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Journal of Function Spaces
Volume 2018, Article ID 3264620, 4 pages
https://doi.org/10.1155/2018/3264620
Research Article

-Meir-Keeler Contraction Mappings in Generalized -Metric Spaces

1Department of Mathematics, Atilim University, Incek, 06836 Ankara, Turkey
2Institute of Mathematics, Silesian University of Technology, Gliwice, Poland
3Department of Mathematics, College of Education of Jubail, Imam Abdulrahman Bin Faisal University, P.O. Box 12020, Jubail 31961, Saudi Arabia
4Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan

Correspondence should be addressed to Hassen Aydi; as.ude.dou@idyamh

Received 29 September 2017; Accepted 20 December 2017; Published 28 January 2018

Academic Editor: Satish Shukla

Copyright © 2018 Erdal Karapinar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We present a fixed point theorem for generalized -Meir-Keeler type contractions in the setting of generalized -metric spaces. The presented results improve, generalize, and unify many existing famous results in the corresponding literature.

1. Introduction and Preliminaries

The idea of a -metric has been introduced in the papers [1, 2]. Very recently, this idea was extended in [3] to a generalized -metric space in the following manner.

Definition 1. Let be a nonempty set and be a fixed constant. A function is called a generalized -metric space (in brief, gbms) if and only if for the conditions are satisfied:(d1)  if and only if .(d2) .(d3) .

A triple is called a generalized -metric space.

On the other hand, Meir and Keeler [4] have proved the following very general result on the existence of fixed points of Meir-Keeler contraction mappings in metric spaces.

Theorem 2 (see [4]). Let be a complete metric space and satisfy the following condition: (d)Given , there exists such that has a unique fixed point . Moreover, for any , where denotes the th iteration of at a point .

This result has been generalized and extended in many directions; see [515]. Using some auxiliary functions, the main purpose of this paper is to extend and generalize this result on generalized -metric spaces.

For the sake of explicitness, we recall some notations. The symbols denote the natural and real numbers, respectively. Furthermore, and .

Berinde [16] characterized comparison functions to define the contraction mappings in the setting of -metric spaces.

Definition 3. Let be a real number. A function is called a -comparison function if (1) is increasing;(2)there exist , , and a convergent nonnegative series such that , for and any .

Denote as the set of -comparison functions. We will need the following essential properties in our further discussion.

Lemma 4 (see [1618]). For a -comparison function , the following statements hold: (1)The series converges for any .(2)The function defined by , , is increasing and continuous at .(3)Each iterate of for is also a -comparison function.(4) is continuous at .(5) for any .

Inspired by Popescu [19], we introduce the concept of generalized -orbital admissible mappings.

Definition 5. Let be a mapping and be a function. We say that is a generalized -orbital admissible if

Notice that each -orbital admissible mapping [19] is generalized -orbital admissible.

Based on the concept of generalized -orbital admissibility, we are the first who establish a fixed point result for a Meir-Keeler type contraction in the setting of generalized -metric spaces.

2. Main Results

We start with this definition.

Definition 6. For an arbitrary constant , let be a self-mapping defined on a generalized -metric space . Then is called an -Meir-Keeler contractive mapping if there exist two auxiliary mappings and such that

Remark 7. For and with , from (4) we derive that

Our main result is as follows.

Theorem 8. Let be a fixed constant and be a complete generalized -metric space. Suppose that a self-mapping is an -Meir-Keeler type contraction. Assume also that (i) is generalized -orbital admissible;(ii)there exists such that ;(iii) is continuous. Then for such , one of the following statements holds: ()For every , ()There exists such that and . In this case, there exists such that .

Proof. On account of assumption (ii), there exists such that . We suppose that case () is not satisfied. Consequently, we have to examine case (). Consequently, there exists such that and . If , the proof is completed. Assume that . By property of and Remark 7, we haveSince is a generalized -orbital admissible mapping, by (ii), we derive that Recursively, we obtain thatApplying (10) in (8), we getThusAgain, on account of (10) and (12) in (8), by induction, one getsConsequently, for , by (13) we haveFinally,for all . By (15) and the fact that , it follows that is a Cauchy sequence of elements of .
Since is complete, there exists with Since is continuous, we get and is a fixed point of , which ends the proof.

Definition 9. Let be a fixed constant. We say that a generalized -metric space is regular if is a sequence in such that for all and as ; then there exists a subsequence of such that and for all .

Theorem 10. Let be a fixed constant and be a complete generalized -metric space. Suppose that a self-mapping is an -Meir-Keeler type contraction. Assume also that (i) is a generalized -orbital admissible mapping;(ii)there exists such that ;(iii) is regular. Then for such , one of the following statements holds: ()For every , ()There exists such that and . In this case, there exists such that .

Proof. In case (), following the proof of Theorem 8, we know that the sequence converges to some . By Definition 9 and condition (iii), there exists a subsequence of such that and for all . Applying (5) for all , we get thatLetting in the above equality, we get ; that is, .

For the uniqueness of a fixed point of an -Meir-Keeler type contraction mapping in , we shall consider the following condition:()For all , we have , where denotes the set of fixed points of .

Theorem 11. By adding condition to the hypotheses of Theorem 8 (resp., Theorem 10), has at most one fixed point in .

Proof. Let be an -Meir-Keeler type contraction. Owing to Theorem 8 (resp., Theorem 10), has a fixed point .
Now, we shall show that has at most one fixed point in . We argue by contradiction. For this, assume that there exist two distinct fixed points and of , where ; that is, We deduce By condition , and since , in view of (5), one writes which is a contradiction, so . This completes the proof.

3. Consequences

3.1. Meir-Keeler Contraction Mappings in gbms

In this section, we present our main result. By letting and , we get the following result.

Theorem 12. Let be a generalized complete -metric space and satisfy the following: given , there exists such that

Let . Then one of the following alternatives holds:()For every ( being the set of all nonnegative integers), ()There exists such that .

In case , we assert the following:(i)The sequence is Cauchy in .(ii)There exists a point such that and .(iii) is the unique fixed point of in .(iv)For every ,

Remark 13. Unfortunately, if is a metric space, we do not get the result of Meir-Keeler [4].

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.

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