Wardowski type contractions and the fixed-circle problem on s-metric spaces

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Research Article

Wardowski Type Contractions and the Fixed-Circle Problem on

𝑆-Metric Spaces

Nabil Mlaiki

,

1

Ufuk Çelik,

2

Nihal Ta

G ,

2

Nihal Yilmaz Özgür

,

2

and Aiman Mukheimer

1

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; nmlaiki2012@gmail.com 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.

In this paper, we present new fixed-circle theorems for self-mappings on an 𝑆-metric space using some Wardowski type

contractions,𝜓-contractive, and weakly 𝜓-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 𝑥₀ and the

radius𝑟.

1. Introduction

Fixed-point theory has many applications in different fields; see [1–10]. Recently, using Wardowski’s technique, some new fixed-point theorems on𝑆-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 𝑆-metric spaces and give new examples. In Section 3, we introduce the notion of an 𝐹_𝑐𝑆-contraction to obtain fixed-circle theorems. By means of this notion, we define new types of an𝐹_𝑐𝑆-contraction such as Hardy-Rogers type 𝐹_𝑐𝑆-contraction and Reich type 𝐹_𝑐𝑆 -contraction and present some fixed-circle results on𝑆-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 for𝜓-contractive and weakly𝜓-contractive self-mappings on 𝑆-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 about𝑆-metric spaces.

Deﬁnition 1 (see [14]). Let 𝑋 be a nonempty set and S :

𝑋 × 𝑋 × 𝑋 󳨀→ [0, ∞) be a function satisfying the following conditions for all𝑥, 𝑦, 𝑧, 𝑎 ∈ 𝑋 :

(𝑆1) S(𝑥, 𝑦, 𝑧) = 0 if and only if 𝑥 = 𝑦 = 𝑧

(𝑆2) S(𝑥, 𝑦, 𝑧) ≤ S(𝑥, 𝑥, 𝑎) + S(𝑦, 𝑦, 𝑎) + S(𝑧, 𝑧, 𝑎) ThenS is called an 𝑆-metric on 𝑋 and the pair (𝑋, S) is called an𝑆-metric space.

Example 2 (see [15]). Let𝑋 = R (or C) and the function S :

𝑋 × 𝑋 × 𝑋 󳨀→ [0, ∞) be defined as

S (𝑥, 𝑦, 𝑧) = |𝑥 − 𝑧| + 󵄨󵄨󵄨󵄨𝑦 − 𝑧󵄨󵄨󵄨󵄨, (1) for all𝑥, 𝑦, 𝑧 ∈ R (or C). Then the function S is an 𝑆-metric onR (or C). This 𝑆-metric is called the usual 𝑆-metric on R (orC).

Volume 2018, Article ID 9127486, 9 pages https://doi.org/10.1155/2018/9127486

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Lemma 3 (see [14]). Let (𝑋, S) be an 𝑆-metric space and

𝑥, 𝑦 ∈ 𝑋. Then we have

S (𝑥, 𝑥, 𝑦) = S (𝑦, 𝑦, 𝑥) . (2) The relationships between a metric and an𝑆-metric were studied in different papers (see [16–18] for more details). In [17], a formula of an𝑆-metric space which is generated by a metric𝑑 was investigated as follows.

Let(𝑋, 𝑑) be a metric space. Then the function S_𝑑: 𝑋 × 𝑋 × 𝑋 󳨀→ [0, ∞) defined by

S_𝑑(𝑥, 𝑦, 𝑧) = 𝑑 (𝑥, 𝑧) + 𝑑 (𝑦, 𝑧) , (3) for all𝑥, 𝑦, 𝑧 ∈ 𝑋, is an 𝑆-metric on 𝑋. The 𝑆-metric S_𝑑 is called the𝑆-metric generated by 𝑑 [18]. We note that there exists an𝑆-metric which is not generated by any metric 𝑑 as seen in the following example.

Example 4. Let𝑋 be a nonempty set, the function 𝑑 : 𝑋 ×

𝑋 󳨀→ [0, ∞) be any metric on 𝑋, and the function S : 𝑋 × 𝑋 × 𝑋 󳨀→ [0, ∞) be defined by

S (𝑥, 𝑦, 𝑧) = min {1, 𝑑 (𝑥, 𝑦)} + min {1, 𝑑 (𝑦, 𝑧)} + min {1, 𝑑 (𝑥, 𝑧)} , (4) for all𝑥, 𝑦, 𝑧 ∈ 𝑋. Then the function S is an 𝑆-metric and (𝑋, S) is an 𝑆-metric space. Indeed,

(𝑆1) for 𝑥, 𝑦, 𝑧 ∈ 𝑋, we have S (𝑥, 𝑦, 𝑧) = 0 ⇐⇒

min{1, 𝑑 (𝑥, 𝑦)} + min {1, 𝑑 (𝑦, 𝑧)} + min {1, 𝑑 (𝑥, 𝑧)} = 0 ⇐⇒

𝑑 (𝑥, 𝑦) = 𝑑 (𝑦, 𝑧) = 𝑑 (𝑥, 𝑧) = 0 ⇐⇒ 𝑥 = 𝑦 = 𝑧

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(𝑆2) using Table 1, we can easily see that the condition (𝑆2) is satisfied.

Also the 𝑆-metric S is not generated by any metric 𝑚. Conversely, suppose that there exists a metric𝑚 such that

S (𝑥, 𝑦, 𝑧) = 𝑚 (𝑥, 𝑧) + 𝑚 (𝑦, 𝑧) , (6) for all𝑥, 𝑦, 𝑧 ∈ 𝑋. Then we get

S (𝑥, 𝑥, 𝑧) = 2𝑚 (𝑥, 𝑧) and so𝑚 (𝑥, 𝑧) = min {1, 𝑑 (𝑥, 𝑧)} (7) and S (𝑦, 𝑦, 𝑧) = 2𝑚 (𝑦, 𝑧) and so 𝑚 (𝑦, 𝑧) = min {1, 𝑑 (𝑦, 𝑧)} . (8) Therefore, we obtain

min{1, 𝑑 (𝑥, 𝑦)} + min {1, 𝑑 (𝑦, 𝑧)} + min {1, 𝑑 (𝑥, 𝑧)} = min {1, 𝑑 (𝑥, 𝑧)} + min {1, 𝑑 (𝑦, 𝑧)} , (9)

which is a contradiction. Consequently,S is not generated by any metric𝑚.

In [19] and [14], a circle and a disc are defined on an 𝑆-metric space as follows, respectively:

𝐶𝑆_𝑥₀_,𝑟= {𝑥 ∈ 𝑋 : S (𝑥, 𝑥, 𝑥₀) = 𝑟} (10) and

𝐷𝑆_x₀_,𝑟= {𝑥 ∈ 𝑋 : S (𝑥, 𝑥, 𝑥0) ≤ 𝑟} . (11)

We give an example.

Example 5. Let𝑋 be a nonempty set, the function 𝑑 : 𝑋 ×

𝑋 󳨀→ [0, ∞) be any metric on 𝑋, and the 𝑆-metric space be defined as Example 4. Let us consider the circle 𝐶𝑆_𝑥

0,𝑟

according to the𝑆-metric: 𝐶𝑆_𝑥₀_,𝑟

= {𝑥 ∈ 𝑋 : S (𝑥, 𝑥, 𝑥₀) = 2 min {1, 𝑑 (𝑥, 𝑥₀)} = 𝑟} . (12) Then we have the following cases:

Case 1. If𝑟 = 2 then 𝐶𝑆_𝑥 0,𝑟= {𝑥 ∈ 𝑋 : 𝑑(𝑥, 𝑥0) ≥ 1}. Case 2. If𝑟 > 2 then 𝐶𝑆_𝑥 0,𝑟= 0. Case 3. If𝑟 < 2 then 𝐶𝑆_𝑥 0,𝑟= 𝐶𝑥0,𝑟/2, where𝐶𝑥0,𝑟/2= {𝑥 ∈ 𝑋 : 𝑑(𝑥, 𝑥₀) = 𝑟/2}.

