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

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 obtainFrom 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 thatClearly, we have . Hence, we obtainIf we put , then we get , since . Repeating this process in condition (28), we findfor 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.