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Journal of Applied Mathematics

Volume 2012 (2012), Article ID 248937, 21 pages

http://dx.doi.org/10.1155/2012/248937

## Some New Common Fixed Point Theorems under Strict
Contractive Conditions in *G*-Metric Spaces

Department of Mathematics, The Hashemite University, P.O. Box 330127, Zarqa 13115, Jordan

Received 7 May 2012; Revised 24 June 2012; Accepted 2 August 2012

Academic Editor: Ya Ping Fang

Copyright © 2012 Zead Mustafa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We introduce some new types of pairs of mappings
on *G*-metric space called *G*-weakly commuting of type () and *G*-*R*-weakly
commuting of type (). We obtain also several common fixed point
results for these mappings under certain contractive condition in *G*-metric
space. Also some examples illustrated to support our results, and comparison
between different types of pairs of mappings are studied.

#### 1. Introduction and Preliminaries

The study of common fixed points of mappings satisfying certain contractive conditions has been at the center of strong research activity and, being the area of the fixed point theory, has very important application in applied mathematics and sciences. In 1976 Jungck [1] proved a common fixed point theorem for commuting maps, but his results required the continuity of one of the maps.

Sessa [2] in 1982 first introduced a weaker version of commutativity for a pair of self-maps, and it is shown in Sessa [2] that weakly commuting pair of maps in metric pace is commuting, but the converse may not be true.

Later, Jungck [3] introduced the notion of compatible mappings in order to generalize the concepts of weak commutativity and showed that weak commuting map is compatible, but the reverse implication may not hold.

In 1996, Jungck [4] defined a pair of self-mappings to be weakly compatible if they commute at their coincidence points.

Therefore, we have one-way implication, namely, commuting maps weakly commuting maps compatible maps weakly Compatible maps. Recently various authors have introduced coincidence points results for various classes of mappings on metric spaces for more detail of coincidence point theory and related results see [5–7].

However, the study of common fixed point of noncompatible mappings has recently been initiated by Pant (see [8, 9]).

In 2002 Amari and El Moutawakil [10] defined a new property called E.A. property which generalizes the concept of noncompatible mappings, and they proved some common fixed point theorem.

*Definition 1.1 (see [10]). *Let and be two self-mappings of a metric space . We say that and satisfy the E.A. property if there exists a sequence such that

In 2005 Zead Mustafa and Brailey Sims introduced the notion of *G*-metric spaces as generalization of the concept of ordinary metric spaces. Based on the notion of *G*-metric space Mustafa et al. [11–15] obtained some fixed point results for mapping satisfying different contractive conditions on complete *G*-metric space, while in [16] the completeness property was omitted and replaced by sufficient conditions, where these conditions do not imply the completeness property.

Chugh et al. [17] obtained some fixed point results for maps satisfying property P in *G*-metric spaces. Saadati et al. [18] studied fixed point of contractive mappings in partially ordered *G*-metric spaces. Shatanawi obtained fixed points of -maps in *G*-metric spaces [19] and a number of fixed point results for the two weakly increasing mappings with respect to partial ordering in *G*-metric spaces [20]. In [21, 22] authors established coupled fixed point theorems in a partially ordered *G*-metric spaces.

Abbas and rhoades [23] proved several common fixed points for noncommuting mappings without continuity in *G*-metric space, and they show that the results 2.3–2.6 generalize Theorems 2.1–2.4 of [11].

In [24] Abbas et al. proved several unique common fixed points for mappings satisfying E.A. property under generalized contraction condition and show that Corollary 3.1 extends the main result in [13] (Theorem 2.1) and Corollary 3.3 is *G*-version of Theorem 2 from [10] in the case of two self-mappings. Also this corollary is in relation with Theorem 2.5 of [23].

In [25] the authors proved some coupled coincidence and common coupled fixed point results for mappings defined on a set equipped with two *G*-metric spaces and these results do not rely on continuity of mappings involved therein as well as they show that Theorem 2.13 is an extension and generalization of (1) Theorem 2.2, Corollary 2.3, Theorem 2.6, Corollaries 2.7 and 2.8 in [26] and (2) Theorem 2.4 and Corollary 2.5 in [27].

Aydi et al. [28] established some common fixed point results for two mappings and on *G*-metric spaces with assumption that is a generalized weakly *G*-contraction mappings of type A and B with respect to .

