Abstract

In this paper, we give a concept of -contraction in the setting of expanded –metric spaces and discuss the existence and uniqueness of a common fixed point. Introduced results generalize well-known fixed point theorems on contraction conditions and in the given spaces.

1. Introduction and Preliminaries

The tremendous applications of fixed point theory had always inspired the growth of this domain. In 1922, Banach formulated his most simple but very natural result which is now popularly referred to as the Banach contraction principle. In the course of the last several decades, this principle has been extended and generalized in many directions with several applications in many branches. Employing simulation functions, Khojasteh et al. [1] initiated the idea of -contractions and utilized the same to cover the varied types of nonlinear contractions in the existing literature. Later, Argoubi et al. [2] and Roldán-López-de-Hierro et al. [3] independently sharpened the notion of simulation functions and also proved some coincidences and common fixed point results. Very recently, Lopez et al. [4] introduced the notion of -contractions in order to extend several nonlinear contractions such as -contractions, manageable contractions, and Meir-Keeler contractions. Indeed, -contractions are associated with -functions that satisfy two independent conditions involving two sequences of nonnegative real numbers. Soon, inspired by -contractions, Shahzad et al. [5] introduced the notion of -contractions which remains an extension of -contractions given in [6] by Roldán-López-de-Hierro and Shahzad wherein the authors proved very interesting results.

Czerwik [7] established a successful generalization of the metric space concept by introducing the notion of -metric space. Following this, a number of authors have introduced respective interesting theorems in -metric, (see [8–14]). Newly, Kamran et al. [15] inspired by the concept of b-metric space, they introduced the concept of extended space and also developed some fixed point theorems for self-mappings defined in such spaces. Their results extend/generalize many of the results already available in the literature. In this paper, we shall define a general contraction condition with the help of some auxiliary functions and investigate the existence and uniqueness of a fixed point for such mappings in the frame of -metric space.

Definition 1 (see [7]). Let be a nonempty set and let satisfy the following for all . (i)(ii)(iii), where The pair is called a -metric space; when , the -metric space becomes a usual metric space.

Example 2 (see [8, 9]). Let . Then, the functional defined by is a -metric space on with .

Example 3 (see [16]). The space (where ) of all real functions such that , together with the functional is a -metric space with .

Example 4 (see [17]). Let and such that Then, Therefore, is a -metric space for all . If , then the ordinary triangle inequality does not hold, and is not a metric space.

Definition 5 (see [15]). Let be a nonempty set and , and let satisfy: (i)(ii)(iii)The pair is called an extended -metric space. If , for , then we reduce to Definition 1.

Example 6 (see [17]). Let and be defined by (i)If , then ,(ii)If and , then and (iii)If , such that with and .

Notice that is not a metric space since . However, it is easy to see that is an extended -metric space for , where

Example 7 (see [15]). Let be the space of all continuous real valued functions defined on , let where , and note that is a complete extended -metric space by considering

In this context, wonderful theorems established by the authors in extended -metric space, for examples, Fahed et al. and Swapna and Phaneendra [18, 19], got some new fixed point results in an extended -metric space. Also, Ullah et al. [20] proved fixed point theorems in complex-valued extended -metric spaces. In this bearing, Mitrović et al. [21] established new results in extended -metric space, in follows that we recollect some fundamental notions, for example, convergence, the notion of the Cauchy sequence, and completeness in an extended - metric space.

Definition 8 (see [22]). Let be an extended -metric space, and then (i)A sequence is said to converge to if, , there exists such that . We write (ii)A sequence in is said to be Cauchy if, , there exists such that

Definition 9 (see [15]). An extended -metric space is complete if every Cauchy sequence in is convergent.

Lemma 10 (see [22]). Let be a complete extended -metric space. If is continuous map, then every convergent sequence in has a unique limit.

Theorem 11 (see [15]). Suppose is an extended -metric space such that is a continuous mapping. Suppose , it fulfills where is such that, for each , we have . Here, Then, has exactly one fixed point , moreover .