Deﬁnition 6 (see [19]). Let(𝑋, S) be an 𝑆-metric space, 𝐶𝑆_𝑥

0,𝑟

be a circle, and𝑇 : 𝑋 󳨀→ 𝑋 be a self-mapping. If 𝑇𝑥 = 𝑥 for every𝑥 ∈ 𝐶𝑆_𝑥

0,𝑟then the circle𝐶

𝑆

𝑥0,𝑟is called the fixed circle of

𝑇.

3. 𝐹

_𝑐𝑆

-Contraction and Hardy-Rogers Type

𝐹

𝑆

𝑐

-Contraction on

𝑆-Metric Spaces

At first, we recall the definition of the following family of functions which was introduced by Wardowski in [20].

Deﬁnition 7 (see [20]). LetF be the family of all functions

𝐹 : (0, ∞) 󳨀→ R such that (𝐹₁) 𝐹 is strictly increasing

(𝐹₂) for each sequence {𝛼_𝑛} in (0, ∞) the following holds: lim𝛼_𝑛= 0 if and only if lim 𝐹(𝛼_𝑛) = −∞

(𝐹₃) there exists 𝑘 ∈ (0, 1) such that lim_𝛼󳨀→0+𝛼𝑘𝐹(𝛼) = 0.

The following is an example of some functions that satisfies conditions(𝐹₁), (𝐹₂), and (𝐹₃) of Definition 7.

Example 8 (see [20]). (1) 𝐹 : (0, ∞) 󳨀→ R defined by 𝐹(𝑥) =

ln(𝑥).

(2) 𝐹 : (0, ∞) 󳨀→ R defined by 𝐹(𝑥) = ln(𝑥) + 𝑥. (3) 𝐹 : (0, ∞) 󳨀→ R defined by 𝐹(𝑥) = −1/√𝑥. (4) 𝐹 : (0, ∞) 󳨀→ R defined by 𝐹(𝑥) = ln(𝑥2_{+ 𝑥).}

Note that these four functions satisfy conditions (𝐹₁), (𝐹2), and (𝐹3) of Definition 7.

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Table 1 1 1 1 ≤ 2 2 2 1 1 1 ≤ 2 2 2𝑑(𝑧, 𝑎) 1 1 1 ≤ 1 1 𝑑(𝑦, 𝑧) ≤ 2 2𝑑(𝑦, 𝑎) 2𝑑(𝑧, 𝑎) 1 1 1 ≤ 𝑑(𝑥, 𝑦) 𝑑(𝑦, 𝑧) 𝑑(𝑥, 𝑧) ≤ 2𝑑(𝑥, 𝑎) 2𝑑(𝑦, 𝑎) 2𝑑(𝑧, 𝑎) 1 1 𝑑(𝑦, 𝑧) ≤ 1 1 1 ≤ 2 2 2 1 1 𝑑(𝑦, 𝑧) ≤ 1 1 1 ≤ 2 2 2𝑑(𝑧, 𝑎) 1 1 𝑑(𝑦, 𝑧) ≤ 2 𝑑(𝑦, 𝑎) 𝑑(𝑧, 𝑎) ≤ 2 2𝑑(𝑦, 𝑎) 2𝑑(𝑧, 𝑎) 1 1 𝑑(𝑦, 𝑧) ≤ 1 𝑑(𝑥, 𝑧) 1 ≤ 2 2𝑑(𝑦, 𝑎) 2𝑑(𝑧, 𝑎) 1 1 𝑑(𝑦, 𝑧) ≤ 𝑑(𝑥, 𝑦) 1 1 ≤ 2 2𝑑(𝑦, 𝑎) 2𝑑(𝑧, 𝑎) 1 1 𝑑(𝑦, 𝑧) ≤ 𝑑(𝑥, 𝑦) 𝑑(𝑥, 𝑧) 𝑑(𝑦, 𝑧) ≤ 2𝑑(𝑥, 𝑎) 2𝑑(𝑦, 𝑎) 2𝑑(𝑧, 𝑎) 1 𝑑(𝑥, 𝑧) 𝑑(𝑦, 𝑧) ≤ 1 1 1 ≤ 2 2 2 1 𝑑(𝑥, 𝑧) 𝑑(𝑦, 𝑧) ≤ 1 1 1 ≤ 2 2 2𝑑(𝑧, 𝑎) 1 𝑑(𝑥, 𝑧) 𝑑(𝑦, 𝑧) ≤ 1 1 𝑑(𝑦, 𝑧) ≤ 2 2𝑑(𝑦, 𝑎) 2𝑑(𝑧, 𝑎) 1 𝑑(𝑥, 𝑧) 𝑑(𝑦, 𝑧) ≤ 𝑑(𝑥, 𝑦) 1 1 ≤ 2𝑑(𝑥, 𝑎) 2𝑑(𝑦, 𝑎) 2 1 𝑑(𝑥, 𝑧) 𝑑(𝑦, 𝑧) ≤ 𝑑(𝑥, 𝑦) 𝑑(𝑥, 𝑧) 𝑑(𝑦, 𝑧) ≤ 2𝑑(𝑥, 𝑎) 2𝑑(𝑦, 𝑎) 2𝑑(𝑧, 𝑎) 𝑑(𝑥, 𝑦) 𝑑(𝑥, 𝑧) 𝑑(𝑦, 𝑧) ≤ 1 1 1 ≤ 2 2 2 𝑑(𝑥, 𝑦) 𝑑(𝑥, 𝑧) 𝑑(𝑦, 𝑧) ≤ 1 1 1 ≤ 2 2 2𝑑(𝑧, 𝑎) 𝑑(𝑥, 𝑦) 𝑑(𝑥, 𝑧) 𝑑(𝑦, 𝑧) ≤ 1 1 𝑑(𝑦, 𝑧) ≤ 2 2𝑑(𝑦, 𝑎) 2𝑑(𝑧, 𝑎) 𝑑(𝑥, 𝑦) 𝑑(𝑥, 𝑧) 𝑑(𝑦, 𝑧) ≤ 2𝑑(𝑥, 𝑎) 2𝑑(𝑦, 𝑎) 2𝑑(𝑧, 𝑎)

Other possibilities can be proved like this table.

Now we introduce the following new contraction type using this family of functions.

Deﬁnition 9. Let(𝑋, S) be an 𝑆-metric space. A self-mapping

𝑇 on 𝑋 is said to be an 𝐹_𝑐𝑆-contraction if there exist𝐹 ∈ F, 𝑡 > 0, and 𝑥₀∈ 𝑋 such that for all 𝑥 ∈ 𝑋 the following holds:

S (𝑇𝑥, 𝑇𝑥, 𝑥) > 0 󳨐⇒

𝑡 + 𝐹 (S (𝑇𝑥, 𝑇𝑥, 𝑥)) ≤ 𝐹 (S (𝑥, 𝑥, 𝑥0)) .

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Now, we present the following proposition.

Proposition 10. Let (𝑋, S) be an 𝑆-metric space. If a

self-mapping𝑇 on 𝑋 is an 𝐹_𝑐𝑆-contraction with𝑥₀ ∈ 𝑋, then we have𝑇𝑥₀= 𝑥₀.

Proof. Assume that𝑇𝑥₀ ̸= 𝑥₀. From the definition of an𝐹_𝑐𝑆 -contraction, we get

S (𝑇𝑥₀, 𝑇𝑥₀, 𝑥₀) > 0 󳨐⇒

𝑡 + 𝐹 (S (𝑇𝑥₀, 𝑇𝑥₀, 𝑥₀)) ≤ 𝐹 (S (𝑥₀, 𝑥₀, 𝑥₀)) . (14) Inequality (14) contradicts with the definition of𝐹 since 𝐹 : (0, ∞) 󳨀→ R and S(𝑥₀, 𝑥₀, 𝑥₀) = 0. Therefore, it should be 𝑇𝑥₀= 𝑥₀.