In this paper, we define new types of self-maps and on -metric space called -weakly commuting of type and --weakly commuting of type . Also we obtain several common fixed point results for these mappings under certain contractive condition in -metric space, and some examples are illustrated to support our results, and a comparison between different types of pairs of mappings are stated.

The following definitions and results will be needed in the sequel.

*Definition 1.2 (see [29]). *A -metric space is a pair , where is a nonempty set, and is a nonnegative real-valued function defined on such that for all we have if ,,,, (symmetry in all three variables),, for all , (rectangle inequality).

The function is called -metric on .

Every -metric on defines a metric on by

*Example 1.3 (see [29]). *Let be a metric space, and define and on to by
for all . Then and are -metric spaces.

*Example 1.4 (see [29]). *Let , and define , by
then is *G*-metric space.

*Definition 1.5 (see [29]). *A sequence in a -metric space is said to converge if there exists such that , and one says that the sequence is -convergent to . We call the limit of the sequence and write or (through this paper we mean by the set of all natural numbers).

Proposition 1.6 (see [29]). *Let be -metric space. Then the following statements are equivalent:*(1)* is -convergent to ,*(2)*, as ,*(3)*, as ,*(4)*, as . *

*Definition 1.7 (see [29]). *In a -metric space , a sequence is said to be -Cauchy if given , there is such that , for all . That is as .

Proposition 1.8 (see [29]). *In a -metric space , the following statements are equivalent:*(1)*the sequence is -Cauchy;*(2)*for every such that , for all . *

*Definition 1.9 (see [29]). *A -metric space is called symmetric -metric space if for all and called nonsymmetric if it is not symmetric.

*Example 1.10. *Let be the set of all natural numbers, and define such that for all : if , , if , , if , if and symmetry in all three variables.

Then, is *G*-metric space and nonsymmetric since if , we have .

Proposition 1.11 (see [29]). *Let be a -metric space; then the function is jointly continuous in all three of its variables. *

*Definition 1.12 (see [29]). *A -metric space is said to be complete if every -Cauchy sequence in is -convergent in .

*Definition 1.13 (see [23]). *Let and be self-maps of a set . If for some , then is called a coincidence point of and , and is called a point of coincidence of and .

Recall that a pair of self-mappings are called weakly compatible if they commute at their coincidence points.

Proposition 1.14 (see [23]). *Let and be weakly compatible self-maps of a set . If and have a unique point of coincidence , then is the unique common fixed point of and . *

In 2001, Abbas et al. [30] introduce a new type of pairs of mappings called *R*-weakly commuting and they proved a unique common fixed point of four *R*-weakly commuting, maps satisfying generalized contractive condition.

*Definition 1.15 (see [30]). *Let be a *G*-metric space, and let and be two self-mappings of ; then and are called R-weakly commuting if there exists a positive real number such that

Very recently, Mustafa et al. [31] introduce some new types of pairs of mappings on *G*-metric space called *G*-weakly commuting of type and *G*-*R*-weakly commuting of type , and they obtained several common fixed point results by using E.A. property.

*Definition 1.16 (see [31]). *A pair of self-mappings of a *G*-metric space is said to be -weakly commuting of type if

*Definition 1.17 (see [31]). *A pair of self-mappings of a -metric space is said to be --weakly commuting of type if there exists some positive real number such that

*Remark 1.18. *The --weakly commuting maps of type are -weakly commuting since , but the converse need not be true.

#### 2. Main Results

##### 2.1. New Concepts and Some Properties

In this section we introduce the concept of G-weakly commuting of type for pairs of mapping and comparison between this concept and Definitions 1.15, 1.16, and 1.17 is studied as well as examples illustrated to show that these types of mappings are different.

First, we introduce the following concepts as follows.

*Definition 2.1. *A pair of self-mappings of a *G*-metric space is said to be *G*-weakly commuting of type if

*Definition 2.2. *A pair of self-mappings (*f,g*) of a *G*-metric space is said to be *G-R*-weakly commuting of type if there exists some positive real number such that

*Remark 2.3. *The -weakly commuting maps of type are --weakly commuting of type . Reciprocally, if , then --weakly commuting maps of type are -weakly commuting of type .

If we interchange and in (2.1) and (2.2), then the pair of mappings is called -weakly commuting of type and --weakly commuting of type , respectively.

