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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.

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 [

A symmetric on a set X is a real valued function d on X × X such that

(i)

Wilson [

(W_{1}) Given

(W_{2}) Given

Hicks and Rhoades [

Different generalizations of the metric space have been introduced by many authors in literature. In particular, Mustafa and Sims [

Definition 1.1 [

(G_{1}) _{2}) _{3}) _{4}) _{5})

Then, the function

Example 1.2 [

or

for all

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

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

The analogue of axioms of Wilson [

(W_{3}) Given

(W_{4}) Given

Definition 1.5: Let

(i)

(ii)

Definition 1.6: Let

The principle of studying the fixed point of contractive maps without continuity at each point of the set was initiated by Kannan [

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

Definition 1.11 [

Lemma 1.12 [

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 [

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 [

Theorem 2.1: Let

For all

Proof: Suppose

Conversely, suppose there exists

Suppose

can be chosen such that

Thus

Letting

This is a contradiction since

Corollary 2.2 [

for all

Remark 2.3: Corollary 2.2 is an analogue of ([

Theorem 2.4: Let

(i)

Suppose

And

Proof: Since

Letting

Using Equation (4) we have

Letting

Combining Equations (5) and (6) yields,

Suppose there exists

Letting

Using Equation (4), we obtain

Combining Equations (7) and (8) gives,

Since

Corollary 2.5: Let

(i) and satisfy property (E.A)

(ii) for all

and

Remarks 2.6: Theorem 2.4 is an extension of ([

The following results are analogue of ([

First we state the following definitions given by Abbas and Rhoades [

Let

(i)

(ii) F is nondecreasing on

Define

Let

(i)

(ii)

Define

Theorem 2.6: Let

and

for each

and

then there is a unique point

Proof: Since the pairs

Case (i)

If max

Case (ii)

If

Case (iii)

If

Case (iv)

If

Combining Equations (13) and (14) gives,

Moreover, if there is another point u such that

Using Equation (12) we get,

Combining Equations (15) and (16) gives,

a contradiction. Therefore

Remarks 2.7: Theorem 2.2 is an analogue of ([

Corollary 2.7: Let

and

for each

and

and

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.

The authors would like to appreciate the Deanship of Scientific Research for supporting this work through their careful editing of this manuscript.