Using this new type contraction, we give the following fixed-circle theorem.

Theorem 11. Let (𝑋, S) be an 𝑆-metric space, 𝑇 be an

𝐹_𝑐𝑆-contractive self-mapping with 𝑥₀ ∈ 𝑋, and 𝑟 =

min{S(𝑇𝑥, 𝑇𝑥, 𝑥) : 𝑇𝑥 ̸= 𝑥}. Then 𝐶𝑆_𝑥

0,𝑟is a ﬁxed circle of

𝑇. 𝑇 especially ﬁxes every circle 𝐶𝑆_𝑥₀_,𝜌where𝜌 < 𝑟. Proof. Let𝑥 ∈ 𝐶𝑆_𝑥

0,𝑟. If𝑇𝑥 ̸= 𝑥, by the definition of 𝑟 we have

S(𝑇𝑥, 𝑇𝑥, 𝑥) ≥ 𝑟. Hence, using the 𝐹_𝑐𝑆-contractive property and the fact that𝐹 is increasing, we obtain

𝐹 (𝑟) ≤ 𝐹 (S (𝑇𝑥, 𝑇𝑥, 𝑥)) ≤ 𝐹 (S (𝑥, 𝑥, 𝑥0)) − 𝑡

< 𝐹 (S (𝑥, 𝑥, 𝑥₀)) = 𝐹 (𝑟) , (15) which also lead to a contradiction. Therefore,S(𝑇𝑥, 𝑇𝑥, 𝑥) = 0 and that is 𝑇𝑥 = 𝑥. Consequently, 𝐶𝑆_𝑥₀_,𝑟is a fixed circle of𝑇.

Now we show that𝑇 also fixes any circle 𝐶𝑆_𝑥

0,𝜌with𝜌 < 𝑟.

Let𝑥 ∈ 𝐶𝑆_𝑥

0,𝜌 and assume thatS(𝑇𝑥, 𝑇𝑥, 𝑥) > 0. By the 𝐹

𝑆 𝑐

-contractive property, we have

𝐹 (S (𝑇𝑥, 𝑇𝑥, 𝑥)) ≤ 𝐹 (S (𝑥, 𝑥, 𝑥0)) − 𝑡 < 𝐹 (𝜌) . (16)

Since𝐹 is increasing, then we find

S (𝑇𝑥, 𝑇𝑥, 𝑥) < 𝜌 < 𝑟. (17) But𝑟 = min{S(𝑇𝑥, 𝑇𝑥, 𝑥) : for all 𝑇𝑥 ̸= 𝑥}, which leads us to a contradiction. Thus,S(𝑇𝑥, 𝑇𝑥, 𝑥) = 0 and 𝑇𝑥 = 𝑥. Hence,𝐶𝑆_𝑥

0,𝜌is a fixed circle of𝑇.

Remark 12. Notice that, in Theorem 11, the 𝐹_𝑐𝑆-contractive self-mapping𝑇 fixes the disc with the center 𝑥₀and the radius 𝑟. Therefore, the center of any fixed circle is also fixed by 𝑇.

In the following example, we see that the converse statement of Theorem 11 is not always true.

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Example 13. Let(𝑋, S) be an 𝑆-metric space, 𝑥₀ ∈ 𝑋 be any

point, and the self-mapping𝑇 : 𝑋 󳨀→ 𝑋 be defined as 𝑇𝑥 ={_{

{

𝑥 ifS (𝑥, 𝑥, 𝑥₀) ≤ 𝑟

𝑥₀ ifS (𝑥, 𝑥, 𝑥₀) > 𝑟, (18) for all𝑥 ∈ 𝑋 with 𝑟 > 0. Then it can be easily seen that 𝑇 is not an𝐹_𝑐𝑆-contractive self-mapping. Indeed, ifS(𝑥, 𝑥, 𝑥₀) > 𝑟 for 𝑥 ∈ 𝑋, then, using Lemma 3 and the 𝐹𝑆

𝑐-contractive property, we get S (𝑇𝑥, 𝑇𝑥, 𝑥) = S (𝑥0, 𝑥0, 𝑥) > 0 󳨐⇒ 𝑡 + 𝐹 (S (𝑥₀, 𝑥₀, 𝑥)) ≤ 𝐹 (S (𝑥, 𝑥, 𝑥₀)) 󳨐⇒ 𝑡 ≤ 0, (19)

which is a contradiction since𝑡 > 0. Hence 𝑇 is not an 𝐹_𝑐𝑆 -contractive self-mapping. But𝑇 fixes every circle 𝐶S_𝑥

0,𝜌where

𝜌 ≤ 𝑟.

Related to the number of the elements of the set𝑋, the number of the fixed circles of an𝐹_𝑐𝑆-contractive self-mapping 𝑇 can be infinite as seen in the following example.

Example 14. Let𝑋 = {𝑥 ∈ Q : 0 ≤ 𝑥 ≤ 2}, the metric 𝑑 :

𝑋 × 𝑋 󳨀→ [0, ∞) be defined as 𝑑 (𝑥, 𝑦) =󵄨󵄨󵄨󵄨_{󵄨󵄨󵄨󵄨} 󵄨 𝑥 1 + |𝑥|− 𝑦 1 + 󵄨󵄨󵄨󵄨𝑦󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨, (20) for all𝑥, 𝑦 ∈ 𝑋, and the 𝑆-metric be defined as in Example 4. Let us define the self-mapping𝑇 : 𝑋 󳨀→ 𝑋 as

𝑇𝑥 ={_{ { 1

8 if 𝑥 = 0

𝑥 otherwise, (21) for all𝑥 ∈ 𝑋. Then the self-mapping 𝑇 is an 𝐹_𝑐𝑆-contractive self-mapping with𝐹 = ln 𝑥+𝑥, 𝑡 = ln 3, and 𝑥₀= 1/2. Indeed, we get S (𝑇𝑥, 𝑇𝑥, 𝑥) = 2₉ > 0 󳨐⇒ S (𝑇𝑥, 𝑇𝑥, 𝑥) = 2₉ < S (𝑥, 𝑥, 𝑥₀) = 2₃ 󳨐⇒ ln(2 9) < ln (23) 󳨐⇒ ln(2 9) + 2 9 < ln ( 2 3) + 2 3 󳨐⇒ ln3 + ln (2 9) + 29 ≤ ln (23) + 23 󳨐⇒ 𝑡 + 𝐹 (S (𝑇𝑥, 𝑇𝑥, 𝑥)) ≤ 𝐹 (S (𝑥, 𝑥, 𝑥0)) . (22) Using Theorem 11, we have

𝑟 = min {S (𝑇𝑥, 𝑇𝑥, 𝑥) : 𝑇𝑥 ̸= 𝑥} = 2₉. (23) Therefore,𝑇 fixes the circle 𝐶𝑆_1/2,2/9= {2/7, 4/5} and the disc 𝐷𝑆

1/2,2/9 = {𝑥 ∈ 𝑋 : S(𝑥, 𝑥, 1/2) ≤ 2/9}. Evidently, the

number of the fixed circles of𝑇 is infinite.

In the following definition, we introduce the notion of a Hardy-Rogers type𝐹_𝑐𝑆-contraction.

