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# New Types of Fc-Contractions and the Fixed-Circle Problem

1
Department of Mathematics, Balıkesir University, 10145 Balıkesir, Turkey
2
Department of Mathematical Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia
*
Author to whom correspondence should be addressed.
Mathematics 2018, 6(10), 188; https://doi.org/10.3390/math6100188
Received: 2 September 2018 / Revised: 26 September 2018 / Accepted: 27 September 2018 / Published: 2 October 2018

## Abstract

In this paper we investigate some fixed-circle theorems using Ćirić’s technique (resp. Hardy-Rogers’ technique, Reich’s technique and Chatterjea’s technique) on a metric space. To do this, we define new types of $F c$ -contractions such as Ćirić type, Hardy-Rogers type, Reich type and Chatterjea type. Two illustrative examples are presented to show the effectiveness of our results. Also, it is given an application of a Ćirić type $F c$ -contraction to discontinuous self-mappings which have fixed circles.

## 1. Introduction

Fixed point theory has become the focus of many researchers lately (see [1,2,3,4]). One of the main important results of fixed point theory is when we show that a self mapping on a metric space under some specific conditions has a unique fixed point. In some cases when we do not have uniqueness of the fixed point, such a map fixes a circle which we call a fixed circle, the fixed-circle problem arises naturally in practice. There exist a lot of examples of self-mappings that map a circle onto itself and fixes all the points of the circle, whereas the circle is not fixed by the self-mapping. For example, let $C , d$ be the usual metric space and $C 0 , 1$ be the unit circle. Let us consider the self-mappings $T 1 : C → C$ and $T 2 : C → C$ defined by
$T 1 z = 1 z ¯ if z ≠ 0 0 if z = 0$
and
$T 2 z = 1 z if z ≠ 0 0 if z = 0 ,$
for all $z ∈ C$ where $z ¯$ is the complex conjugate of the complex number z. Then, we have $T i ( C 0 , 1 ) = C 0 , 1$ ($i = 1 , 2$), but $C 0 , 1$ is the fixed circle of $T 1$ while it is not the fixed circle of $T 2$ (especially $T 2$ fixes only two points of the unit circle). Thus, a natural question arises as follows:
What is (are) the necessary and sufficient condition(s) for a self-mapping T that make a given circle as the fixed circle of T? Therefore, it is important to investigate new fixed-circle results.
Various fixed-circle theorems have been obtained using different approaches on metric and some generalized metric spaces (see [5,6,7,8,9] for more details). For example, in , fixed-circle results were proved using the Caristi’s inequality on metric spaces. In , it was given a fixed-circle theorem for a self-mapping that maps a given circle onto itself. In , it was extended known fixed-circle results in many directions and introduced a new notion called as an $F c$-contraction. In addition, some generalized fixed-circle theorems were investigated on an S-metric space (see [6,7]).
Motivated by the above studies, we present some new fixed-circle theorems using the ideas given in [10,11]. In , it was proved some fixed-point results using an F-contraction of the Hardy-Rogers-type and in , it was obtained a fixed-point theorem using a Ćirić type generalized F-contraction. We generate some fixed-circle results from these types of contractions using Wardowski’s technique. For some fixed-point results obtained by this technique, one can consult the references [10,11,12,13]. In Section 2, we define the notions of a Ćirić type $F c$-contraction, Hardy-Rogers type $F c$-contraction, Reich type $F c$-contraction and Chatterjea type $F c$-contraction. Using these concepts, we prove some results related to the fixed-circle problem. In Section 3, we present an application of our obtained results to a discontinuous self-mapping that has a fixed circle.

