Journal of Mathematics
Volume 2016 (2016), Article ID 4746732, 6 pages
http://dx.doi.org/10.1155/2016/4746732

Research Article

On Parametric -Metric Spaces and Fixed-Point Type Theorems for Expansive Mappings

Nihal Taş and Nihal Yılmaz Özgür

Department of Mathematics, Balıkesir University, 10145 Balıkesir, Turkey

Received 29 July 2016; Revised 11 October 2016; Accepted 16 October 2016

Academic Editor: Kaleem R. Kazmi

Copyright © 2016 Nihal Taş and Nihal Yılmaz Özgür. 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 the notion of a parametric -metric space as generalization of a parametric metric space. Using some expansive mappings, we prove a fixed-point theorem on a parametric -metric space. It is important to obtain new fixed-point theorems on a parametric -metric space because there exist some parametric -metrics which are not generated by any parametric metric. We expect that many mathematicians will study various fixed-point theorems using new expansive mappings (or contractive mappings) in a parametric -metric space.

1. Introduction and Backgrounds

Contractive conditions have been started by studying Banach’s contraction principle. These conditions have been used in various fixed-point theorems for some generalized metric spaces. Then expansive conditions were introduced [1] and new fixed-point results were obtained using expansive mappings.

Recently, the notion of an -metric has been studied by some mathematicians. This notion was introduced by Sedghi et al. in [2] as follows.

Definition 1 (see [2]). Let be a nonempty set and let be a function. is called an -metric on if,

  if and only if , ,

for each . The pair is called an -metric space.

Using the notion of an -metric space, various meaningful fixed-point studies were obtained by some researchers (see [2–6] for more details).

The relationship between a metric and an -metric was studied and an example of an -metric which is not generated by any metric was given in [3, 4].

Later, the notion of a parametric metric space was introduced and some basic concepts such as a convergent sequence and a Cauchy sequence were defined in [7]. We recall the following definitions.

Definition 2 (see [7]). Let be a nonempty set and let be a function. is called a parametric metric on if,

  if and only if , , ,

for each and all . The pair is called a parametric metric space.

Definition 3 (see [7]). Let be a parametric metric space and let be a sequence in :

(1) converges to if and only if there exists such that

for all and all ; that is,

It is denoted by .(2) is called a Cauchy sequence if, for all ,

(3) is called complete if every Cauchy sequence is convergent.

In the following definition, the concept of a parametric -metric space as generalization of a parametric metric space was given.

Definition 4 (see [8]). Let be a nonempty set, let be a real number, and let be a function. is called a parametric -metric on if,

  if and only if , , ,

for each and all . The pair is called a parametric -metric space.

Notice that a parametric -metric is sometimes called a parametric -metric according to a real number in the above definition (see [9]).

Some fixed-point theorems have been still investigated using the notions of a parametric metric space and a parametric -metric space for various contractive or expansive mappings (see [7–10] for more details). For example, Hussain et al. proved some fixed-point theorems on complete parametric metric spaces and triangular intuitionistic fuzzy metric spaces [7]. Also, Hussain et al. introduced the notion of parametric -metric space and investigated some fixed-point results [8]. Jain et al. established some fixed-point, common fixed-point, and coincidence point theorems for expansive type mappings on parametric metric spaces and parametric -metric spaces [10]. Rao et al. obtained two common fixed-point theorems on parametric -metric spaces [9].

The aim of this paper is to introduce the concept of a parametric -metric and give some basic facts. We give two examples of a parametric -metric which is not generated by any parametric metric. We prove some fixed-point results under various expansive mappings in a parametric -metric space. Also, we verify our results with some examples.

2. Parametric -Metric Spaces

In this section, we introduce the notion of “a parametric -metric space” and give some basic properties of this space. Also, we investigate a relationship between a parametric metric and a parametric -metric (resp., a parametric -metric and a parametric -metric).

Definition 5. Let be a nonempty set and let be a function. is called a parametric -metric on if,

  if and only if , ,

for each and all . The pair is called a parametric -metric space.

Now we give the following examples of parametric -metric spaces.

Example 6. Let and let the function be defined by

for each and all . Then is a parametric -metric and the pair is a parametric -metric space.

Example 7. Let and let the function be defined by

for each and all , where is a continuous function. Then is a parametric -metric and the pair is a parametric -metric space.

Example 8. Let and let the function be defined by

for each and all , where is a continuous function. Then is a parametric -metric and the pair is a parametric -metric space.

We prove the following lemma which can be considered as the symmetry condition in a parametric -metric space.

Lemma 9. Let be a parametric -metric space. Then we have

for each and all .

Proof. Using the condition , we obtain

From inequalities (8), we have

Now we give the relationship between a parametric metric and a parametric -metric in the following lemma.

