Common Fixed Point Results for Occasionally Weakly Compatible Maps in G-Symmetric Spaces ()
1. 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
(i) 
and 
if and only if
;
(ii) 
Wilson [3] also gave two more axioms of a symmetric d on X as:
(W1) Given
, 
and 
in
, 
and 
imply that
;
(W2) Given
,
and
; 
and 
imply that
.
Hicks and Rhoades [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 
be a nonempty set, and let 
be a function satisfying:
(G1) 
if
(G2) 
for all 
with
(G3) 
for all 
with
(G4) 
(symmetry in all three variables)(G5) 
for all 
(rectangle inequality).
Then, the function 
is called a generalized metric, or more specifically a G-metric on
, and the pair 
is a G-metric space.
Example 1.2 [5] : Let 
be a metric space. The function
, defined by
or
for all
, is a G-metric on
.
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 
is a function 
such that for all
, the following conditions are satisfied:
and
, if
;
for all 
with 
, for all 
with 
,×××, (symmetry in all three variables).
It should be observed that our notion of a G-symmetric space is the same as that of G-metric space (Definition 1.1) without the rectangular property
.
Example 1.4: Let 
equipped with a G-symmetric defined by:
for all
. Then, the pair 
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:
(W3) Given
, 
and 
in
; 
and 
imply that
.
(W4) Given 
and an 
in
; 
and 
imply that
.
Definition 1.5: Let 
be a G-symmetric space.
(i) 
is 
-complete if for every 
-Cauchy sequence
, there exists 
in 
with
.
(ii) 
is 
-continuous if
Definition 1.6: Let 
be a nonempty subset of
. 
is said to be 
-bounded if and only if
.
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 
and 
in a G-symmetric space 
are said to be weakly compatible if they commute at their points of coincidence, that is, if 
for some
, then
.
Definition 1.11 [10] : Two self maps 
and 
of a set X are occasionally weakly compatible if and only if there is a point 
in 
which is a coincidence point of 
and 
at which f and g commute.
Lemma 1.12 [13] : Let 
be a set, 
, 
occasionally weakly compatible self maps of
. If 
and 
have a unique point of coincidence, 
, then 
is the unique common fixed point of 
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.
2. Main Results
Theorem 2.1: Let 
be a bounded G-symmetric for
. Suppose 
is 
-complete and 
is 
-continuous. Then 
has a fixed point if and only if there exists 
and a 
-continuous function 
which is compatible with 
and satisfies 
and
(1)
For all
. Moreover, suppose
, 
are occasionally weakly compatible, then 
and 
have a unique common fixed point.
Proof: Suppose 
for some
, put 
for all
. Then the conditions of the theorem are satisfied.
Conversely, suppose there exists 
and 
so that Equation (1) holds. Let
.
Suppose 
is arbitrarily chosen. 
can be chosen such that
. Continuing in this process, 
can be chosen such that
. Using Equation (1) and the sequence
,
Thus 
is a 
-Cauchy sequence and since 
is 
-complete, there exists 
with
. Since g is 
-continuous, it implies that 
Also 
yields
. 
is 
-continuous implying that
. The compatibility of 
and 
gives
, that is 
which implies that
. Suppose there exists another point in 
saying 
such that
. Now we claim that
. Suppose
, then using Equation (1) gives
Letting 
yields
This is a contradiction since
, hence
. Therefore 
is the unique point of coincidence 
and
. By Lemma (1.12), 
is the unique common fixed point of 
and 
Corollary 2.2 [15] : Let 
be a bounded 
-symmetric for X that satisfies 
Suppose that 
is 
-complete and 
is 
-continuous. Then 
has a fixed point if and only if there exists 
and a 
-continuous function 
which commutes with 
and satisfies 
and
, (2)
for all
. Indeed, 
and 
have a unique common fixed point if Equation (2) holds.