Deﬁnition 15. Let(𝑋, S) be an 𝑆-metric space and 𝑇 be a

self-mapping on𝑋. If there exist 𝐹 ∈ F, 𝑡 > 0, and 𝑥₀ ∈ 𝑋 such that for all𝑥 ∈ 𝑋 the following holds:

S (𝑇𝑥, 𝑇𝑥, 𝑥) > 0 󳨐⇒ 𝑡 + 𝐹 (S (𝑇𝑥, 𝑇𝑥, 𝑥)) ≤ 𝐹 (𝛼S (𝑥, 𝑥, 𝑥0) + 𝛽S (𝑇𝑥, 𝑇𝑥, 𝑥) + 𝛾S (𝑇𝑥0, 𝑇𝑥0, 𝑥0) + 𝛿S (𝑇𝑥₀, 𝑇𝑥₀, 𝑥) + 𝜂S (𝑇𝑥, 𝑇𝑥, 𝑥₀)) , (24) where 𝛼 + 𝛽 + 𝛾 + 𝛿 + 𝜂 = 1, 𝛼, 𝛽, 𝛾, 𝛿, 𝜂 ≥ 0 and𝛼 ̸= 0, (25)

then the self-mapping𝑇 is called a Hardy-Rogers type 𝐹_𝑐𝑆 -contraction on𝑋.

self-mapping𝑇 on 𝑋 is a Hardy-Rogers type 𝐹_𝑐𝑆-contraction with

𝑥₀∈ 𝑋 then we have 𝑇𝑥₀= 𝑥₀.

Proof. Suppose that 𝑇𝑥₀ ̸= 𝑥₀. Using the hypothesis, we

obtain S (𝑇𝑥0, 𝑇𝑥0, 𝑥0) > 0 󳨐⇒ 𝑡 + 𝐹 (S (𝑇𝑥₀, 𝑇𝑥₀, 𝑥₀)) ≤ 𝐹 (𝛼S (𝑥₀, 𝑥₀, 𝑥₀) + 𝛽S (𝑇𝑥₀, 𝑇𝑥₀, 𝑥₀) + 𝛾S (𝑇𝑥₀, 𝑇𝑥₀, 𝑥₀) + 𝛿S (𝑇𝑥₀, 𝑇𝑥₀, 𝑥₀) + 𝜂S (𝑇𝑥₀, 𝑇𝑥₀, 𝑥₀)) = 𝐹 ((𝛽 + 𝛾 + 𝛿 + 𝜂) S (𝑇𝑥₀, 𝑇𝑥₀, 𝑥₀)) < 𝐹 (S (𝑇𝑥₀, 𝑇𝑥₀, 𝑥₀)) , (26)

(5)

which is a contradiction since 𝑡 > 0. Therefore, we get 𝑇𝑥₀= 𝑥₀.

Remark 17. Using Proposition 16, a Hardy-Rogers type𝐹_𝑐𝑆 -contraction condition can be changed as follows:

S (𝑇𝑥, 𝑇𝑥, 𝑥) > 0 󳨐⇒ 𝑡 + 𝐹 (S (𝑇𝑥, 𝑇𝑥, 𝑥)) ≤ 𝐹 (𝛼S (𝑥, 𝑥, 𝑥0) + 𝛽S (𝑇𝑥, 𝑇𝑥, 𝑥) + 𝛿S (𝑇𝑥0, 𝑇𝑥0, 𝑥) + 𝜂S (𝑇𝑥, 𝑇𝑥, 𝑥₀)) , (27) where 𝛼 + 𝛽 + 𝛿 + 𝜂 ≤ 1, 𝛼, 𝛽, 𝛿, 𝜂 ≥ 0 and𝛼 ̸= 0. (28) Now using the Hardy-Rogers type𝐹_𝑐𝑆-contraction condi-tion, we prove the following fixed-circle theorem.

Theorem 18. Let (𝑋, S) be an 𝑆-metric space, 𝑇 be a

Hardy-Rogers type𝐹_𝑐𝑆-contractive self-mapping with𝑥₀∈ 𝑋, and 𝑟 be deﬁned as in Theorem 11. IfS(𝑇𝑥, 𝑇𝑥, 𝑥₀) = 𝑟, then 𝐶𝑆_𝑥

0,𝑟is a

ﬁxed circle of𝑇. 𝑇 especially ﬁxes every circle 𝐶𝑆_𝑥

0,𝜌where𝜌 < 𝑟.

Proof. Let𝑥 ∈ 𝐶𝑆_𝑥

0,𝑟and 𝑇𝑥 ̸= 𝑥. Using the Hardy-Rogers

type𝐹_𝑐𝑆-contraction property, Proposition 16, Lemma 3, and the fact that𝐹 is increasing, we get

𝐹 (𝑟) ≤ 𝐹 (S (𝑇𝑥, 𝑇𝑥, 𝑥)) ≤ 𝐹 (𝛼S (𝑥, 𝑥, 𝑥0) + 𝛽S (𝑇𝑥, 𝑇𝑥, 𝑥) + 𝛿S (𝑇𝑥0, 𝑇𝑥0, 𝑥) + 𝜂S (𝑇𝑥, 𝑇𝑥, 𝑥0)) − 𝑡

< 𝐹 (𝛼S (𝑥, 𝑥, 𝑥₀) + 𝛽S (𝑇𝑥, 𝑇𝑥, 𝑥) + 𝛿S (𝑇𝑥0, 𝑇𝑥0, 𝑥) + 𝜂S (𝑇𝑥, 𝑇𝑥, 𝑥0)) = 𝐹 ((𝛼 + 𝛿 + 𝜂) 𝑟 + 𝛽S (𝑇𝑥, 𝑇𝑥, 𝑥))

≤ 𝐹 ((𝛼 + 𝛽 + 𝛿 + 𝜂) S (𝑇𝑥, 𝑇𝑥, 𝑥)) ≤ 𝐹 (S (𝑇𝑥, 𝑇𝑥, 𝑥)) ,

(29)

which is a contradiction. Hence S(𝑇𝑥, 𝑇𝑥, 𝑥) = 0 and so 𝑇𝑥 = 𝑥. Consequently, 𝐶𝑆

𝑥0,𝑟 is a fixed circle of 𝑇. By the

similar arguments used in the proof of Theorem 11, 𝑇 also fixes any circle𝐶𝑆_𝑥

0,𝜌where𝜌 < 𝑟.

Corollary 19. (1) Let (𝑋, S) be an 𝑆-metric space, 𝑇 be a

Hardy-Rogers type𝐹_𝑐𝑆-contractive self-mapping with𝑥₀ ∈ 𝑋, and𝑟 be deﬁned as in Theorem 11. If S(𝑇𝑥, 𝑇𝑥, 𝑥₀) = 𝑟 for all

𝑥 ∈ 𝐶𝑆

𝑥0,𝑟then𝑇 ﬁxes the disc 𝐷

𝑆 𝑥0,𝑟.

(2) If we consider𝛼 = 1 and 𝛽 = 𝛾 = 𝛿 = 𝜂 = 0 in

Deﬁnition 15, then we obtain the concept of an𝐹_𝑐𝑆-contractive mapping.

In Definition 15, if we get𝛿 = 𝜂 = 0 then we have the following definition.

S (𝑇𝑥, 𝑇𝑥, 𝑥) > 0 󳨐⇒ 𝑡 + 𝐹 (S (𝑇𝑥, 𝑇𝑥, 𝑥)) ≤ 𝐹 (𝛼S (𝑥, 𝑥, 𝑥0) + 𝛽S (𝑇𝑥, 𝑇𝑥, 𝑥) + 𝛾S (𝑇𝑥0, 𝑇𝑥0, 𝑥0)) , (30) where 𝛼 + 𝛽 + 𝛾 < 1 and 𝛼, 𝛽, 𝛾 ≥ 0, (31) then the self-mapping𝑇 is called a Reich type 𝐹_𝑐𝑆-contraction on𝑋.

self-mapping𝑇 on 𝑋 is a Reich type 𝐹_𝑐𝑆-contraction with𝑥₀ ∈ 𝑋 then we get𝑇𝑥₀= 𝑥₀.

Proof. The proof follows easily since𝛽 + 𝛾 < 1.

