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Journal of Mathematics
Volume 2017 (2017), Article ID 6018054, 7 pages
https://doi.org/10.1155/2017/6018054
Research Article

Common Fixed Points via -Sequences in -Metric Spaces

1Institut de Mathématiques et de Sciences Physiques (IMSP), UAC, 01 BP 613, Porto-Novo, Benin
2École Normale Supérieure (ENS) de Natitingou/UNSTIM Abomey, BP 72, Natitingou, Benin

Correspondence should be addressed to Yaé Ulrich Gaba

Received 8 June 2017; Accepted 25 October 2017; Published 19 November 2017

Academic Editor: Kaleem R. Kazmi

Copyright © 2017 Yaé Ulrich Gaba. 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 use -sequences in this article to derive common fixed points for a family of self-mappings defined on a complete -metric space. We imitate some existing techniques in our proofs and show that the tools employed can be used at a larger scale. These results generalize well known results in the literature.

1. Introduction and Preliminaries

The generalization of the Banach contraction mapping principle has been a heavily investigated branch of research. In recent years, several authors have obtained fixed point and common fixed point results for various classes of mappings in the setting of many generalized metric spaces. One of them, the -metric space, is our focus in this paper, and fixed point results, in this setting, presented by authors like Abbas et al. [1], Gaba [2, 3], Mustafa and Sims [4], Sihag et al. [5], and many more, are enlightening on the subject. Moreover, in [6], we introduced the concept of -sequence which extended the idea of -series proposed by Sihag et al. in [5]. The present article exclusively presents natural extensions of some results already given by Abbas et al. [1] and Sihag et al. [5] and therefore generalizes some recent results regarding fixed point theory in -metric spaces. We also show how the idea of -sequence is used in proving some of these results. The method builds on the convergence of an appropriate series of coefficients. We also make use of a special class of homogeneous functions. Recent and similar work can also be read in [2, 3, 6, 7].

We recall here some key results that will be useful in the rest of this manuscript. The basic concepts and notations attached to the idea of -metric spaces can be read extensively in [4], but for the convenience of the reader, we discuss the most important ones.

Definition 1 (compare [4, Definition ]). Let be a nonempty set, and let the function satisfy the following properties: (G1) if whenever .(G2) whenever with .(G3) whenever with .(G4) (symmetry in all three variables).(G5)for any points .

Then, is called a -metric space.

The property (G3) is crucial and shall play a key role in our proofs.

Proposition 2 (compare [4, Proposition ]). Let be a -metric space.
Then, for a sequence , the following are equivalent: (i) is -convergent to (ii)(iii)(iv)

Proposition 3 (compare [4, Proposition ]). In a -metric space , the following are equivalent: (i)The sequence is -Cauchy.(ii)For each , there exists such that for all .

Definition 4 (compare [4, Definition ]). A -metric space is said to be complete if every -Cauchy sequence in is -convergent in .

Definition 5 (compare [4, Definition ]). A -metric space is said to be symmetric if

Definition 6 (compare [3, Definition ]). A sequence in a metric space is a -sequence if there exist and such that

Definition 7 (compare [2, Definition ]). A sequence in a -metric space is a -sequence if there exist and such that

Definition 8 (compare [5, Definition ]). For a sequence of nonnegative real numbers, the series is an -series if there exist and such that

Remark 9. For a given -sequence in a -metric space , the sequence of nonnegative real numbers, defined by is an -series.

Moreover, any nonincreasing -sequence of elements of endowed with the max (the max metric refers to ) metric is also an -series. Therefore, -sequences generalize -series, but to ease computations, we shall consider, throughout the paper, -series (however, the reader can convince himself that using -sequences does not add to the complexity of the problem).

2. First Generalizations Results

We begin with the following generalization of [5, Theorem ], the main result of Sihag et al.

Let be the class of continuous, nondecreasing, subadditive, and homogeneous functions such that .

Theorem 10. Let be a complete -metric space and be a family of self-mappings on such thatfor all with , , and some , homogeneous with degree . Ifis an -series, then have a unique common fixed point in .

Proof. We will proceed in two main steps.
Claim  1. have a common fixed point in .
For any , we construct the sequence by setting
We assume without loss of generality that for all . Using (7), we obtain, for the triplet ,By property (G3) of , one knows thatHence,that is,Also, we getRepeating the above reasoning, we obtainIf we setwe have thatTherefore, for all ,Using the fact that is subadditive, we writeNow, let and be as in Definition 8; then, for and using the fact that the geometric mean of nonnegative real numbers is at most their arithmetic mean, it follows thatAs , we deduce that Thus, is a -Cauchy sequence.
And since is complete, there exists such that -converges to .
Moreover, for any positive integers , we have Letting and using property (G3), we obtainand this is a contradiction, unless , since . Then, is a common fixed point of .
Claim  2. is the unique common fixed point of .
Finally, we prove the uniqueness of the common fixed point . To this aim, let us suppose that is another common fixed point of ; that is, . Then, using (7) again, we havewhich yields , since . So, is the unique common fixed point of .