## 2. New Fixed-Circle Results via Some Classical Techniques

Let $( X , d )$ be a metric space and $T : X → X$ be a self-mapping in the whole paper. Now we investigate some new fixed-circle theorems using the ideas of some classical fixed-point theorems.
At first, we recall some necessary definitions and a theorem related to fixed circle. A circle and a disc are defined on a metric space as follows, respectively:
$C u 0 , r = u ∈ X : d ( u , u 0 ) = r$
and
$D u 0 , r = u ∈ X : d ( u , u 0 ) ≤ r .$
Definition 1
().Let $C u 0 , r$ be a circle on X. If $T u = u$ for every $u ∈ C u 0 , r$ then the circle $C u 0 , r$ is said to be a fixed circle of T.
Definition 2
().Let $F$ be the family of all functions $F : ( 0 , ∞ ) → R$ such that
• $( F 1 )$ F is strictly increasing,
• $( F 2 )$ For each sequence $α n$ in $0 , ∞$ the following holds
$lim n → ∞ α n = 0 if and only if lim n → ∞ F ( α n ) = − ∞ ,$
• $( F 3 )$ There exists $k ∈ ( 0 , 1 )$ such that $lim α → 0 + α k F ( α ) = 0$.
Definition 3
().If there exist $t > 0$, $F ∈ F$ and $u 0 ∈ X$ such that for all $u ∈ X$ the following holds:
$d ( u , T u ) > 0 ⇒ t + F ( d ( u , T u ) ) ≤ F ( d ( u 0 , u ) ) ,$
then T is said to be an $F c$-contraction on X.
Theorem 1
().Let T be an $F c$-contractive self-mapping with $u 0 ∈ X$ and
$r = min d ( u , T u ) : u ≠ T u .$
Then $C u 0 , r$ is a fixed circle of T. Especially, T fixes every circle $C u 0 , ρ$ where $ρ < r$.
Now we define new contractive conditions and give some fixed-circle results.
Definition 4.
If there exist $t > 0$, $F ∈ F$ and $u 0 ∈ X$ such that for all $u ∈ X$ the following holds:
$d ( u , T u ) > 0 ⟹ t + F ( d ( u , T u ) ) ≤ F ( m ( u , u 0 ) ) ,$
where
$m ( u , v ) = max d ( u , v ) , d ( u , T u ) , d ( v , T v ) , 1 2 d ( u , T v ) + d ( v , T u ) ,$
then T is said to be a Ćirić type $F c$-contraction on X.
Proposition 1.
If T is a Ćirić type $F c$-contraction with $u 0 ∈ X$ then we have $T u 0 = u 0$.
Proof.
Assume that $T u 0 ≠ u 0$. From the definition of a Ćirić type $F c$-contraction, we get
$d ( u 0 , T u 0 ) > 0 ⟹ t + F ( d ( u 0 , T u 0 ) ) ≤ F ( m ( u 0 , u 0 ) ) = F max d ( u 0 , u 0 ) , d ( u 0 , T u 0 ) , d ( u 0 , T u 0 ) , 1 2 d ( u 0 , T u 0 ) + d ( u 0 , T u 0 ) = F ( d ( u 0 , T u 0 ) ) ,$
a contradiction because of $t > 0$. Then we have $T u 0 = u 0$.☐
Theorem 2.
Let T be a Ćirić type $F c$-contraction with $u 0 ∈ X$ and r be defined as in (1). If $d ( u 0 , T u ) = r$ for all $u ∈ C u 0 , r$ then $C u 0 , r$ is a fixed circle of T. Especially, T fixes every circle $C u 0 , ρ$ with $ρ < r$.
Proof.
Let $u ∈ C u 0 , r$. Since $d ( u 0 , T u ) = r$, the self-mapping T maps $C u 0 , r$ into (or onto) itself. If $T u ≠ u$, by the definition of r, we have $d ( u , T u ) ≥ r$. So using the Ćirić type $F c$-contractive property, Proposition 1 and the fact that F is increasing, we get
$F ( r ) ≤ F ( d ( u , T u ) ) ≤ F ( m ( u , u 0 ) ) − t < F ( m ( u , u 0 ) ) = F max d ( u , u 0 ) , d ( u , T u ) , d ( u 0 , T u 0 ) , 1 2 d ( u , T u 0 ) + d ( u 0 , T u ) = F max r , d ( u , T u ) , 0 , r = F ( d ( u , T u ) ) ,$
a contradiction. Therefore, $d ( u , T u ) = 0$ and so $T u = u$. Consequently, $C u 0 , r$ is a fixed circle of T.
Now we show that T also fixes any circle $C u 0 , ρ$ with $ρ < r$. Let $u ∈ C u 0 , ρ$ and assume that $d ( u , T u ) > 0$. By the Ćirić type $F c$-contractive property, we have
