-Metric Generalization of Some Fixed Point Theorems

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 thatThen 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 bySuppose that there existssuch that We show that is a Cauchy sequence in . Using (4), we havewhere Therefore, for all , we can write Now, for any , we haveTherefore, 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 conditionwhere are nonnegative reals withIf 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 getThus, 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 soIf , then by (14) we haveAs , we have and the inequality is possible only if By using (24), we haveCase 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 conditionwhere are nonnegative reals withIf 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 soIf , by (34), we get the following.
As , we have and the above inequality is possible only if . Therefore, by (46) we getAgain, if , by (34) we have the following.
Taking limit as , we haveand 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.