*Example 2.4. *Let , with the *G*-metric , for all . Define by, ; then as an easy calculation one can show that . Then the pair is *G*-Weakly commuting of type and *G-R*-Weakly commuting of type .

*Example 2.5. *Let , with the *G*-metric , for all . Define by, , then for we see that and . Therefore the pair is not -weakly commuting of type or , but it is *G-R*-weakly commuting of type (and ) for .

The following examples show a pair of mappings that -weakly commuting of type need not be -weakly commuting of type .

*Example 2.6. *Let , with the *G*-metric , for all . Define ; then we see that and , while as an easy calculation one can show that for we have . Therefore the pair is not -weakly commuting of type , but it is G-weakly commuting of type .

The following example shows that(1)a pair of mappings that is -weakly commuting of type need not be -weakly commuting of type ;(2)a pair of mappings that is -weakly commuting of type need not be -weakly commuting of type ;(3)a pair of mappings that is --weakly commuting of type need not be -weakly commuting;

*Example 2.7. *Let and for all . Define the mappings by

Then,

Moreover,

If , we have .

If , we have

If , then

Thus, and are weakly commuting of type , but for , we have

Therefore, the pair is not *G*-weakly commuting of type , but it is *G*-weakly commuting of type .

Also for , we have

Therefore, the pair is not *G*-weakly commuting of type .

AS an easy calculation one can see that are --weakly commuting of type for ; but for we have , hence is NOT -weakly commuting for .

Lemma 2.8. *If and are -weakly commuting of type or --weakly commuting of type , then and are weakly compatible. *

*Proof. *Let be a coincidence point of and , that is, ; then if the pair is -weakly commuting of type , we have

It follows ; then they commute at their coincidence point.

Similarly, if the pair is --weakly commuting of type , we have

Thus ; then the pair is weakly compatible.

The following example shows that(1)the converse of Lemma 2.8 fails (for the case of -weakly commutativity),(2)a pair of mappings that is -weakly commuting need not be --weakly commuting of type ,(3)a pair of mappings that is -weakly commuting need not be --weakly commuting of type .

*Example 2.9. *Let and . Define by and . We see that is the only coincidence point and and , so and are weakly compatible.

But, by an easy calculation, one can see that for we have,
Therefore, and are not -weakly commuting of type .

Also, we see that ; therefore the mappings are -weakly commuting for , but for we have ; hence are not --weakly commuting of type for and ; hence are not --weakly commuting of type for .

Now, we rewrite Definition 1.1 on -metric spaces setting.

*Definition 2.10. *Let and be two self-mappings of a -metric space . We say that and satisfy the E.A. property if there exists a sequence such that and -converge to for some ; that is, thanks to Proposition 1.6,

*Remark 2.11. *In view of (1.2) and Example 1.3, Definition 1.1 is equivalent to Definition 2.10.

In the following example, we show that if and satisfy the E.A. property, then the pair need not be -weakly commuting of type .

*Example 2.12. *We return to Example 2.9. Let . We have , and , therefore, . Then and satisfy the E.A. property, but we know that the pair is not -weakly commuting of type .

Following Matkowski (see [32]), let be the set of all functions such that be a nondecreasing function with for all . If , then is called map. If is map, then it is easy to show that(1) for all ,(2).

##### 2.2. Some Common Fixed Point Results

We start this section with the following theorem.

Theorem 2.13. *Let be a -metric space; suppose mappings satisfy the following condition: *(1)* and be -weakly commuting of type , *(2)*,*(3)* is -complete subspace of , *(4)* Then and have a unique common fixed point. *

* Proof. *Let , and then choose such that and where ; then by induction we can define a sequence as follows:
We will show that the sequence is -cauchy sequence:
where

We will have different cases. Case (1): if , then , which is contradiction. Case (2): if , then in this case we have, which implies that

but from *G*-metric property (G5) we have

Thus, from (2.18) and (2.19) we see that case (2) is impossible.

Then, we must have the case

Thus, for and from (2.16) we have,

Given, since , and , there is an integer , such that

Hence, we have

Now for ; , we claim that

We will prove (2.24) by induction on .

Inequality (2.24) holds for , by using (2.23) and the fact that .

Assume (2.24) holds for . For , we have

By induction on , we conclude that (2.24) holds for all .

Hence, the sequence is -cauchy sequence in ; since is -complete, then there exists such that .

Thus, there exists such that , also .

We will show that . Supposing that , then condition (4) implies that, where

Taking the limit as and using the fact that the function is continuous we get
Therefore,
which is contradiction; hence .