$F ( d ( u , T u ) ) ≤ F ( m ( u , u 0 ) ) − t < F ( m ( u , u 0 ) ) = F ( d ( u , T u ) ) ,$
a contradiction. Thus we obtain $d ( u , T u ) = 0$ and $T u = u$. So, $C u 0 , ρ$ is a fixed circle of T.☐
Corollary 1.
Let T be a Ćirić type $F c$-contractive self-mapping with $u 0 ∈ X$ and r be defined as in (1). If $d ( u 0 , T u ) = r$ for all $u ∈ C u 0 , r$ then T fixes the disc $D u 0 , r$.
Definition 5.
If there exist $t > 0$, $F ∈ F$ and $u 0 ∈ X$ such that for all $u ∈ X$ the following holds:
$d ( u , T u ) > 0 ⟹ t + F ( d ( u , T u ) ) ≤ F α d ( u , u 0 ) + β d ( u , T u ) + γ d ( u 0 , T u 0 ) + δ d ( u , T u 0 ) + η d ( u 0 , T u ) ,$
where
$α + β + γ + δ + η = 1 , α , β , γ , δ , η ≥ 0 and α ≠ 0 ,$
then T is said to be a Hardy-Rogers type $F c$-contraction on X.
Proposition 2.
If T is a Hardy-Rogers type $F c$-contraction with $u 0 ∈ X$ then we have $T u 0 = u 0$.
Proof.
Assume that $T u 0 ≠ u 0$. From the definition of a Hardy-Rogers type $F c$-contraction, we get
$d ( u 0 , T u 0 ) > 0 ⟹ t + F ( d ( u 0 , T u 0 ) ) ≤ F α d ( u 0 , u 0 ) + β d ( u 0 , T u 0 ) + γ d ( u 0 , T u 0 ) + δ d ( u 0 , T u 0 ) + η d ( u 0 , T u 0 ) = F β + γ + δ + η d ( u 0 , T u 0 ) < F ( d ( u 0 , T u 0 ) ) ,$
a contradiction because of $t > 0$. Then we have $T u 0 = u 0$.☐
Using Proposition 2, we rewrite the condition (3) as follows:
$d ( u , T u ) > 0 ⟹ t + F ( d ( u , T u ) ) ≤ F α d ( u , u 0 ) + β d ( u , T u ) + δ d ( u , T u 0 ) + η d ( u 0 , T u ) ,$
where
$α + β + δ + η ≤ 1 , α , β , δ , η ≥ 0 and α ≠ 0 .$
Using this inequality, we obtain the following fixed-circle result.
Theorem 3.
Let T be a Hardy-Rogers type $F c$-contraction with $u 0 ∈ X$ and r be defined as in (1). If $d ( u 0 , T u ) = r$ for all $u ∈ C u 0 , r$ then $C u 0 , r$ is a fixed circle of T. Especially, T fixes every circle $C u 0 , ρ$ with $ρ < r$.
Proof.
Let $u ∈ C u 0 , r$. Using the Hardy-Rogers type $F c$-contractive property, Proposition 2 and the fact that F is increasing, we get
$F ( r ) ≤ F ( d ( u , T u ) ) ≤ F α d ( u , u 0 ) + β d ( u , T u ) + δ d ( u , T u 0 ) + η d ( u 0 , T u ) − t < F ( α r + β d ( u , T u ) + δ r + η r ) ≤ F ( ( α + β + δ + η ) d ( u , T u ) ) ≤ F ( d ( u , T u ) ) ,$
a contradiction. Therefore, $d ( u , T u ) = 0$ and so $T u = u$. Consequently, $C u 0 , r$ is a fixed circle of T. By the similar arguments used in the proof of Theorem 2, T also fixes any circle $C u 0 , ρ$ with $ρ < r$.☐
Corollary 2.
Let T be a Hardy-Rogers type $F c$-contractive self-mapping with $u 0 ∈ X$ and r be defined as in (1). If $d ( u 0 , T u ) = r$ for all $u ∈ C u 0 , r$ then T fixes the disc $D u 0 , r$.
Remark 1.
If we consider $α = 1$ and $β = γ = δ = η = 0$ in Definition 5, then we get the notion of an $F c$-contractive mapping.
In Definition 5, if we choose $δ = η = 0$, then we obtain the following definition.
Definition 6.
If there exist $t > 0$, $F ∈ F$ and $u 0 ∈ X$ such that for all $u ∈ X$ the following holds:
$d ( u , T u ) > 0 ⟹ t + F ( d ( u , T u ) ) ≤ F α d ( u , u 0 ) + β d ( u , T u ) + γ d ( u 0 , T u 0 ) ,$
where
$α + β + γ < 1 and α , β , γ ≥ 0 ,$
then T is said to be a Reich type $F c$-contraction on X.
Proposition 3.
If a self-mapping T on X is a Reich type $F c$-contraction with $u 0 ∈ X$ then we have $T u 0 = u 0$.
Proof.
From the similar arguments used in the proof of Proposition 2, the proof follows easily since $β + γ < 1$.☐
Using Proposition 3, we rewrite the condition (4) as follows:
