Symmetric Spaces

The notion of a G-symmetric space is introduced and the common fixed points for some pairs of occasionally weakly compatible maps satisfying some contractive conditions in a G-symmetric space are proved. The results extend and improve some results in literature.


Introduction
The notion of metric spaces is widely used in fixed point theory and applications. Different authors had generalized the notions of metric spaces. Recently, Eke and Olaleru [1] introduced the concept of G-partial metric spaces by introducing the non-zero self-distance to the notion of G-metric spaces. The G-partial metrics are useful in modeling partially defined information which often appears in Computer Science. The concept of symmetric spaces in which the triangle inequality of a metric space is not included was introduced by Cartan [2] and defined as: A symmetric on a set X is a real valued function d on X × X such that  [4] observed that the use of the triangle inequality is not necessary in certain proof of metric theorems. Based on this idea, they proved some common fixed point results in symmetric spaces.
Different generalizations of the metric space have been introduced by many authors in literature. In particular, Mustafa and Sims [5] generalized the concept of a metric space by assigning a real number to every triplet of an arbitrary set. Thus, it is defined as: Definition 1.1 [5]: Let X be a nonempty set, and let : G X X X R + × × → be a function satisfying: x y X ∈ with x y ≠ , (symmetry in all three variables), for all , , , x y z a X ∈ (rectangle inequality). Then, the function G is called a generalized metric, or more specifically a G-metric on X , and the pair ( ) , X G is a G-metric space.
In this work, we generalize the symmetric spaces by omitting the rectangle inequality axiom of G-metric space. This leads to our introduction of the notion of a G-symmetric space defined as follows: Definition 1.3: A G-symmetric on a set X is a function : such that for all , , x y z X ∈ , the following conditions are satisfied: is a G-symmetric space. This does not satisfy the rectangle inequality property of a G-metric space, hence it is not a G-metric space.
The analogue of axioms of Wilson [3] in G-symmetric space is as follows: , d X G be a G-symmetric space.
The principle of studying the fixed point of contractive maps without continuity at each point of the set was initiated by Kannan [6] in 1968. The establishment of a common fixed point for a contractive pair of commuting maps was proved by Jungck [7]. Thereafter, Sessa [8] introduced the notion of weakly commuting maps. Jungck [9] introduced the concept of compatible maps which is more general than the weakly commuting maps. Jungck further weakened the notion of compatibility by introducing weakly compatibility. Al-Thagafi and Shahzad [10] defined the notion of occasionally weakly compatible maps which is more general than that of weakly compatible maps. Pant [11] further introduced the concept of non-compatible maps. The importance of non-compatibility is that it permits the existence of the common fixed points for the class of Lipschitz type mapping pairs without assuming continuity of the mappings involved or completeness of the space. In 2002, Aamri and El Moutawakil [12] introduced the (E-A) property and thus generalized the concept of non-compatible maps.
This work proves the existence of a unique common fixed point for pairs of occasionally weakly compatible maps defined on a G-symmetric space satisfying some strict contractive conditions. The work generalized many known results in literature.
The following definitions are important for our study. Definition 1.9: Two selfmaps f and g in a G-symmetric space ( ) are said to be weakly compatible if they commute at their points of coincidence, that is, if fx gx = for some x X ∈ , then fgx gfx = . Definition 1.11 [10]: Two self maps f and g of a set X are occasionally weakly compatible if and only if there is a point x in X which is a coincidence point of f and g at which f and g commute. Lemma 1.12 [13]: Let X be a set, f , g occasionally weakly compatible self maps of X . If f and g have a unique point of coincidence, : w fx gx = = , then w is the unique common fixed point of f and g. The existence of some common fixed point results for two generalized contractive maps in a symmetric (semimetric) space satisfying certain contractive conditions were proved by Hicks and Rhoades [4] and Imdad et al. [14]. Jungck and Rhoades [13] proved the existence of common fixed points for two pairs of occasionally weakly compatible mappings defined on symmetric spaces by using a short process of obtaining the unique common fixed point of the maps. Bhatt et al. [15] proved the existence and uniqueness of a common fixed point for pairs of maps defined on symmetric spaces without using the (E-A) property and completeness, under a relaxed condition by assuming symmetry only on the set of points of coincidence. Abbas and Rhoades [16] proved the existence of a unique common fixed point for a class of operators called occasionally weakly compatible maps defined on a symmetric space satisfying a generalized contractive condition.
In this work, the existence of common fixed points for two occasionally weakly compatible maps satisfying certain contractive conditions in a G-symmetric space is proved. Our results are analogue of the result of Abbas and Rhoades [16] and an improvement of the results of Imdad et al. [14] and others in literature.

Main Results
For all , x y X ∈ . Moreover, suppose g , f are occasionally weakly compatible, then f and g have a unique common fixed point.
Then the conditions of the theorem are satisfied.
Conversely, suppose there exists k and g so that Equation (1) for all , x y X ∈ . Indeed, f and g have a unique common fixed point if Equation (2) holds.

Theorem 2.4: Let ( )
, d X G be a d G -symmetric space that satisfies ( ) 3 . W Let f and g be two selfmappings of X such that (i) f and g satisfy property (E-A), If f and g are occasionally weakly compatible, then f and g have a unique common fixed point.
Combining Equations (5) and (6) , , , Using Equation (4), we obtain Combining Equations (7) and (8) gives, Since 0 1 k ≤ < , we obtain ga gb = . Therefore ga fa gb fb = = = . Hence w is the unique point of coincidence of f and g . By Lemma 1.12, w is the unique common fixed point of f and g.
W . Let f and g be two self-mappings of X such that (i) f and g satisfy property (E.A) . Suppose that f and g are weakly compatible, then f and g have a unique common fixed point. Remarks 2.6: Theorem 2.4 is an extension of ( [14], Theorems 2.1, 2.2, 2.3) to G-symmetric spaces from symmetric spaces.
The following results are analogue of ([16] Theorem 1). First we state the following definitions given by Abbas and Rhoades [16]. Let for each , , Combining Equations (13) and (14) gives,