Since and are -weakly commuting of type , then .

Thus, ; it follows that .

Finally, we will show that is common fixed point of and .

Supposing that , then
where
Since , and , therefore (2.30) implies that
Hence, (2.29) becomes
Similarly we get,
So,
a contradiction which implies that . Then is a common fixed point.

To prove uniqueness suppose we have and such that , and ; then condition (4) implies that
Therefore,
Similarly, ; thus , a contradiction which implies that . Then is a unique common fixed point of and .

Now we give an example to support our result.

*Example 2.14. *Let , and define by and by and . Then,(a) is -complete subspace of ,(b)(c) and are -weakly commuting of type , (d) and satisfy condition (4) of Theorem 2.13.

It is clear that (a) and (b) are satisfied.

To show (c), as an easy calculation one can show that ; we have . Then and are G-weakly commuting of type .

To show (d), for we have

Therefore, all hypotheses of Theorem 2.13 are satisfied and unique common fixed point of and .

Corollary 2.15. *Let be a -metric space, and suppose mappings satisfy the following conditions: *(1)* and be G -weakly commuting of type , *(2)*,
*(3)* is -complete subspace of , *(4)* for all , where ; then and have a unique common fixed point. *

* Proof. *It suffices to take in Theorem 2.13.

Theorem 2.16. *Let be a -metric space. Suppose the mappings are -weakly commuting of type and satisfy the following condition: *(1)* and satisfy E.A. property, *(2)* is closed subspace of , *(3)* for all , where ; then and have a unique common fixed point. *

*Proof. *Since and satisfy E.A. property, there exists in a sequence satisfying for some .

Since is closed subspace of and , there exists such that , also.

We will show that supposing that , then condition (3) implies that
where,
Taking the limit as and using the fact that the function is continuous, we get
which is contradiction since , so . Since and are -weakly commuting of type , then
Therefore, ; then
Finally, we will show that is common fixed point of and .

Supposing that , then
where
But and . Thus,
Hence,

Adjusting similarly, we get
Therefore,
a contradiction which implies that , but . Then is a common fixed point of and .

To prove uniqueness, suppose we have and such that , and ; then
Hence, . Similarly, . Therefore, a contradiction which implies that . Then is a unique common fixed point of and .

Now we give an example to support our result.

*Example 2.17. *Let , define by

and let by , .

Then,(a) is closed subspace of ,(b) and are *G*-weakly commuting of type ,(c) and satisfy E.A. property. (d) and satisfy condition (4) for .

*Proof. *(a) is obvious.

To show (b), as an easy calculation one can show that ; we have . Then and are -weakly commuting of type .

To show (c), if we consider the sequence , then and as . Thus, and satisfy the E.A. property.

To show (d), for we have

Then

Therefore, all hypotheses of Theorem 2.16 are satisfied for and , a unique common fixed point of and .

Theorem 2.18. *Let be a -metric space, and suppose mappings be --weakly commuting of type . Suppose that there exists a mapping such that *(1)*, *(2)* is -complete subspace of , *(3)*,
** for all ; then and have a unique common fixed point. *

*Proof. *Let , and then choose such that and where ; then by induction we can define a sequence as follows:
We will show that the sequence is -cauchy sequence:

Consider , then

Thus, the sequence is nonincreasing and bounded below by ; hence is convergent sequence.

On the other hand we have, from (G5) and (2.57), that for

Therefore, the sequence is *G*-cauchy sequence in .

Since is -complete subspace, then there exists such that ; having there exists such that , also .

We will show that ; supposing that , then condition (4) implies that
Taking the limit as , we get
hence,
Adjusting similarly, we get

Therefore,

Thus, a contradiction implies .

Since and are -weakly commuting of type , then

Thus, , then .

Finally, we will show that is common fixed point of and .

Suppose that , so

Since and , therefore (2.65) implies that

Similarly, we have .

A contradiction implies that . Then is a common fixed point.

To prove uniqueness suppose we have and such that where and ; then as an easy calculation one can get
Similarly, , a contradiction which implies that . Then, is a unique common fixed point of and .

Now we give an example to support our result.

*Example 2.19. *Let , such that and . Define by and .

Then, (a), (b) is -complete subspace of , (c),(d) and satisfy condition (4) of Theorem 2.18.

Then as an easy calculation one can see that