Journal of Function Spaces
Volume 2018 (2018), Article ID 2658653, 9 pages
https://doi.org/10.1155/2018/2658653

Research Article

-Metric Generalization of Some Fixed Point Theorems

Mumtaz Ali and Muhammad Arshad

Department of Mathematics, International Islamic University, H-10, Islamabad 44000, Pakistan

Correspondence should be addressed to Mumtaz Ali; mumtaz6767@yahoo.com

Received 27 December 2017; Revised 7 March 2018; Accepted 11 March 2018; Published 15 April 2018

Academic Editor: Pasquale Vetro

Copyright © 2018 Mumtaz Ali and Muhammad Arshad. 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 prove common fixed point theorems in terms of -metric spaces with new contractions. The results presented in this paper include -metric generalizations of some fixed point theorems of Fisher, Pachpatte, and Sahu and Sharma.

1. Introduction

Fixed point theory is a fascinating topic for research in both analysis and topology. In this direction the Banach contraction mapping theorem of 1922 popularly known as Banach contraction mapping principle is a rewarding result in fixed point theory. It has widespread applications in both pure and applied mathematics. The well-known Banach [1] contraction mapping principle states that if “is a complete metric space andis a contraction mapping ofinto itself thenhas unique fixed point in.” This celebrated principle has been generalized by several authors. In 1989, Bakhtin [2] introduced the concept of -metric space which is generalization of renowned Banach contraction mapping principle. Czerwik [3, 4] extended the concept of -metric space in 1993. Bakhtin’s concept of -metric spaces has been extensively generalized and improved by several mathematicians for fixed points in several different ways, namely, Boriceanu [5], Bota et al. [6], Chen et al. [7], Hussain and Shah [8], Kutbi et al. [9], and Shukla [10] to name a few. In this paper, our main concern is to study common fixed point theorems in complete -metric spaces for three self-mappings. The obtained results are generalizations of -metric variant of fixed point theorems of Fisher, Pachpatte, and Sahu and Sharma.

The following fixed point theorems were proved in [11–13].

Theorem 1 (see [11]). Letbe a mapping of the complete metric space into itself satisfying the inequality

Theorem 2 (see [12]). Ifis a mapping of the complete metric space into itself satisfying the inequality

for all , where and then has a unique fixed point.

Theorem 3 (see [13]). If is a mapping of the complete metric space into itself satisfying the inequality

for all in, where and , then has a unique fixed point.

2. Preliminaries

In this section we recall some basic definitions and necessary results from existing literature that will be used in the sequel.

Definition 4 (see [3]). Let be a nonempty set and be a given real number. A function is said to be a -metric on if the following conditions hold:
(i) if and only if
(ii) for all
(iii) for all
The pair is called -metric space.

It is clear from the definition of -metric that every metric space is -metric for , but the converse need not be true. The following example illustrates the above remarks.

Example 5 (see [5]). Let . Define by , and for The function defined above is a -metric space but is not a metric space for .

Proposition 6 (see [14]). Let be a nonempty set and the mappings have a unique point of coincidence in . If and are weakly compatible self-maps of , then have a unique common fixed point.

Definition 7. A sequence in a -metric space is called Cauchy sequence if and only if .

Definition 8. A sequence in a -metric space is said to converge to a point if and only if . We denote this by .

Definition 9. A -metric space is said to be complete if and only if every Cauchy sequence in converges to a point of .

Definition 10 (see [15]). Let and be self-mappings of a set . If for some in , then is called a coincidence point of and and is called a point of coincidence of and .

Definition 11 (see [16]). The mappings are weakly compatible, if, for every , the following holds: whenever .

Definition 12. A point is said to be a fixed point of a self-map if .

3. Main Results

In this section we obtain coincidence points and common fixed point theorems for three maps in complete -metric spaces. In order to start our main results we begin with a simple but useful Lemma.

Lemma 13. Let be a complete -metric space with the coefficient and let be self-mappings from into itself satisfying the following conditions:
(i) .
(ii)

for all ; , such that

Then every sequence with initial point is a Cauchy sequence in .

