Abstract

In this article, certain fixed point results for Geraghty-type contractions in the setting of -Complete extended fuzzy -metric spaces are established. An example is elaborated, and a supporting application is also established. Many known existing results are special cases of our obtained results.

1. Introduction

Kramosoil and Michalek [1] defined the notion of a fuzzy metric space (FMS) by using the concept of fuzzy sets introduced by Zadeh [2]. Grabiec [3] gave the concept of weak Cauchy sequences, which is called a -Cauchy sequence and proved the Banach contraction principle (BCP) [4] in the setting of a FMS. George and Veermani [5] modified the definition of a FMS given by Kramosil and Michalek [1] and established some fixed point results. For more works in FMSs, see [6–10].

In 1989, Bakhtin [11] introduced the notion of a -metric space (BMS). Later on, the concept of a BMS was further used by Czerwick [12] to establish different fixed point results on this platform. The study of -metric space endows an imperative place in fixed point theory with multiple aspects. Many mathematicians (Abdeljawad et al. [13, 14], Akkouchi [15], Chifu and Karapinar [16], Kadelburg and Radenović [17], Chauhan and Gupta [18], Kamran et al. [19] and Gupta [20], etc) led the foundation to improve fixed point theory in BMSs. Another innovative task has been achieved by Kamran et al. [21] in 2017 by introducing the notion of an extended -metric space (EBMS), which generalizes the notion of a BMS. Some fixed point results are proved in this new setting. See for instance, the works shown in [22, 23].

By considering an auxiliary function, Geraghty [24] established a generalisation of the Banach contraction principle in the complete metric spaces. Later on, Gupta et al. [25] proved the fixed point theorems for -Geraghty contraction type maps in ordered metric spaces. For more results using Geraghty contraction type maps in metric spaces can be seen in [26–30].

Nǎdǎban [31] generalized the notion of -metric space (BMS) by introducing the concept of fuzzy -metric space. The idea of an extended fuzzy -metric space (EFBMS) was introduced by Mahmood et al. in [32]. In the present article, some fixed point results for Geraghty-type contractions in -complete EFBMS are established. Our results are generalizations of many existing results on FMS. See, for example, [32–35]. At the end, by applying our results, we give a real application.

2. Preliminaries

Recently, the concept of an EFBMS in [32] has been introduced as follows:

Definition 1 (see [32]). Let be a non empty set. Given and let be a continuous -norm. A fuzzy set in is called an extended fuzzy -metric on if for all , the following conditions hold: ;  if and only if ; ; ;  is left continuous, and .Here, is called an EFBMS.

Remark 1. Taking , the notion of a FBMS defined in [31] is obtained and by taking , the notion of a FMS defined in [1] is obtained.

Example 1 (see [32]). Let and define by . Clearly, is a BMS. Define the mapping byLet be defined byand take the continuous -norm , that is, . Then is an EFBMS.
The notions of convergence, Cauchyness and completeness in an EFBMS can be generalized naturally as follows:

Definition 2 (see [32]). Let be an EFBMS.(i)A sequence in is said to be convergent if there exits such that Grabiec [3] gave the concept of weak Cauchyness which we will denote by -Cauchyness as follows:(ii)A sequence in is said to be a -Cauchy sequence if for all and (iii)An EFBMS in which every -Cauchy sequence is convergent is called a -complete EFBMS.

Lemma 1 (see [34]). Let be a complete FBMS and for all and , then .
Let be a complete FBMS. Throughout this article, letwhere .

3. Main Results

The BCP in the setting of -complete EFBMSs, is established as follows:

Theorem 1. Let be a -complete EFBMS with . Let be a mapping satisfyingfor all , where and . Then has a unique fixed point.

Proof. Let . Generate a sequence by the iterative process . For all , by (5), we haveSo, we haveFor any , taking and using repeatedly, one can writeUsing (7) and , we getThenSince for all , we have , taking limit as , we getHence, is a -Cauchy sequence. Since is a -complete EFBMS, there exists such thatWe want to show that is a fixed point of . ConsiderThat is, is a fixed point.

3.1. Uniqueness

Assume for some . Then

Hence, the fixed point is unique.

Example 2. Let and
It is easy to verify that is a -complete EFBMS. Define a mapping such that . Now, for all , we haveRecall thatThis implies thatAlso, is the unique fixed point of .

Remark 2. Taking , we get Theorem 3.1 of [36].

Remark 3. Taking , the main result of [32] is obtained and for and for , the main result of [3] is obtained.

Theorem 2. Consider a -complete EFBMS with . Let be a mapping satisfying the conditionfor all , where and . Then has a unique fixed point.

Proof. Starting by the same way as in Theorem 1, we haveHenceIfthen (20) impliesBy Lemma 1, has a fixed point.
Ifthen from (20), we haveThe rest of the proof follows from Theorem 1.

Remark 4. Taking we get Theorem 3.6 of [36].
The following result is an extension of the main result of Gupta at el. [34].

Theorem 3. Consider a -complete EFBMS with . Let be a mapping satisfyingfor all , where and , then has a unique fixed point.

Proof. ConsiderIfthen (26) impliesThen the result is trivial by Lemma 1.Ifthen from (26) we haveContinuing in this way, one writesTherefore, proceeding as in Theorem 1 after inequality (7), the desired result is obtained.

Theorem 4. Let be a -complete EFBMS with . Let be a mapping satisfyingfor all , where and , then has a unique fixed point.

Proof. For , we choose a sequence in and start by . For all , we haveSo, we haveIfthen from (34)Then result is trivial by Lemma 1. Ifthen from (34)Continuing in this direction, we getThe desired result is then established by the same procedure as in Theorem 1. The following result is the extension of the main result of Roshan et al. [35].

Theorem 5. Consider a -complete EFBMS with . Let be a mapping satisfyingfor all , where , wherethen has a unique fixed point.

Proof. Let . Choose a sequence in by starting an iterative process . For all , we haveNowThat isOne writesWe obtainIf then from (42)Since the range of is , he result is obvious by Lemma 1. If then from (42)Continuing in this way, we haveThen the result is trivial by Lemma 1.

Theorem 6. Let be a -complete EFBMS with . Let be a mapping satisfying the conditionwherefor all , where and , then has a unique fixed point.

Proof. Consider