Fixed Points of Generalized -Meir-Keeler Contraction Mappings in -Metric Spaces

Abstract

In this note, we define Meir-Keeler contraction in -metric spaces. Further, by adding the concept of -admissible mappings, we define generalized -Meir-Keeler contraction and used it for examining the existence and uniqueness of fixed points. Various results are also given as a consequence of our results.

1. Introduction and Preliminaries

The Banach contraction principle has been an important instrument for the study of a fixed point. It has been widely used in different areas like nonlinear analysis, applied mathematics, economics, and physics. Due to its importance, the result has been generalized in different ways. Meir and Keeler [1] introduce a generalization of the Banach contraction principle. According to them, self-mapping in a metric space is called Meir-Keeler contraction if for an there exists such that implies for all . They also state and prove that if a self-mapping in a complete metric space satisfies Meir-Keeler contraction, then there is a unique fixed point for the mapping . There are a large number of works on Meir-Keeler contraction of which some of the recent works are mentioned here.

Pourhadi et al. [2] introduced the concept of Meir-Keeler expansive mappings and obtained Krasnosel’skii-type fixed point theorem in Banach spaces. A new fixed point theorem was obtained by Du and Rassias [3] for a Meir-Keeler type condition as a generalization of the Banach contraction principle, Kannan’s fixed point theorem, Chatterjea’s fixed point theorem, etc., simultaneously.

The idea of -metric space [4–6] is defined by combining definitions of -metric space [7] and -metric space [8]. Samet et al. [9] introduced the concept of -admissible mapping. This concept was further extended to -metric space, -metric space, -metric space, etc. (for details, see [10–14]). There are various recent results on Meir-Keeler type and related topics which will be helpful to the readers for more information. Some of them can be seen in [15–21].

In this article, we give the concept of -admissible and Meir-Keeler contraction in -metric space. The new contraction will be known as generalized -Meir-Keeler contraction. By using generalized -Meir-Keeler contraction mappings, we study the existence and uniqueness of the fixed point in -metric space.

The following definitions and properties will be needed.

Definition 1 (see [8]). In a set , suppose is a real number and is a function satisfying (1) if and only if for all (2) for all (3) for all

Then, is called -metric on and the pair is called a -metric space with coefficient .

Definition 2 (see [7]). In a set , suppose is a function satisfying (1) if and only if for all (2), for all

Then, the pair is said to be an -metric space.

Definition 3 (see [5]). In a set , suppose is a real number and is a function satisfying (i) if and only if (ii), for all in

Here, is said to be a -metric and is said to be a -metric space.

Definition 4 (see [4]). A -metric satisfying for all is called a symmetric -metric.

Definition 5 (see [5]). In a -metric space , a sequence is called (i)convergent if and only if as where and is expressed as (ii)Cauchy if and only if as , where (iii)complete -metric space if every Cauchy sequence is convergent and converging to in

We recall some types of -admissible mappings in a metric space .

Definition 6 (see [9]). Let and be functions. Here, is said to be -admissible if implies for all .

Definition 7 (see [15]). Let and are functions. Here, the pair of mappings is said to be an-admissible if implies and for all .

Definition 8 (see [16]). Let and be functions. Here, is known as triangular -admissible, if (i), which implies (ii), , which implies

Definition 9 (see [15]). Let and be functions. Here, the pair is said to be a triangular -admissible, if (i), which implies and (ii), , which implies for all

We extend the concept of -admissible mapping to be suitable for -metric and -metric spaces. Here, we consider as -metric space or -metric space.

Definition 10. Let and are functions, then is called -admissible, if , implies .

Example 11. Consider and define and by for all and Then, is an -admissible mapping.

Definition 12. Let and be three functions. The pair is called -admissible if such that , then we have and .

2. Main Result

Here, we give various types of Meir-Keeler contractive mappings in order to extend various results of Gülyaz et al. [17] in -metric space. Throughout this paper, assume is a -metric space, is a real number, and is a mapping.

Definition 13. An -admissible mapping in is known as -Meir-Keeler contraction mapping of type I, if there esists for all such that implies for all .

Definition 14. An -admissible mapping in is known as -Meir-Keeler contraction mapping of type II, if there exists for all such that implies for all .

Remark 15. (i)If is an -Meir-Keeler contraction of type I, then for all and equality is true, when (ii)If is an -Meir-Keeler contraction of type II, then for all and equality is true, when

Now, we introduce the following generalization of Meir-Keeler mappings.

Definition 16. An -admissible mapping in is known as generalized -Meir-Keeler contraction mapping of type AI, if there exists for all such that implies where for all .

Definition 17. An -admissible mapping in is known as generalized -Meir-Keeler contraction mapping of type AII, if there exists for all such that implies where for all .

Definition 18. An -admissible mapping in is known as generalized -Meir-Keeler contraction mapping of type BI, if there exists for all such that implies where for all .

Definition 19. An -admissible mapping in is known as generalized -Meir-Keeler contraction mapping of type BII, if there exists for all such that implies where for all .