For our objectives, we recall the definition of orbital admissible maps introduced by Popescu [23].

Definition 12. Let be a self-map on and . We say that is an -orbital admissible if for all , we have

Remark 13 (see [23]). Every -admissible mapping is an -orbital admissible mapping.

Definition 14 (see [24]). For a nonempty set , suppose and are mappings. One says that self-mapping on is -admissible if for , one has

Definition 15 (see [17]). Let be the family of functions satisfying the following conditions (i) is nondecreasing(ii)

2. Main results

Definition 16. Let be an extended -metric space and let and such that where

Then, and are -contraction for all , where .

The headmost principal result of this paper is as follows:

Theorem 17. Let be a complete extended - metric space, and let be an -contraction mappings. Let for all where . Assume also that (i) and are -orbital admissible(ii)There exists such that (iii) and are continuousThen, and possess a unique coincidence fixed point ; that is, .

Proof. By a supposition, for some , we have . Suppose that . Let that . Since and are -admissible, we get Repeatedly, we obtain for all On regard of (15) and (11), we get where Now, if for some , thus which is a contradiction. On the other hand, if , then for all , we have Sequentially for all , we get Thus, there exists such that Taking to inequality (19), we obtain Therefore, when , we get We will show that is a Cauchy sequence, as follows: Also, And so, until the inequality (24) reaches to we conclude that The series Suppose , then by (13) where . Then, (28) converges by [25]. As result, in perspective of (27), we have Then, is a Cauchy sequence, and since is a complete extended quasimetric space, there exists such that By condition (ii), we obtain Hence, we deduce that . Furthermore, let such that where . So, by (12), we obtain where Then, Therefore, ; thus, . Hence, and possess a unique coincidence fixed point in .

To mitigation the continuity case on the given self-mappings, we will modify Definition 16 as follows:

Definition 18. Let be an extended quasimetric space, and we say that and are -contraction such that and if for all fulfilled and

By remove continuity of the given mappings, we get the following major result.

Theorem 19. Let be a complete extended quasimetric space, and let be -contraction mappings. Let (13) and conditions (i) and (ii) of Theorem 17 be satisfied. Assume also that (iii)If is a sequence in such that and as , then there exists a subsequence such that Then, and possess a coincidence fixed point , that is,

Proof. By inequality (11) and condition (iii) in Theorem 19, there exists such that . Applying inequality (11), we obtain that And Also, we have Then, When then neither nor for some when . If , then by inequality (39), we obtain . Thus, from Definition 18, we get Also, by inequalities (38) and (42), we have Thus, Which is an ambivalence. Therefore, and . Likewise, we can get that . Hence, is a common fixed point for and in , that is, .
Let be two common fixed points of and such that then by (38) and (39), we get where Now, if , then , which implies by (46) that but this is a contradiction. Hence, , i.e., . This proves the uniqueness of the common fixed point of given mappings.

Example 20. Suppose that , where and . Consider the extended -metric space on as follows:

It is apparent that the triangle inequality on is not fulfilled. Actually,

Observes that the condition (iii) in Definition 5 is satisfied. Suppose that are defined as

Taking

and are an -contractive mappings with . Furthermore, there exists such that .

Actually, we have for .

Now, suppose that with . It implies that .

By Definition 1, we have , and then and are -admissible mappings. Also, and are clearly not continuous mappings.

Otherwise, if such that, , then for all . Assume the sequence is considered iteratively as for all . Considering the arbitrary point , it is located in either or ; so, we have two cases:

In case , thus is constant sequence and . Then, for all , we have , which implies that .

In case , thus is constant sequence and . Then, for all , we have , which implies that .

Consequently, and fulfill the conditions of Theorem 19; hence, and have a unique common fixed point on , which is .

3. Conclusion

By replacing or with a proper one, we can conclude several results from the showed prime result in this paper on different sides. For example, we can obtain results in this frame of periodic contractions and partially ordered spaces.

Data Availability

No data was used in this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.