Remark 22. Using Proposition 21, a Reich type 𝐹_𝑐𝑆

-contraction condition can be changed as follows: S (𝑇𝑥, 𝑇𝑥, 𝑥) > 0 󳨐⇒ 𝑡 + 𝐹 (S (𝑇𝑥, 𝑇𝑥, 𝑥)) ≤ 𝐹 (𝛼S (𝑥, 𝑥, 𝑥₀) + 𝛽S (𝑇𝑥, 𝑇𝑥, 𝑥)) , (32) where 𝛼 + 𝛽 < 1 and𝛼, 𝛽 ≥ 0. (33)

Theorem 23. Let (𝑋, S) be an 𝑆-metric space, 𝑇 be a Reich

type𝐹_𝑐𝑆-contractive self-mapping with𝑥₀∈ 𝑋, and 𝑟 be deﬁned as in Theorem 11. Then𝐶𝑆_𝑥

0,𝑟is a ﬁxed circle of𝑇. Also, 𝑇 ﬁxes

every circle𝐶𝑆_𝑥

0,𝜌where𝜌 < 𝑟. In other words, 𝑇 ﬁxes the disc

𝐷𝑆 𝑥0,𝑟.

Proof. The proof follows easily since

𝐹 (𝑟) ≤ 𝐹 (S (𝑇𝑥, 𝑇𝑥, 𝑥)) ≤ 𝐹 ((𝛼 + 𝛽) S (𝑇𝑥, 𝑇𝑥, 𝑥)) < 𝐹 (S (𝑇𝑥, 𝑇𝑥, 𝑥)) . (34)

(6)

In Definition 15, if we get𝛼 = 𝛽 = 𝛾 = 0 and 𝛿 = 𝜂, then we have the following definition.

S (𝑇𝑥, 𝑇𝑥, 𝑥) > 0 󳨐⇒ 𝑡 + 𝐹 (S (𝑇𝑥, 𝑇𝑥, 𝑥)) ≤ 𝐹 (𝜂 (S (𝑇𝑥₀, 𝑇𝑥₀, 𝑥) + S (𝑇𝑥, 𝑇𝑥, 𝑥₀))) , (35) where 𝜂 ∈ (0, 1 2) , (36)

then the self-mapping 𝑇 is called a Chatterjea type 𝐹_𝑐𝑆 -contraction on𝑋.

self-mapping𝑇 on 𝑋 is a Chatterjea type 𝐹_𝑐𝑆-contraction with𝑥₀∈

𝑋 then we get 𝑇𝑥0= 𝑥0.

Proof. The proof follows easily.

Theorem 26. Let (𝑋, S) be an 𝑆-metric space, 𝑇 be a

Chat-terjea type𝐹_𝑐𝑆-contractive self-mapping with𝑥₀∈ 𝑋, and 𝑟 be deﬁned as in Theorem 11. IfS(𝑇𝑥, 𝑇𝑥, 𝑥₀) = 𝑟 for all 𝑥 ∈ 𝐶𝑆_𝑥

0,𝑟

then𝐶𝑆_𝑥

0,𝑟is a ﬁxed circle of𝑇. Also, 𝑇 ﬁxes every circle 𝐶

𝑆 𝑥0,𝜌

where𝜌 < 𝑟. In other words, 𝑇 ﬁxes the disc 𝐷𝑆_𝑥

0,𝑟.

Proof. The proof follows easily by the similar arguments used

in the proofs of Theorems 11 and 18.

Now we give the following illustrative example.

Example 27. Let C be the set of all complex numbers.

Consider the set

𝑋_𝑧= {0, 4, 𝑧, 𝑧2, 𝑧4, 𝑧8, 𝑧8− 2, 𝑧8+ 2, 𝑧16, 𝑧16− 2, 𝑧16

+ 2} ⊂ C, (37)

where𝑧 is any complex number with |𝑧| = 2 and the 𝑆−metric is defined as in [18] such that

S (𝑥, 𝑦, 𝑡) = |𝑥 − 𝑡| + 󵄨󵄨󵄨󵄨𝑥 + 𝑡 − 2𝑦󵄨󵄨󵄨󵄨, (38) for all𝑥, 𝑦, 𝑡 ∈ 𝑋_𝑧. Let us define the self-mapping𝑇 : 𝑋_𝑧󳨀→ 𝑋_𝑧as

𝑇𝑥 ={_{ {

𝑧 if 𝑥 = 0

𝑥 otherwise, (39) for all𝑥 ∈ 𝑋_𝑧. Then the self-mapping𝑇 is an 𝐹_𝑐𝑆-contractive self-mapping with𝐹 = −1/√𝑥, 𝑡 = 1/28and𝑥₀= 𝑧16. Indeed, we obtain S (𝑇𝑥, 𝑇𝑥, 𝑥) = 4 > 0, (40) for𝑥 = 0, and S (𝑥, 𝑥, 𝑥₀) = 217. (41) Then we have 𝑡 + S (𝑇𝑥, 𝑇𝑥, 𝑥) =₂1₈ −1 2 ≤ − 128_√2. (42) Also we obtain 𝑟 = min {S (𝑇𝑥, 𝑇𝑥, 𝑥) : 𝑇𝑥 ̸= 𝑥} = 4. (43) Therefore, the self-mapping𝑇 fixes the circle 𝐶𝑆_𝑧16_,4 = {𝑧16−

2, 𝑧16_{+ 2} and the disc 𝐷}𝑆

𝑧16_,4= {𝑧16− 2, 𝑧16, 𝑧16+ 2}.

Also the self-mapping 𝑇 is a Hardy-Rogers type 𝐹_𝑐𝑆 -contractive self-mapping(resp., a Reich type 𝐹_𝑐𝑆-contractive self-mapping and a Chatterjea type 𝐹_𝑐𝑆-contractive self-mapping) on 𝑋_𝑧 with 𝛼 = 1, 𝛽 = 𝛿 = 𝜂 = 0 (resp., 𝛼 = (216 _{− 2}14_{+ 2}8_)/217₍₂14 _{− 2}8 _{+ 1), 𝛽 = 1/4 and 𝜂 =}

5/(217_{+ 4(1 − 2}15_))).

4. 𝜓-Contractive and Weakly 𝜓-Contractive

Self-Mappings on

𝑆-Metric Spaces

First, in this section we present this well-known interesting class of functions.

Deﬁnition 28. Denote by Ψ the family of nondecreasing

functions 𝜓 : [0, +∞) 󳨀→ [0, +∞) such that +∞ ∑ 𝑛=1𝜓 𝑛_{(𝑡) < +∞ for each 𝑡 > 0,} (44)

where𝜓𝑛is the𝑛-th iterate of 𝜓.

Lemma 29. For every function 𝜓 : [0, +∞) 󳨀→ [0, +∞) the

following holds: if𝜓 is nondecreasing, then, for each 𝑡 > 0,

lim_{𝑛󳨀→+∞}𝜓𝑛(𝑡) = 0 implies that 𝜓(𝑡) < 𝑡.

Now, we define the𝜓-contractive self-mapping in an 𝑆-metric space.

Deﬁnition 30. Let𝑇 be a self-mapping on an 𝑆-metric space

(𝑋, S). We say that 𝑇 is 𝜓-contractive self-mapping if there exist𝑥₀∈ 𝑋 and 𝜓 ∈ Ψ such that for all 𝑥, 𝑦, 𝑧 ∈ 𝑋 we have

S (𝑇𝑦, 𝑇𝑧, 𝑥) ≤ 𝜓 (S (𝑥, 𝑥, 𝑥₀))

− min {𝜓 (S (𝑇𝑦, 𝑇𝑦, 𝑥₀)) , 𝜓 (S (𝑇𝑧, 𝑇𝑧, 𝑥₀))} . (45)

Theorem 31. Let 𝑇 be a 𝜓-contractive self-mapping with 𝑥₀∈ 𝑋 on an 𝑆-metric space (𝑋, S), and consider the circle 𝐶𝑆

𝑥0,𝑟.

