Journal of Mathematics

Volume 2018, Article ID 9127486, 9 pages

https://doi.org/10.1155/2018/9127486

## Wardowski Type Contractions and the Fixed-Circle Problem on -Metric Spaces

^{1}Department of Mathematics and General Sciences, Prince Sultan University Riyadh 11586, Saudi Arabia^{2}Balikesir University, Department of Mathematics, 10145 Balikesir, Turkey

Correspondence should be addressed to Nabil Mlaiki; moc.liamg@2102ikialmn

Received 18 February 2018; Accepted 5 September 2018; Published 10 October 2018

Academic Editor: Ming-Sheng Liu

Copyright © 2018 Nabil Mlaiki 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

In this paper, we present new fixed-circle theorems for self-mappings on an -metric space using some Wardowski type contractions, -contractive, and weakly -contractive self-mappings. The common property in all of the obtained theorems for Wardowski type contractions is that the self-mapping fixes both the circle and the disc with the center and the radius .

#### 1. Introduction

Fixed-point theory has many applications in different fields; see [1–10]. Recently, using Wardowski’s technique, some new fixed-point theorems on -metric spaces [11] and some new fixed-circle theorems on metric spaces [12, 13] have been obtained. Our aim in this paper is to obtain various fixed-circle results using this technique. In Section 2, we recall some necessary background on -metric spaces and give new examples. In Section 3, we introduce the notion of an -contraction to obtain fixed-circle theorems. By means of this notion, we define new types of an -contraction such as Hardy-Rogers type -contraction and Reich type -contraction and present some fixed-circle results on -metric spaces. Also, we give an illustrative example of a self-mapping satisfying all of the conditions of the obtained theorems. In Section 4, we prove the existence along with the conditions that give us uniqueness of a fixed circle for -contractive and weakly -contractive self-mappings on -metric spaces. In Section 5, we give an application of fixed-circle results obtained by Wardowski technique to integral type contractive self-mappings.

#### 2. Preliminaries

In this section, we recall some necessary notions, relations, and results about -metric spaces.

*Definition 1 (see [14]). *Let be a nonempty set and be a function satisfying the following conditions for all if and only if

Then is called an -metric on and the pair is called an -metric space.

*Example 2 (see [15]). *Let (or ) and the function be defined as for all (or ). Then the function is an -metric on (or ). This -metric is called the usual -metric on (or ).

Lemma 3 (see [14]). *Let be an -metric space and . Then we have *

The relationships between a metric and an -metric were studied in different papers (see [16–18] for more details). In [17], a formula of an -metric space which is generated by a metric was investigated as follows.

Let be a metric space. Then the function defined byfor all , is an -metric on . The -metric is called the -metric generated by [18]. We note that there exists an -metric which is not generated by any metric as seen in the following example.

*Example 4. *Let be a nonempty set, the function be any metric on , and the function be defined by for all . Then the function is an -metric and is an -metric space. Indeed,

for , we have

using Table 1, we can easily see that the condition is satisfied.