Discussions on Proinov--Contraction Mapping on -Metric Space

Abstract

In the present paper, we introduce the notion of Proinov--contraction mapping and we discuss it within the most interesting abstract structure, namely, -metric spaces. We further take into consideration the necessary conditions to guarantee the existence and uniqueness of fixed points for such mappings, as well as indicate the validity of the main results by providing illustrative examples.

1. Introduction and Preliminaries

The fixed point theory focuses on investigating the necessary and sufficient conditions on the operator as well as the abstract structure within which the operator is defined. Many research papers, on fixed point theory, aim to bring forth a new condition on the operator (contraction criteria) or suggest a new abstract structure, or both. The present paper highlights a new contraction condition, namely, a Proinov--contraction, on the most interesting abstract structure of -metric spaces.

The notion of -metric has been approached by several researchers such as Bakhtin [1] and Czerwik [2, 3]. For instance, Berinde [4, 5] named this structure as “quasimetric.” To be more precise, by -metric, we understand the natural successful extension of metric by weakening “the triangle inequality” with “the extended triangle inequality.” In other words, the condition of metric turns into the new condition for all and for a real number . Evidently, in case of , these two notions coincide. Despite the high similarities of the definitions of the notion of metric and -metric, there topological properties may differ. For instance, it is known that metric is a continuous map, but, as a mapping, -metric is not necessarily continuous. Moreover, an open ball is not open and a closed ball is not a closed set. These differences make this structure very interesting to investigate. In particular, in [6], the authors characterized the weak -contractions in setting of -metric spaces. In [7], the existence of the fixed point of certain set-valued mappings was discussed in the context of -metric spaces. Additionally, Ulam Stability of the fixed point, in the framework of -metric spaces, has been considered in [8]. On the other hand, in [9–12], the authors focused on the existence of distinct multivalued operators in the context of -metric spaces. In [13], Pacurar dealt with a fixed point for -contractions in the same structures. Another fact worth mentioning is that Shukla [14] defined partial -metric spaces while considering the fixed point theorem.

The notion of Proinov--contraction mapping is based on two aspects: “Proinov-type mappings” [15] and “simulation functions” [16, 17]. Proinov [15] proved that several existing results are consequences of Skof’s result [18] reported in 1977. On the other hand, the simulation function also helps to get a very general contraction condition whose consequences involve several existing fixed point theorems, including Banach’s.

Throughout the paper, we presume that is a nonempty set.

The notion of simulation function, introduced by Joonaghany et al. [16], combine several existing results.

Definition 1 (see [16]). A function is called a simulation function if
(ζ₁)
(ζ₂) for all
(ζ₃) are sequences in such that , then The set of all simulation functions will be denoted by . On account of , we observe that We also notice that in [17], it was shown that is superfluous.

Definition 2 (see [16]). Let be a metric space and . We say that a self-mapping on is a -contraction with respect to , if

Considering with and , it follows that the Banach contraction forms a -contraction with respect to .

Theorem 3. On a complete metric space, every -contraction has a unique fixed point.

Definition 4. On a nonempty set , let be a function such that the following conditions hold:
(b₁) if and only if
(b₂) for all
(b₃) for all , with
Then, we say that function is a -metric. In this case, the tripled forms a -metric space.

Of course, for , the above function defines a distance (or metric) on .

An illustrative example of -metric would be the following:

Example 1. Let the space Then, the function , where is a -metric, with .

The concepts of convergent and Cauchy sequences on -metric spaces can be defined in a similar way to the case of ordinary metric spaces.

Definition 5. Let be a sequence in the -metric space . We say that the sequence is
(c) convergent ⟺ there exists such that for any , there exists such that , for all
This means, ; we write , or .
(C) Cauchy ⟺ for any , there exists such that , for all

In case every Cauchy sequence in is convergent, we say that the -metric space is complete.

Lemma 6 (see [19]). Let be a -metric space and be a sequence of elements in such that there exists such that for every . Then, is a Cauchy sequence.

Definition 7. Let , , be a -metric space and a function satisfying the following:
(ζ_b1) for all
(ζ_b2) If are two sequences in , such that for then Thus, is said to be a --simulation function. We shall denote by the family of all -simulation functions.

(See, e.g., [16, 20, 21], for more details and examples.)

In [22], the authors considered several fixed point theorems, in the setting of -metric spaces, for a family of contractions (called multiparametric contractions) depending on two functions (that are not defined in ) and some parameters.

Definition 8 (see [22]). Let be a -metric space and be a mapping. Let be a set of five nonnegative real numbers, and we denote by the function defined, for all , by We say that is a -multiparametric contraction on if where are two auxiliary functions and .