Thus, for every𝑥 ∈ 𝐶𝑆_𝑥

0,𝑟,𝑇 either ﬁxes 𝑥 or maps 𝑥 to the

interior of 𝐶𝑆_𝑥

0,𝑟. Moreover, if for every 𝑥 ∈ 𝐶

𝑆

𝑥0,𝑟 we have

(7)

Proof. If𝑥 ∈ 𝐶𝑆_𝑥

0,𝑟, then since𝑇 is 𝜓-contractive we have

S (𝑇𝑥, 𝑇𝑥, 𝑥) ≤ 𝜓 (S (𝑥, 𝑥, 𝑥0)) − 𝜓 (S (𝑇𝑥, 𝑇𝑥, 𝑥0))

= 𝜓 (𝑟) − 𝜓 (S (𝑇𝑥, 𝑇𝑥, 𝑥0)) .

(46) IfS(𝑇𝑥, 𝑇𝑥, 𝑥₀) < 𝑟, then we are in the case where 𝑇 maps 𝑥 to the interior of𝐶𝑆_𝑥

0,𝑟. If S(𝑇𝑥, 𝑇𝑥, 𝑥0) ≥ 𝑟, then by using the

fact that𝜓 is a nondecreasing function we have

S (𝑇𝑥, 𝑇𝑥, 𝑥) ≤ 𝜓 (𝑟) − 𝜓 (S (𝑇𝑥, 𝑇𝑥, 𝑥0)) . (47)

Now, ifS(𝑇𝑥, 𝑇𝑥, 𝑥₀) > 𝑟, then the above inequality implies thatS(𝑇𝑥, 𝑇𝑥, 𝑥) < 0 which leads to a contradiction. Hence, in this case we must haveS(𝑇𝑥, 𝑇𝑥, 𝑥₀) = 𝑟. Thus,

S (𝑇𝑥, 𝑇𝑥, 𝑥) ≤ 𝜓 (𝑟) − 𝜓 (S (𝑇𝑥, 𝑇𝑥, 𝑥0))

= 𝜓 (𝑟) − 𝜓 (𝑟) = 0, (48) and that is𝑇𝑥 = 𝑥.

Therefore,𝑇 either fixes 𝑥 or maps 𝑥 to the interior of 𝐶𝑆_𝑥

0,𝑟

as required.

To prove the second part of our theorem, we may assume thatS(𝑇𝑥, 𝑇𝑥, 𝑥₀) = 𝑟, for all 𝑥 ∈ 𝐶𝑆_𝑥

0,𝑟. Now, we only need to

show that if there exists𝑥 ∈ 𝑋 where 𝑇𝑥 = 𝑥, then 𝑥 ∈ 𝐶𝑆_𝑥

0,𝑟,

and that will prove the uniqueness. So, first let𝑥 ∈ 𝐶𝑆_𝑥

0,𝑟, and

that is𝑇𝑥 = 𝑥, and also let 𝑦 ∈ 𝑋 be an arbitrary fixed point of𝑇 (i.e., 𝑇𝑦 = 𝑦) we have two cases.

Case 1. IfS(𝑦, 𝑦, 𝑥₀) ≥ 𝑟 then by using the fact that 𝜓 is a

nondecreasing function we have S (𝑦, 𝑦, 𝑥) = S (𝑇𝑦, 𝑇𝑦, 𝑥)

≤ 𝜓 (𝑟) − 𝜓 (S (𝑇𝑦, 𝑇𝑦, 𝑥0)) .

(49) Now, ifS(𝑇𝑦, 𝑇𝑦, 𝑥₀) > 𝑟 then the above inequality implies thatS(𝑦, 𝑦, 𝑥) < 0 which leads to a contradiction. Hence, in this case we must haveS(𝑦, 𝑦, 𝑥₀) = 𝑟.

S (𝑦, 𝑦, 𝑥) = S (𝑇𝑦, 𝑇𝑦, 𝑥)

≤ 𝜓 (S (𝑥, 𝑥, 𝑥0)) − 𝜓 (S (𝑇𝑦, 𝑇𝑦, 𝑥0))

= 𝜓 (𝑟) − 𝜓 (𝑟) = 0,

(50)

and that is𝑥 = 𝑦.

Case 2. IfS(𝑦, 𝑦, 𝑥₀) < 𝑟 then once again by using the fact

that𝜓 is a nondecreasing function we have

S (𝑥, 𝑥, 𝑦) ≤ 𝜓 (S (𝑦, 𝑦, 𝑥₀)) − 𝜓 (S (𝑇𝑥, 𝑇𝑥, 𝑥₀)) = 𝜓 (S (𝑦, 𝑦, 𝑥₀)) − 𝜓 (S (𝑥, 𝑥, 𝑥₀)) = 𝜓 (S (𝑦, 𝑦, 𝑥₀)) − 𝜓 (𝑟) < 𝜓 (𝑟) − 𝜓 (𝑟) = 0,

(51)

which leads us to a contradiction. Therefore, 𝐶𝑆_𝑥

0,𝑟 is the unique fixed circle of𝑇 in 𝑋 as

desired.

Next, we give the definition of a weakly𝜓-contractive self-mapping.

Deﬁnition 32. Let𝑇 be a self-mapping on an 𝑆-metric space

(𝑋, S). We say that 𝑇 is a weakly 𝜓-contractive self-mapping with𝑥₀ ∈ 𝑋 if there exist 𝑥₀∈ 𝑋 and 𝜓 ∈ Ψ such that for all 𝑥, 𝑦, 𝑧 ∈ 𝑋 we have

S (𝑇𝑦, 𝑇2_{𝑧, 𝑥)}

≤ 𝜓 (S (𝑥, 𝑥, 𝑥0))

− min {𝜓 (S (𝑇𝑦, 𝑇𝑦, 𝑥₀)) , 𝜓 (S (𝑇𝑧, 𝑇𝑧, 𝑥₀))} . (52)

Theorem 33. Let 𝑇 be a weakly 𝜓-contractive self-mapping

with𝑥₀ ∈ 𝑋 on an 𝑆-metric space (𝑋, S) and consider the circle𝐶𝑆_𝑥

0,𝑟. Thus, for every 𝑥 ∈ 𝐶

𝑆

𝑥0,𝑟 𝑇 either ﬁxes 𝑥 or maps 𝑥

to the interior of𝐶𝑆_𝑥

0,𝑟. Moreover, if for every 𝑥 ∈ 𝐶

𝑆

𝑥0,𝑟, we have

S(𝑇𝑥, 𝑇𝑥, 𝑥₀) = 𝑟, then 𝐶𝑆

𝑥0,𝑟is a unique ﬁxed circle of𝑇 in 𝑋.

Proof. If𝑥 ∈ 𝐶𝑆_𝑥

0,𝑟, then since𝑇 is weakly 𝜓-contractive we

have

S (𝑇𝑥, 𝑇2𝑥, 𝑥) ≤ 𝜓 (S (𝑥, 𝑥, 𝑥₀)) − 𝜓 (S (𝑇𝑥, 𝑇𝑥, 𝑥₀)) = 𝜓 (𝑟) − 𝜓 (S (𝑇𝑥, 𝑇𝑥, 𝑥0)) .

(53)

IfS(𝑇𝑥, 𝑇𝑥, 𝑥₀) < 𝑟, then we are in the case where 𝑇 maps 𝑥 to the interior of𝐶𝑆_𝑥

0,𝑟. If S(𝑇𝑥, 𝑇𝑥, 𝑥0) ≥ 𝑟, then by using the

fact that𝜓 is a nondecreasing function we have

S (𝑇𝑥, 𝑇2𝑥, 𝑥) ≤ 𝜓 (𝑟) − 𝜓 (S (𝑇𝑥, 𝑇𝑥, 𝑥0)) . (54)

Now, ifS(𝑇𝑥, 𝑇𝑥, 𝑥₀) > 𝑟, then the above inequality implies thatS(𝑇𝑥, 𝑇2𝑥, 𝑥) < 0 which leads to a contradiction. Hence, in this case we must haveS(𝑇𝑥, 𝑇𝑥, 𝑥₀) = 𝑟. Thus,

S (𝑇𝑥, 𝑇2𝑥, 𝑥) ≤ 𝜓 (𝑟) − 𝜓 (S (𝑇𝑥, 𝑇𝑥, 𝑥0))

= 𝜓 (𝑟) − 𝜓 (𝑟) = 0, (55) and that is𝑇𝑥 = 𝑥.

