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Common Fixed Point Theorems for Occasionally Weakly Compatible Mappings in G-metric Spaces

Reena C*, Balbir Singh

Department of Mathematics, Starex University, Gurugram, 122413, Haryana, India

Abstract: In this paper, we prove some common fixed point results for two pairs of mappings using the concept of occasionally weakly compatible, (E.A)/ common property (E.A) and an inequality involving quadratic terms in the settings of G-metric spaces.

Keywords: common fixed points, occasionally weakly compatible mappings, G-metric spaces, coincidence points

Received September 02, 2021; Revised October 05, 2021; Accepted October 13, 2021

Citation: Reena C, Balbir Singh, Common Fixed Point Theorems for Occasionally Weakly Compatible Mappings in G-metric Spaces. Turkish Journal of Analysis and Number Theory, 9 (2). 25-29. January 2021. doi:10.12691/tjant-9-2-2

1. Introduction

In 2006, Mustafa and Sims [1] introduced the notion of G-metric spaces as a generalization of the metric spaces. After that, many authors studied fixed and common fixed point in generalized metric spaces, see [2,3,4,5,6,7,8,9,10]. These results provide the basis for carrying out analysis in G-metric spaces, in particular for the development of G-metric fixed point theory for mappings satisfying a variety of contractive type conditions. In G-metric spaces, fixed point theory is indispensable due to its wide application. The classical result of Banach continues to be the source of inspiration for many researchers working in the area of G-metric fixed point theory. A G-metrical common fixed point theorem generally involves conditions on commutativity, continuity, and completeness and suitable containment of ranges of the involved mappings besides an appropriate contraction conditions and researcher in this domain are aimed at weakening one or more of these conditions.

Now, we give some preliminaries and basic definitions which we use throughout the paper.

Definition 1.1. (G-metric spaces, see [1]).

In 2006, Mustafa and Sims introduced the concept of G-metric spaces as follows:

Let be a nonempty set and be a function satisfying the following:

(G1)

(G2)

(G3)

(G4) (symmetry in all three variables),

(G5) for all x, y, z, a in rectangle inequality).

Then the function is called a generalized metric or, more specifically, a G-metric on and the pair is called a G-metric spaces.

Definition 1.2. [1] A G-metric space -is called a symmetric G-metric if

Definition 1.3. [11] Let f and g be two self mappings on G-metric spaces The mappings f and g are said to be compatible if

whenever is a sequence in such that

for some

Definition 1.4. [12] Two maps f and g are said to be weakly compatible if they commute at coincidence points.

Definition 1.5. [11] Let f and g be two self mappings of a G- metric space Then the pair is said to be satisfy the property E.A. if there exists a sequence in such that

for some

Definition 1.6. [12] A pair of self-mappings of a G-metric space is said to be non compatible if there exists a sequence in such that

for some But is either non zero or non – existent.

Definition 1.7. [10] A pair of self-mappings of a G-metric space is said to be occasionally weakly compatible if

Definition 1.8. [10] Let f and g be self mappings of a G-metric space. If for some Then is called a coincidence point of f and g and the set of coincidence point is denoted by and is called a point of coincidence of f and g.

2. Main Results

In this section, we prove some common fixed point theorems for four self-maps in the setting of G-metric spaces.

Theorem 2.1. Let A, B, S and T be four self-mappings of a symmetric G-metric space such that

[picture] | (2.1)

for all

[picture] | (2.2)

the pair satisfies property (E.A) and is a closed subspace of

OR

the pair satisfies property (E.A) and is a closed subspace of

Then

Moreover, if both the pairs are occasionally weak compatible mapping in then the mappings have a unique common fixed point.

Proof: Since the pair satisfy the property E.A., there exists a sequence in such that

[picture] | (2.3)

Since then there exists a sequence in such that and hence

[picture] | (2.4)

Now, we prove that To prove it, we use the inequality (2.1)

Using equations (2.3) and (2.4) we have

Since is a symmetric G-metric space, we have

which is a contradiction.

Hence = 0.

So, we have.

[picture] | (2.5)

It is assumed that is a closed subspace of by equation (2.3), we have.

[picture] | (2.6)

Now we claim that If Then we have.

Now taking and using equations (2.3), (2.4) and (2.5), we have a contradiction.

Hence

[picture] | (2.7)

Hence So we have

[picture] | (2.8)

We know that Then there exists a in such that

[picture] | (2.9)

Now, we claim that If this does not happen, that is, Then using (2.8) and (2.9), we have

a contradiction. Hence

[picture] | (2.10)

Hence, we have from (2.9) and (2.10),

This implies that

[picture] | (2.11)

So proof of the theorem holds well when we follow the condition (2.3).

Now as our supposition the pair is occasionally weak compatible, then there exists so we have

[picture] | (2.12)

and

[picture] | (2.13)

From equation (2.12) and (2.13) we have.

[picture] | (2.14)

Again as our supposition the pair is occasionally weak compatible, then there exists so we have

[picture] | (2.15)

and

[picture] | (2.16)

From equations (2.15) and (2.16), we have

[picture] | (2.17)

Now, we claim that If this does not happen that is Then using equations (2.14) and (2.17), we have

a contradiction. So we have

[picture] | (2.18)

So from equations (2.14) and (2.18), we have

[picture] | (2.19)

Now we claim that If this does not happen that is Then from equations (2.12) and (2.19) we have

which is a contradiction. So we can conclude that

[picture] | (2.20)

Now from equations (2.17), (2.19) and (2.20), we have

[picture] | (2.21)

and

[picture] | (2.22)

Now, we claim that If this does not happen, then from equations (2.15) and (2.22), we have

which is a contradiction. So we have

[picture] | (2.23)

So from equations (2.22) and (2.23), we have

[picture] | (2.24)

So from equations (2.21) and (2.24), we have.

Hence is a common fixed point of the mappings

Uniqueness: Let be another common fixed point of the mappings other than So, we have,

Then from condition (2.1), we have

Since is a symmetric G-metric space, so

which is a contradiction. Hence This implies that is a unique common fixed point of the mappings

Theorem 2.2. Let A, B, S and T be four self-mappings of a symmetric G-metric space such that

[picture] | (2.25)

for all

[picture] | (2.26)

The pair satisfies property (E.A) and is a closed subspace of

OR

The pair satisfies property (E.A) and is a closed subspace of

Then

Moreover, if both the pairs are occasionally weakly compatible mapping in then the mappings have a unique common fixed point.

Proof: Since the pair satisfy the property E.A. Then there exists two sequences and in such that.

[picture] | (2.27)

As our supposition that are a closed subspaces of then,

[picture] | (2.28)

Now, we claim that If this does not happen then by inequality (2.25), we have

On taking and using equations (2.27) and (2.28), we have a contradiction.

Hence

[picture] | (2.29)

Hence So we have

[picture] | (2.30)

Now we claim that If this does not happen that is Then using (2.27) and (2.28), we have

which is a contradiction. Hence

Hence we have

This implies that

Now, remaining part of the proof follows as that of Theorem 2.1.

References

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