Remark 2.3: Corollary 2.2 is an analogue of ([15] , Theorem 2.1) in the setting of G-symmetric space. Theorem 2.1 is an improvement of Bhatt et al. ([15] , Theorem 2.1) since occasionally weakly compatible maps are more general than commuting maps and the concept of a 
-symmetric space extends that of a symmetric space.
Theorem 2.4: Let 
be a 
-symmetric space that satisfies 
Let 
and 
be two selfmappings of 
such that
(i) 
and 
satisfy property (E-A)(ii) for all 
Suppose
(3)
And
(4)
Suppose 
is a 
-closed subset of X with 
If 
and 
are occasionally weakly compatible, then 
and 
have a unique common fixed point.
Proof: Since 
and 
satisfy property (E-A), there exists a sequence 
in X such that 
for some 
Also 
is closed implying that there exist some 
such that
. This yields that 
by 
We claim that 
Suppose
then using Equation (3) we get,
Letting 
we have,
(5)
Using Equation (4) we have
Letting 
gives,
(6)
Combining Equations (5) and (6) yields,
Suppose there exists 
such that 
Suppose 
then using Equation (3) we have,
Letting 
yields,
(7)
Using Equation (4), we obtain
(8)
Combining Equations (7) and (8) gives,
Since
, we obtain
. Therefore
. Hence w is the unique point of coincidence of 
and
. By Lemma 1.12, w is the unique common fixed point of 
and g.
Corollary 2.5: Let 
be a 
-symmetric space that satisfies
. Let f and g be two self-mappings of 
such that
(i) 
and 
satisfy property (E.A)
(ii) for all 
(9)
and
(10)
Assume 
is 
-closed subsets of 
with
. Suppose that 
and 
are weakly compatible, then 
and 
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 
Let 
satisfy
(i) 
and 
for each 
and
(ii) F is nondecreasing on 
Define 
Let 
satisfy
(i) 
for each 
and
(ii) 
is nondecreasing.
Define
Theorem 2.6: Let 
be a set with 
-symmetric
. Let 
Suppose that 
and 
are self-maps of 
and that the pairs 
and 
are each occasionally weakly compatible. If for each 
for which 
we have
(11)
and
(12)
for each 
and 
where 
if 
and 
if 
and
and
then there is a unique point 
such that 
and a unique point 
such that 
Moreover, 
so that there is a unique common fixed point of 
and 
Proof: Since the pairs 
and 
are each occasionally weakly compatible, then there exist 
such that 
and 
We claim that 
On the contrary, suppose 
then
Case (i)
If max
then Equation (11) becomes
Case (ii)
If 
then Equation (11) becomes
(13)
Case (iii)
If 
then Equation (12) becomes,
Case (iv)
If 
then Equation (13) becomes,
(14)
Combining Equations (13) and (14) gives,
—a contradiction. Therefore 
That is, 
Moreover, if there is another point u such that 
then, using Equations (12) and (13) it follows that 
or 
and 
is a unique point of coincidence of 
and
. By Lemma 1.12, 
is the only common fixed point of f and S. That is 
Similarly there is a unique point 
such that 
Suppose that 
then using Equation (12) we have,
(15)
Using Equation (12) we get,
(16)
Combining Equations (15) and (16) gives,
a contradiction. Therefore 
and w is a common fixed point of
, 
and 
Following the preceding argument, it is clear that 
is unique.
Remarks 2.7: Theorem 2.2 is an analogue of ([16] Theorem 1) in the setting of G-symmetric spaces.
Corollary 2.7: Let 
be a set with 
-symmetric
. Let
Suppose that 
and 
are self-maps of 
and that the pairs 
and 
are each occasionally weakly compatible (owc). If for each 
for which 
we have
(17)
and
(18)
for each 
and 
where 
if 
and 
if 
and
and
and 
then 
and 
have a unique common fixed point.
Proof: Since Equations (17) and (18) are special cases of Equations (11) and (12), then the proof of the corollary follows immediately from Theorem 2.6.
Acknowledgements
The authors would like to appreciate the Deanship of Scientific Research for supporting this work through their careful editing of this manuscript.