Theorem 11. Let be a complete -metric space and be a family of self-mappings on such thatfor all with , , some positive integer , and some , homogeneous with degree . Ifis an -series, then have a unique common fixed point in .

Proof. It follows from Theorem 10 that the family have a unique common fixed point . Now, for any positive integers , that is, and are also fixed points for and (remember that any fixed point of is a fixed point of for ; cf. Theorem 10). Since the common fixed point of is unique, we deduce that

The next result, corollary of Theorem 10, corresponds to the result presented by Sihag et al. [5, Theorem ].

Corollary 12 (compare [5, Theorem ]). Let be a complete -metric space and be a family of self-mappings on such thatfor all with , Ifis an -series, then have a unique common fixed point in .

Proof. In Theorem 10, take (the identity map on ), and .

Example 13. Let and whenever and for . Clearly, is a complete -metric space.

Following the notation in the definition, we set the following.

, where and . We also define for all and and . Then, is continuous, nondecreasing, subadditive, and homogeneous of degree and . Assume and . Hence, we have

Therefore, condition (27) is satisfied for all with . Moreover, since is homogeneous of degree , the sequence satisfies the condition on (8); that is, it is an -series.

Then, by Corollary 12, has a common fixed point, which is this case

3. Second Generalizations Results

The next generalization is that of [1, Theorem ], the main result of Abbas et al. Instead of considering three maps, we consider a family of maps like in the previous case. Moreover, to show the reader that -sequences do not add to the complexity of the problem, we shall use them in the next statement.

Theorem 14. Let be a complete -metric space and be a sequence of self-mappings on . Assume that there exist three sequences , , and of elements of such thatfor all with , where , , and . If the sequence , whereis a nonincreasing -sequence of endowed with the max (the max metric refers to ) metric, then have a unique common fixed point in . Moreover, any fixed point of is a fixed point of for .

Proof. We will proceed in two main steps.
Claim  1. Any fixed point of is also a fixed point of and for .
Assume that is a fixed point of and suppose that and . Then,which is a contradiction unless
Claim  2. For any , we construct the sequence by setting We assume without loss of generality that for all . Using (31), we obtainBy property (G3), one can writeAgain, sincewe obtainthat is, Hence,Also, we getRepeating the above reasoning, we obtainIf we setwe have thatTherefore, for all ,Now, let and be as in Definition 8; then, for and using the fact that the geometric mean of nonnegative real numbers is at most their arithmetic mean, it follows thatAs , we deduce that Thus, is a -Cauchy sequence.
Moreover, since is complete, there exists such that -converges to .
If there exists such that , then, by Claim  1, the proof of existence is complete.
Otherwise, for any positive integers , we have Letting and using property (G3), we obtainand this is a contradiction, unless .
Finally, we prove the uniqueness of the common fixed point . To this aim, let us suppose that is another common fixed point of ; that is, . Then, using (31), we havewhich yields . So, is the unique common fixed point of .

Following the same lines of the proof of Theorem 11, one can prove the next theorem.

Theorem 15. Let be a complete -metric space and be a sequence of self-mappings on . Assume that there exist three sequences , , and of elements of such thatfor all with , some positive integer , where , , and . If the sequence , whereis a nonincreasing -sequence of endowed with the max (the max metric refers to ) metric, then have a unique common fixed point in . Moreover, any fixed point of is a fixed point of for .

The next result, corollary of Theorem 14, corresponds to the result presented by Abbas et al. [1, Theorem ].

Corollary 16. Let be a complete -metric space , mappings on . Assume that there exist three nonnegative real numbers , , and such thatfor all with
Then, have a unique common fixed point in . Moreover, any fixed point of is a fixed point of and and vice versa.

Proof. In Theorem 14, take , , . Also, set Hence, we haveThe sequence is constant, so, in Definition 8, if we choose and , it is clear that is an -series. Indeed, sincetherefore, for any ,

We conclude this manuscript with the following result, whose proof is straightforward, following the steps of the proofs of the earliest results.

Theorem 17. Let be a complete -metric space and be a sequence of self-mappings on . Assume that there exist three sequences , , and of elements of such thatfor all with , some positive integer , and some , homogeneous with degree , where , , and . If the sequence whereis a nonincreasing -sequence of endowed with the (the max metric refers to ) metric, then have a unique common fixed point in . Moreover, any fixed point of is a fixed point of for .

In addition to the examples provided by Abbas et al. and Sihag et al., illustrations of all the above results can be read in [2, Example ] and [6, Example ].

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this article.

Acknowledgments

This work was carried out with financial support from the government of Canada’s International Development Research Centre (IDRC) and within the framework of the AIMS Research for Africa Project.

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