Therefore,𝑇 either fixes 𝑥 or maps 𝑥 to the interior of 𝐶𝑆_𝑥

0,𝑟

as required.

To prove the second part of our theorem, we may assume thatS(𝑇𝑥, 𝑇𝑥, 𝑥₀) = 𝑟, for all 𝑥 ∈ 𝐶𝑆_𝑥

0,𝑟. Now, we only need to

show that if there exists𝑥 ∈ 𝑋, where 𝑇𝑥 = 𝑥, then 𝑥 ∈ 𝐶𝑆_𝑥

0,𝑟,

and that will prove the uniqueness. So, first let𝑥 ∈ 𝐶𝑆_𝑥

0,𝑟, and

that is𝑇𝑥 = 𝑥, and also let 𝑦 ∈ 𝑋 be an arbitrary fixed point (i.e.,𝑇𝑦 = 𝑦) we have two cases.

Case 1. IfS(𝑦, 𝑦, 𝑥₀) ≥ 𝑟 then by using the fact that 𝜓 is a

nondecreasing function we have S (𝑦, 𝑦, 𝑥) = S (𝑇𝑦, 𝑇2𝑦, 𝑥)

≤ 𝜓 (𝑟) − 𝜓 (S (𝑇𝑦, 𝑇𝑦, 𝑥0)) .

(8)

Now, ifS(𝑇𝑦, 𝑇𝑦, 𝑥₀) > 𝑟, then the above inequality implies thatS(𝑦, 𝑦, 𝑥) < 0 which leads to a contradiction. Hence, in this case we must haveS(𝑦, 𝑦, 𝑥₀) = 𝑟.

S (𝑦, 𝑦, 𝑥) = S (𝑇𝑦, 𝑇2𝑦, 𝑥)

≤ 𝜓 (S (𝑥, 𝑥, 𝑥₀)) − 𝜓 (S (𝑇𝑦, 𝑇𝑦, 𝑥₀)) = 𝜓 (𝑟) − 𝜓 (𝑟) = 0,

(57)

and that is𝑥 = 𝑦.

Case 2. IfS(𝑦, 𝑦, 𝑥₀) < 𝑟 then once again by using the fact

that𝜓 is a nondecreasing function we have S (𝑥, 𝑥, 𝑦) = S (𝑇𝑥, 𝑇2_{𝑥, 𝑦)} ≤ 𝜓 (S (𝑦, 𝑦, 𝑥₀)) − 𝜓 (S (𝑇𝑥, 𝑇𝑥, 𝑥₀)) = 𝜓 (S (𝑦, 𝑦, 𝑥₀)) − 𝜓 (S (𝑥, 𝑥, 𝑥₀)) = 𝜓 (S (𝑦, 𝑦, 𝑥₀)) − 𝜓 (𝑟) < 𝜓 (𝑟) − 𝜓 (𝑟) = 0, (58)

which leads us to a contradiction. Therefore, 𝐶𝑆_𝑥

0,𝑟 is the unique fixed circle of𝑇 in 𝑋 as

desired.

5. An Application to Integral Type Contractive

Self-Mappings

We assume that 𝜑 : [0, ∞) 󳨀→ [0, ∞) is a Lebesgue-integrable mapping which is summable (that is, with finite integral) on each compact subset of[0, ∞), nonnegative, and such that, for each𝜀 > 0,

∫𝜀

0 𝜑 (𝑡) 𝑑𝑡 > 0. (59)

Now we give the following definition.

Deﬁnition 34. Let (𝑋, S) be an 𝑆-metric space and 𝜑 :

[0, ∞) 󳨀→ [0, ∞) be defined as in (59). A self-mapping 𝑇 on 𝑋 is said to be an integral type 𝐹𝑆

𝑐-contraction if there exist

𝐹 ∈ F, 𝑡 > 0, and 𝑥₀∈ 𝑋 such that for all 𝑥 ∈ 𝑋 the following holds: S (𝑇𝑥, 𝑇𝑥, 𝑥) > 0 󳨐⇒ 𝑡 + ∫𝐹(S(𝑇𝑥,𝑇𝑥,𝑥)) 0 𝜑 (𝑡) 𝑑𝑡 ≤ ∫ 𝐹(S(𝑥,𝑥,𝑥0)) 0 𝜑 (𝑡) 𝑑𝑡. (60)

Proposition 35. Let (𝑋, S) be an 𝑆-metric space and 𝜑 :

[0, ∞) 󳨀→ [0, ∞) be deﬁned as in (59). If a self-mapping 𝑇

on𝑋 is an integral type 𝐹_𝑐𝑆-contraction with𝑥₀ ∈ 𝑋 then we get𝑇𝑥₀= 𝑥₀.

Proof. Suppose that 𝑇𝑥₀ ̸= 𝑥₀. From the definition of an integral type𝐹_𝑐𝑆-contraction, we have

S (𝑇𝑥₀, 𝑇𝑥₀, 𝑥₀) > 0 󳨐⇒ 𝑡 + ∫𝐹(S(𝑇𝑥0,𝑇𝑥0,𝑥0)) 0 𝜑 (𝑡) 𝑑𝑡 ≤ ∫ 𝐹(S(𝑥0,𝑥0,𝑥0)) 0 𝜑 (𝑡) 𝑑𝑡. (61)

Inequality (61) contradicts with the definition of𝐹 since 𝐹 : (0, ∞) 󳨀→ R and S(𝑥₀, 𝑥₀, 𝑥₀) = 0. Hence, it should be 𝑇𝑥₀= 𝑥₀.

Using this new definition, we get the following fixed-circle result.

Theorem 36. Let (𝑋, S) be an 𝑆-metric space, 𝜑 : [0, ∞) 󳨀→

[0, ∞) be deﬁned as in (59), 𝑇 be an integral type 𝐹_𝑐𝑆 -contraction with𝑥₀ ∈ 𝑋, and 𝑟 be deﬁned as in Theorem 11. Then𝐶𝑆_𝑥

0,𝑟is a ﬁxed circle of𝑇.

Proof. Let𝑥 ∈ 𝐶𝑆_𝑥

0,𝑟. Assume that 𝑇𝑥 ̸= 𝑥. Then, by the

definition of𝑟, we get

𝑟 ≤ S (𝑇𝑥, 𝑇𝑥, 𝑥) . (62) Using the fact that𝐹 is increasing property, we have

𝐹 (𝑟) ≤ 𝐹 (S (𝑇𝑥, 𝑇𝑥, 𝑥)) (63) and ∫𝐹(𝑟) 0 𝜑 (𝑡) 𝑑𝑡 ≤ ∫ 𝐹(S(𝑇𝑥,𝑇𝑥,𝑥)) 0 𝜑 (𝑡) 𝑑𝑡. (64)

From inequality (64) and the definition of integral type𝐹_𝑐𝑆 -contractivity, we obtain ∫𝐹(𝑟) 0 𝜑 (𝑡) 𝑑𝑡 ≤ ∫ 𝐹(S(𝑇𝑥,𝑇𝑥,𝑥)) 0 𝜑 (𝑡) 𝑑𝑡 ≤ ∫𝐹(S(𝑥,𝑥,𝑥0)) 0 𝜑 (𝑡) 𝑑𝑡 − 𝑡 < ∫𝐹(S(𝑥,𝑥,𝑥0)) 0 𝜑 (𝑡) 𝑑𝑡 = ∫ 𝐹(𝑟) 0 𝜑 (𝑡) 𝑑𝑡, (65)

which is a contradiction. Therefore, we find 𝑇𝑥 = 𝑥. Consequently,𝐶𝑆_𝑥

0,𝑟is a fixed circle of𝑇.

