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

Kanayo Stella Eke, Johnson Olaleru

Department of Computer and Information Science/Mathematics, Covenant University, Ota, Nigeria.

Department of Mathematics, University of Lagos, Lagos, Nigeria.

**DOI: **10.4236/am.2014.54071
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Department of Computer and Information Science/Mathematics, Covenant University, Ota, Nigeria.

Department of Mathematics, University of Lagos, Lagos, Nigeria.

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.

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Eke, K. and Olaleru, J. (2014) Common Fixed Point Results for Occasionally Weakly Compatible Maps in G-Symmetric Spaces. *Applied Mathematics*, **5**, 744-752. doi: 10.4236/am.2014.54071.

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:

(W_{1}) Given, and in, and imply that;

(W_{2}) 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:

(G_{1}) if(G_{2}) for all with(G_{3}) for all with(G_{4}) (symmetry in all three variables)(G_{5}) 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:

(W_{3}) Given, and in; and imply that.

(W_{4}) 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.

Conflicts of Interest

The authors declare no conflicts of interest.

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