Proof. Let and choose a point such that and for there exists such that , and continuing this process we construct sequences and in given by

Suppose that there existssuch that

We show that is a Cauchy sequence in . Using (4), we have

where

Therefore, for all , we can write

Now, for any , we have

Therefore, we have

Thus,

Hence is a Cauchy sequence in -metric space .

The next theorem is -metric variant of Theorem 1.3 in [13].

Theorem 14. Let be a complete -metric space with the coefficient . Suppose the self-maps satisfy the condition

where are nonnegative reals with

If and is a complete subspace of , then the maps , and have a coincidence point in . Moreover, if and are weakly compatible pairs. Then , and have a unique common fixed point in .

Proof. Letbe an arbitrary point in and define the sequence in such that and . Now, we show that is a Cauchy sequence. So, by (4), we have

Similarly, we can show that

where

Therefore, for all with , we can get

Thus, as . It follows from Lemma 13 that is a Cauchy sequence and, by the completeness of , converges to some . Therefore,

Thus, implies either or .
Case 1. Let . Since is a complete subspace of and implies is closed, hence, there exist such that . If , then, by using (14), we get

Taking limit as yields

and the above inequality is possible only if which implies that . It follows that

Since and are weakly compatible, we have and so

If , then by (14) we have

As , we have

and the inequality is possible only if

By using (24), we have

Case 2. If , again there exists points such that if , and then by using (14) we get

It follows that

which is possible only if :

Since and are weakly compatible and hence and so and by (28), we have

Thus, is the common fixed point of self-mappings , and . This completes the proof of the theorem.

Uniqueness. In order to prove uniqueness, let be two distinct common fixed points of the self-maps , and then we have by (14)

which is possible only if which gives us uniqueness of .

Remark 15. If we put , then we get Theorem 1.2 in [12].

Now we present the modified form of Theorem 3 in terms of -metric spaces.

Theorem 16. Let be a complete -metric space with the coefficient and suppose the self-maps satisfy the condition

where are nonnegative reals with

If and is a complete subspace of , then the maps , and have a coincidence point in . Moreover, if and are weakly compatible, then , and have a unique common fixed point in .

Proof. Let and define a sequence of points in as follows, and. By using (34), we have

Or

where

Similarly, it can be shown that

Therefore, for all we can get

Now, for any , we have

Therefore, from Lemma 13, we have

where . It follows that the sequence is a Cauchy sequence and, by the completeness of , converges to some .
Therefore,

Since is a complete subspace of , there exists such that . If , using (34), we get

and, on taking limit as ,

which is possible only if . It follows that . Since the pair of mappings is weakly compatible, we have and so

If , by (34), we get the following.
As , we have

and the above inequality is possible only if . Therefore, by (46) we get

Again, if , by (34) we have the following.
Taking limit as , we have

and this is possible only if . Since and are weakly compatible, :

By (48) and (50) we have

Thusis the unique common fixed point of , and .
Uniqueness. In order to see the uniqueness of the common fixed point, let and be two distinct common fixed points of , and such that . Then by using (34) we get

and the inequality is possible only if which implies that and the common fixed point is unique.

Theorem 16 yields the following corollaries.

Corollary 17. Let be a complete -metric space with coefficient and be a self-mapping of into itself satisfying

for all ; and such that then , and have a unique common fixed point in .

Proof. Putting and , and in Theorem 16, we get the required result.

Corollary 18. Let be a complete -metric space and be self-maps of into itself satisfying the inequality

for all , such that , and then have a unique common fixed point in .

Proof. Putting in Theorem 16 we get the required result.

Remark 19. Corollary 17, is the result of Pachpatte [12].

Remark 20. Corollary 18, is the result of Fisher [11].

Example 21. Let , and is defined as follows:

Then, it is easy to verify that is a -metric space with parameter . Define the mappings by

Then, for , the self-maps , and satisfy all the conditions of Theorems 14 and 16 with 5 as the only common fixed point in .

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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