Remark 20. (i)Let be a generalized -Meir-Keeler contraction of type AI or BI. Thenfor all , where the equality holds only when (ii)Let be a generalized -Meir-Keeler contraction of type AII or BII. Thenfor all , where the equality holds only when

Lemma 21. Let be a -metric space and be a sequence satisfying (i) for all , (ii), for all Then, is a Cauchy sequence in .

Proof. In order to show that sequence is Cauchy, we must prove that for any .
From (ii), we have Applying limit as , we get Now, Thus, is a Cauchy sequence in -metric space .☐

Theorem 22. Let be a complete -metric space and be a mapping. Let satisfy the following: (i) is a generalized -Meir-Keeler contraction mapping of type AI(ii) is -admissible(iii)There is so that (iv) is continuousThen, there exists a fixed point of in .

Proof. Suppose and . Define the sequence in as Suppose for some that is implies that is a fixed point of . Thus, assume that for all . From (ii), we have implies that continuing on the same lines, we have Here, we need to show that sequence satisfies the conditions of Lemma 21. If we put and in (9), for all , there is satisfying implies where From Remark 20(ii), we have due to the fact that , we see that equality does not hold, hence, If for some , then (11) implies which is not possible. Then, for all , so that (11) yields which shows that Lemma 21(ii) is true.☐

Next, we consider the case for for all .

If possible, let for some . We have for some . In general, let

We have ; by inequality (12), we have becomes impossible. Thus, for some is not true, and hence, it must be for all . So, due to Lemma 21, is a Cauchy sequence in . Thus, converges to , i.e.,

By the continuity of , we have so converges to . Since the limit is unique, .

Theorem 23. Let be a complete -metric space and be a mapping. Let be a mapping such that (v)for a pair of fixed points of , together with the four conditions of Theorem 22, then has a unique fixed point in

Proof. The existence of a fixed point is proved in Theorem 22. Now, for uniqueness, consider and as two different fixed points of in .
By (9), we have implies where By (v), , since , Remark 20(ii) becomes which is a contradiction, hence, , i.e., . Thus, the fixed point of is unique.☐

Definition 24. In -metric space , is a mapping. Then, -metric space is known as an -regular if for any sequence , and for all ; we have for all .

Theorem 25. In a complete -metric space is a parameter and is an -admissible mapping. Let be a generalized -Meir-Keeler contraction of type AI satisfying the following: (i)There is so that (ii)The -metric space is an -regular, then there exists a fixed point of in (iii)For all pairs of fixed points, , Then, has unique fixed point.

Proof. Suppose such that . Define a sequence such that for all and converges to uniquely.
As is -regular,
By (9), we have implies where On the other hand, from Remark 20(ii), we have We have Also, Taking the limit as in (46), we have which conclude that .☐

The uniqueness part is identical to Theorem 23.

Note: Theorems 22, 23, and 25 will be true for generalized -Meir-Keeler contraction mapping of type BI and BII.

Example 26. Let be endowed with -metric Define by Clearly, mapping is -admissible and continuous mapping. Let without loss of generality, assume that , then Now, to calculate in our case, if we take , then after a simple calculation, we have Now, suppose that for Now, observe that and and assume that , then we have which implies that Since for all ; otherwise, , and we have Hence, satisfies the conditions of generalized -Meir-Keeler contraction mapping of type AI. Also, all the conditions of Theorem 22 are satisfied, and hence, is the unique fixed point of mapping .

3. Consequences

Here, we consider some consequences of Theorems 22, 23, and 25.

Corollary 27. Let be complete -metric space and be an -admissible mapping satisfying the following: (i)For all , there exists such thatimplies where for all (ii)There exists such that (iii) is continuous or -metric space is -regularThen, has a fixed point in .
Also, (iv)for every pair of fixed points of , if Then, the fixed point of is unique in .

Proof. As for all , the proof is obvious from Theorems 22, 23, and 25.☐

Corollary 28. Let be complete -metric space and be an -Meir-Keeler contraction of type I; that is, there exists for every such that implies for all .
If is continuous or -metric space is -regular, then has a fixed point. Further, with condition (v) in Theorem 23, the fixed point of is unique.

Proof. The proof follows easily from the relation for all .☐

Taking in Theorem 25, we get the following.

Corollary 29. Let be a complete -metric space and be a continuous mapping. If there exists for every such that implies where for all . Then, the fixed point of is unique.

Corollary 30. Let be a complete -metric space and be a continuous mapping. If there exists for every such that implies where for all . Then, has a unique fixed point.
The Meir-Keeler contraction can be stated on -metric spaces as follows.

Corollary 31. Let be a complete -metric space and be a continuous Meir-Keeler mapping. If there exists for every such that becomes for all . Then has a unique fixed point.

4. Conclusion

In this article, we define Meir-Keeler contraction in -metric spaces using the concept of -admissible mapping. Further, we define generalized -Meir-Keeler contraction. Using these definitions of contractive mappings, we prove theorems for the existence and uniqueness of fixed points. We show that obtained results are potential generalizations of various results in the literature.

Data Availability

No data is used in this research.

Conflicts of Interest

The authors declare not having competing interests.

Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.