Inspired by some results in [15], we will consider a pair of two functions that satisfy the following:

(p₁) for any

(p₂) is nondecreasing

Let be the set of such pair of functions; that is,

2. Main Results

Definition 9. Let be a -metric space. A mapping is a Proinov--contraction mapping of type if there exist , , a number , and nonnegative real numbers , with , such that for all with , we have where

Remark 10. We mention that following Corollary 11 in [22], we have that, for , either admits at least one fixed point or , , for all distinct .

Theorem 11. On a complete -metric space , any continuous Proinov--contraction mapping of type has a unique fixed point provided that .

Proof. Starting with a point , we can consider the sequence in , build as follows: We observe that if there is some such that , it follows that , so is a fixed point of the mapping . With this in mind, we will presume that for all . Thus, since by (12), which is equivalent, taking into account, with Moreover, since the above inequality becomes Since the pair , it follows Consequently, Let . Consequently, Moreover, by Lemma 6, it follows that the sequence is Cauchy, and taking into account the completeness of the -metric space , we find that there exists such that But, the mapping was supposed to be continuous, so that Thereupon, ; that is, is a fixed point of the mapping .
Supposing that there exists another point , such that , we have Thus, where We have in this case or, since is nondecreasing, which is a contradiction. Therefore, the mapping admits a unique fixed point.

Example 2. Let , the function , and be a -metric with , and let be a continuous mapping, where Let the pair , with , , for any , and , , for . Thus, choosing , , , and , we have For such that , we have and For , such that , we have and Therefore, is a continuous Proinov--contraction mapping of type , and from Theorem 11, it follows that has a unique fixed point.

Corollary 12. Let be a complete -metric space and be a continuous mapping such that there exist , , a number , and nonnegative real numbers such that for all with , we have where Then, has a unique fixed point provided that .

Theorem 13. On a complete -metric space any Proinov--contraction mapping of type has a unique fixed point provided that .

Proof. Let be the sequence in defined by (14), with , for all . Thus, by (12), Thus, using , it follows where Since , proceeding in the previous proof, it follows that is a convergent sequence in . Thus, there exists , such that
We shall show that . First of all, we claim that or By contradiction, if we suppose that there exists such that neither (39) nor (40) hold, we have which is a contradiction. Consequently, at least one of (39) or (40) holds, so that we can find a subsequence of , such that Therefore, keeping (12) in mind, which is equivalent with Moreover, since , and then, But, Consequently, there exists , and we have On the other hand, Therefore, which is a contradiction. Thus, . Supposing that this point is not unique, we can find another point , such that . In this case, We have, and, taking into account, which is a contradiction, because So, the mapping possesses a unique fixed point.

Corollary 14. Let be a complete -metric space and be a continuous mapping such that there exist , , a number , and nonnegative real numbers such that for all with , we have where Then, has a unique fixed point provided that .

Theorem 15. On a complete -metric space , any Proinov--contraction mapping of type has a unique fixed point provided that and .

Proof. Let be the sequence in defined by (14), with , for all . Thus, by (12), Thus, using , it follows or, equivalent (keeping in mind and ) where Assuming that there exists such that , we have Therefore, by (58) and (59), together with , we get and taking into account, it follows which is a contradiction.
Consequently, , for any , and is a nonincreasing sequence; so, we can find such that . Moreover, and then, from (58) and , which leads us to Letting , we get , for any . Thus, Lemma 6 ensure that the sequence is Cauchy, that is, Moreover, the -metric space is supposed to be complete, so, we can find such that Further, from the proof of Theorem 13, we know that at least one of (39) or (40) holds, and for this reason, there exists a subsequence of such that which implies Therefore, and, by , Now, since taking into account (66) and (67), But, which combined with (73) showing that But, this is a contradiction, so,
We claim that is the only fixed point of . Suppose that, on the contrary, there exists , such that and . Thus, and moreover, which is a contradiction.

Example 3. Let and a function , defined as follows: It is easy to check that is a -metric, with . Let the mapping , where Let the pair , where , for any , and , . Choosing and and , we have We consider the following cases (which respect the condition ): (i), , ,which means (ii), ,which means Consequently, the mapping is a Proinov--contraction mapping of type and, by Theorem 15, it follows that has a unique fixed point.

Corollary 16. Let be a complete -metric space and be a c mapping such that there exist , , a number , and nonnegative real numbers such that for all with , we have where Then, has a unique fixed point provided that and .

3. Conclusion

In this paper, we extend the renowned Proinov’s result [15] in several directions: First of all, we investigate the contractions involving interesting rational forms. Secondly, the abstracted structure is chosen as a -metric space that is one of the natural and novel generalizations of the concept of metric spaces. Thirdly, we use auxiliary simulation functions to improve Proinov’s results [15].

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

The authors declare that the study was realized in collaboration with equal responsibility. All authors read and approved the final manuscript.