Lemma 10. Let be a parametric metric space and let the function be defined by

for each and all . Then is a parametric -metric and the pair is a parametric -metric space.

Proof. It can be easily seen from Definitions 2 and 5.

We call the parametric metric as the parametric -metric generated by . Notice that there exist parametric -metrics satisfying for all parametric metrics. We give some examples.

Example 11. Let and let the function be defined by

for each and all . Then is a parametric -metric and the pair is a parametric -metric space. We have ; that is, is not generated by any parametric metric .

Example 12. Let and let the function be defined by

for each and all . Then is a parametric -metric and the pair is a parametric -metric space. We have ; that is, is not generated by any parametric metric .

In the following lemma, we see the relationship between a parametric -metric and a parametric -metric.

Lemma 13. Let be a parametric -metric space and let the function be defined by

for each and all . Then is a parametric -metric and the pair is a parametric -metric space.

Proof. Using condition , we see that conditions and are satisfied. Now we show that condition is satisfied. Using condition and Lemma 9, we have

which implies that

Then is a parametric -metric with .

Remark 14. Notice that the minimum value of is . So it should be ; that is, does not define a parametric metric in Lemma 13.

Definition 15. Let be a parametric -metric space and let be a sequence in :

(1) converges to if and only if there exists such that

for all and all ; that is,

It is denoted by .(2) is called a Cauchy sequence if, for all ,

(3) is called complete if every Cauchy sequence is convergent.

Lemma 16. Let be a parametric -metric space. If converges to , then is unique.

Proof. Let and let with . Then there exists such that

for each , all , and . If we take , then, using condition and Lemma 9, we get

for each . Therefore and .

Lemma 17. Let be a parametric -metric space. If converges to , then is Cauchy.

Proof. By the similar arguments used in the proof of Lemma 16, we can easily see that is a Cauchy sequence.

As a consequence of Lemma 10 and Definition 15, we obtain the following corollary.

Corollary 18. Let be a parametric metric space and let be a parametric -metric space, where is generated by parametric metric . Then we have the following:

(1) in if and only if in .(2) is Cauchy in if and only if is Cauchy in .(3) is complete if and only if is complete.

Definition 19. Let be a parametric -metric space and let be a self-mapping of . is said to be a continuous mapping at in if

for any sequence in and all such that

3. Some Fixed-Point Results

In this section, we give some fixed-point results for expansive mappings in a complete parametric -metric space.

Definition 20. Let be a parametric -metric space and let be a self-mapping of .
There exist real numbers satisfying and such that

for each and all .

Theorem 21. Let be a complete parametric -metric space and let be a surjective self-mapping of . If satisfies condition , then has a unique fixed point in .

Proof. Using the hypothesis, it can be easily seen that is injective. Indeed, if we take , then, using condition , we get

for all and so ; that is, we have since .
Let us denote the inverse mapping of by . Let and define the sequence as follows:

Suppose that for all . Using condition and Lemma 9, we have

which implies that

Clearly, we have . Hence, we obtain

If we put , then we get , since . Repeating this process in condition (28), we find

for all .
Let with . Using inequality (29) and condition , we have

If we take limit for , we obtain

Therefore is Cauchy. Then there exists such that

since is a complete parametric -metric space. Using the surjectivity hypothesis, there exists a point such that . From condition , we have

If we take limit for , we obtain

which implies that and .
Now we show the uniqueness of . Let be another fixed point of with . Using condition and Lemma 9, we get

which implies that , since . Consequently, has a unique fixed point .

We give some examples which satisfy the conditions of Theorem 21.

Example 22. Let be the complete -metric space with the -metric defined in Example 8. Let us define the self-mapping as

for all with , and the function as

for all . Then satisfies the conditions of Theorem 21 with and . Then has a unique fixed point in .

Example 23. Let be the complete -metric space with the -metric defined in Example 8. Let us define the self-mapping as

for all with , and the function as

for all . Then satisfies the conditions of Theorem 21 with and . Then has a unique fixed point in .

If we take in condition , then we obtain the following corollary.

Corollary 24. Let be a complete parametric -metric space and let be a surjective self-mapping of . If there exist real numbers satisfying and such that

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

If we take and and and in Theorem 21 and Corollary 24, respectively, then we obtain the following corollaries.

Corollary 25. Let be a complete parametric -metric space and let be a surjective self-mapping of . If there exists a real number such that

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

Corollary 26. Let be a complete parametric -metric space and let be a surjective self-mapping of . If there exist a positive integer and a real number such that

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

Proof. From Corollary 25, by a similar way used in the proof of Theorem 21, it can be easily seen that has a unique fixed point in . Also we have

and so we obtain that is a fixed point for . We get , since is the unique fixed point.

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

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