Remark 37. (1) An integral type 𝐹_𝑐𝑆-contractive self-mapping 𝑇 fixes also the disc 𝐷𝑆

𝑥0,𝑟.

(2) If we set the function 𝜑 : [0, ∞) 󳨀→ [0, ∞) in Theorem 36 as 𝜑(𝑡) = 1 for all 𝑡 ∈ [0, ∞), then we get Theorem 11.

(3) By the similar argument used in Definition 34, the notions of an integral Hardy-Rogers type𝐹_𝑐𝑆-contractive self-mapping, an integral Reich type𝐹_𝑐𝑆-contractive self-mapping, an integral Chatterjea type𝐹_𝑐𝑆-contractive self-mapping, and obtained corresponding fixed-circle theorems can be defined.

(9)

Finally, we give the following example.

Example 38. Let𝑋 = {𝑒, 2𝑒, 𝑒+ 1/2, 2𝑒− 1/2, 2𝑒+ 1/2} ⊂ R be

the𝑆-metric space with the usual 𝑆-metric and the function 𝜑 : [0, ∞) 󳨀→ [0, ∞) be defined by

𝜑 (𝑡) = 2𝑡 + 1, (66) for all𝑡 ∈ [0, ∞). Let us define the self-mapping 𝑇 : 𝑋 󳨀→ 𝑋 as 𝑇𝑥 ={_{ { 𝑒 + 1 2 if𝑥 = 𝑒 𝑥 otherwise, (67)

for all𝑥 ∈ 𝑋. The self-mapping 𝑇 is an integral type 𝐹_𝑐𝑆 -contractive self-mapping with𝐹 = ln 𝑥, 𝑡 = 1, and 𝑥₀ = 2𝑒. Indeed, we get

S (𝑇𝑥, 𝑇𝑥, 𝑥) = S (𝑒 + 1₂, 𝑒 + 1₂, 𝑒) = 2󵄨󵄨󵄨󵄨_{󵄨󵄨󵄨𝑒 +}1₂ − 𝑒󵄨󵄨󵄨󵄨_󵄨󵄨󵄨 = 1 > 0,

(68)

for𝑥 = 𝑒. Then we have

S (𝑇𝑥, 𝑇𝑥, 𝑥) = 1 < S (𝑥, 𝑥, 𝑥0) = 2𝑒 󳨐⇒ ln1 = 0 < ln (2𝑒) = ln 2 + 1 󳨐⇒ ∫0 0 (2𝑡 + 1) 𝑑𝑡 = 0 < ∫ ln 2+1 0 (2𝑡 + 1) 𝑑𝑡 = ln22 + ln 8 + 2 󳨐⇒ 1 ≤ ln22 + ln 8 + 2. (69) Also we obtain 𝑟 = min {S (𝑇𝑥, 𝑇𝑥, 𝑥) : 𝑇𝑥 ̸= 𝑥} = 1. (70) Consequently,𝑇 fixes the circle 𝐶𝑆_2𝑒,1 = {2𝑒 − 1/2, 2𝑒 + 1/2} and the disc𝐷𝑆_2𝑒,1= {2𝑒 − 1/2, 2𝑒, 2𝑒 + 1/2}.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The first author would like to thank Prince Sultan Univer-sity for funding this work through research group Nonlin-ear Analysis Methods in Applied Mathematics (NAMAM) Group no. RG-DES-2017-01-17.

References

[1] W. Shatanawi and M. B. Hani, “A coupled fixed point theorem in b-metric spaces,” International Journal of Pure and Applied Mathematics, vol. 109, no. 4, pp. 889–897, 2016.

[2] W. Shatanawi, A. Pitea, and R. Lazovi´c, “Contraction conditions using comparison functions on b-metric spaces,” Fixed Point Theory and Applications, vol. 2014, no. 1, article 135, 2014. [3] W. Shatanawi, “Fixed and common fixed point for mappings

satisfying some nonlinear contractions in b-metric spaces,” Journal of Mathematical Analysis, vol. 7, no. 4, pp. 1–12, 2016. [4] H. Monfared, M. Azhini, and M. Asadi, “Fixed point results on

M-metric spaces,” Journal of Mathematical Analysis, vol. 7, no. 5, pp. 85–101, 2016.

[5] A. Felhi, S. Sahmim, and H. Aydi, “Ulam-hyers stability and

well-posedness of fixed point problems for𝛼-𝜆-metric spaces,”

Fixed Point Theory and Applications, vol. 2016, no. 1, 20 pages, 2016.

[6] H. Aydi, M.-F. Bota, E. Karapinar, and S. Mitrovi´c, “A fixed point theorem for set-valued quasi-contractions in b-metric spaces,” Fixed Point Theory and Applications, vol. 2012, article 88, 2012. [7] E. Karapinar, S. Czerwik, and H. Aydi, “(𝛼, Ψ)-Meir-Keeler

Contraction Mappings in Generalized b-Metric Spaces,” Journal of Function Spaces, vol. Volume, Article ID 3264620, p. 4, 2018. [8] H. Aydi, M.-F. Bota, E. Karapinar, and S. Moradi, “A common fixed point for weak f-contractions on b-metric spaces,” Fixed Point Theory and Applications, vol. 13, no. 2, pp. 337–346, 2012. [9] H. Aydi, A. Felhi, and S. Sahmim, “Common fixed points in

rectangular b-metric spaces using (E.A) property,” Journal of Advanced Mathematical Studies, vol. 8, no. 2, pp. 159–169, 2015. [10] H. Aydi, “Alpha-implicit contractive pair of mappings on quasi b-metric spaces and an application to integral equations,” Journal of Nonlinear and Convex Analysis, vol. 17, no. 12, pp. 2417–2433, 2016.

[11] S. Chaipornjareansri, “Fixed point theorems for F_𝑤

-contrac-tions in complete s-metric spaces,” Thai Journal of Mathematics, no. Special issue, pp. 98–109, 2016.

[12] N. Tas, N. Y. ¨Ozg¨ur, and N. Mlaiki, “New fixed-circle results

related to Fc-contractive and Fc -expanding mappings on metric spaces,” Submitted to Mathematical Notes.

[13] N. Tas, N. Y. ¨Ozg¨ur, and N. Mlaiki, “New types of F_𝑐

-con-tractions and the fixed-circle problem,” Mathematics, vol. 6, no. 10, 188 pages, 2018.

[14] S. Sedghi, N. Shobe, and A. Aliouche, “A generalization of fixed point theorems in S-metric spaces,” Matematicki Vesnik, vol. 64, no. 3, pp. 258–266, 2012.

[15] S. Sedghi and N. V. Dung, “Fixed point theorems on S-metric spaces,” Matematicki Vesnik, vol. 66, no. 1, pp. 113–124, 2014. [16] A. Gupta, “Cyclic Contraction on S-Metric Space,” International

Journal of Analysis and Applications, vol. 3, pp. 119–130, 2013. [17] N. T. Hieu, N. T. Thanh Ly, and N. V. Dung, “A generalization

of ciric quasi-contractions for maps on S-metric spaces,” Thai Journal of Mathematics, vol. 13, no. 2, pp. 369–380, 2015.

[18] N. Y. ¨Ozg¨ur and N. Tas, “Some new contractive mappings on

S-metric spaces and their relationships with the mapping (S25),” Mathematical Sciences, vol. 11, no. 1, pp. 7–16, 2017.

[19] N. Y. ¨Ozg¨ur and N. Tas¸, “Some fixed-circle problem on

S-metric spaces with a geoS-metric viewpoint,” https://arxiv.org/ abs/1704.08838.

[20] D. Wardowski, “Fixed points of a new type of contractive mappings in complete metric spaces,” Fixed Point Theory and Applications, vol. 2012, article 94, 2012.

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