$d ( u , T u ) > 0 ⟹ t + F ( d ( u , T u ) ) ≤ F α d ( u , u 0 ) + β d ( u , T u ) ,$
where
$α + β < 1 and α , β ≥ 0 .$
Using this inequality, we obtain the following fixed-circle result.
Theorem 4.
Let T be a Reich type $F c$-contraction with $u 0 ∈ X$ and r be defined as in (1). Then $C u 0 , r$ is a fixed circle of T. Especially, T fixes every circle $C u 0 , ρ$ with $ρ < r$.
Proof.
It can be easily seen since
$F ( r ) ≤ F ( d ( u , T u ) ) ≤ F ( ( α + β ) d ( u , T u ) ) < F ( d ( u , T u ) ) .$
Corollary 3.
Let T be a Reich type $F c$-contractive self-mapping with $u 0 ∈ X$ and r be defined as in (1). Then T fixes the disc $D u 0 , r$.
In Definition 5, if we choose $α = β = γ = 0$ and $δ = η$, then we obtain the following definition.
Definition 7.
If there exist $t > 0$, $F ∈ F$ and $u 0 ∈ X$ such that for all $u ∈ X$ the following holds:
$d ( u , T u ) > 0 ⟹ t + F ( d ( u , T u ) ) ≤ F η ( d ( u , T u 0 ) + d ( u 0 , T u ) ) ,$
where
$η ∈ 0 , 1 2 ,$
then T is said to be a Chatterjea type $F c$-contraction on X.
Proposition 4.
If a self-mapping T on X is a Chatterjea type $F c$-contraction with $u 0 ∈ X$ then we have $T u 0 = u 0$.
Proof.
From the similar arguments used in the proof of Proposition 2, it can be easily proved.☐
Theorem 5.
Let T be a Chatterjea type $F c$-contraction with $u 0 ∈ X$ and r be defined as in (1). If $d ( u 0 , T u ) = r$ for all $u ∈ C u 0 , r$ then $C u 0 , r$ is a fixed circle of T. Especially, T fixes every circle $C u 0 , ρ$ with $ρ < r$.
Proof.
By the similar arguments used in the proof of Theorem 3 and Definition 7, it can be easily checked.☐
Corollary 4.
Let T be a Chatterjea type $F c$-contractive self-mapping with $u 0 ∈ X$ and r be defined as in (1). If $d ( u 0 , T u ) = r$ for all $u ∈ C u 0 , r$ then T fixes the disc $D u 0 , r$.
Now we give two illustrative examples of our obtained results.
Example 1.
Let $X = 1 , 2 , e 3 − 1 , e 3 , e 3 + 1$ be the metric space with the usual metric. Let us define the self-mapping $T : X → X$ as
$T u = 2 if u = 1 u otherwise ,$
for all $u ∈ X$.
The Ćirić type $F c$-contractive self-mapping T: The self-mapping T is a Ćirić type $F c$-contractive self-mapping with $F = ln u$, $t = ln ( e 3 − 1 )$ and $u 0 = e 3$. Indeed, we get
$d ( u , T u ) = d ( 1 , T 1 ) = d ( 1 , 2 ) = 1 > 0$
for $u = 1$ and
$m ( u , u 0 ) = m ( 1 , e 3 ) = max d ( 1 , e 3 ) , d ( 1 , 2 ) , 1 2 d ( 1 , e 3 ) + d ( e 3 , 2 ) = max e 3 − 1 , 1 , e 3 − 3 2 = e 3 − 1 .$
Then, we have
$t + F ( d ( u , T u ) ) = ln ( e 3 − 1 ) + ln ( d ( 1 , 2 ) ) = ln ( e 3 − 1 ) ≤ ln ( d ( m ( u , u 0 ) ) ) = ln ( e 3 − 1 ) .$
The Hardy-Rogers type $F c$-contractive self-mapping T: The self-mapping T is a Hardy-Rogers type $F c$-contractive self-mapping with $F = ln u$, $t = ln ( e 3 ) − ln 3$, $α = β = 1 3$, $δ = η = 0$ and $u 0 = e 3$. Indeed, we get
$d ( u , T u ) = d ( 1 , T 1 ) = d ( 1 , 2 ) = 1 > 0$
for $u = 1$ and
$α d ( u , u 0 ) + β d ( u , T u ) + δ d ( u , T u 0 ) + η d ( u 0 , T u ) = 1 3 d ( 1 , e 3 ) + d ( 1 , 2 ) = 1 3 e 3 − 1 + 1 = e 3 3 .$
Then, we have
$t + F ( d ( u , T u ) ) = ln ( e 3 ) − ln 3 + ln ( d ( 1 , 2 ) ) = ln ( e 3 ) − ln 3 ≤ ln ( d ( α d ( u , u 0 ) + β d ( u , T u ) + δ d ( u , T u 0 ) + η d ( u 0 , T u ) ) ) = ln ( e 3 ) − ln 3 .$
The Reich type $F c$-contractive self-mapping T: The self-mapping T is a Reich type $F c$-contractive self-mapping with $F = ln u$, $t = ln ( e 3 ) − ln 4$, $α = β = 1 4$ and $u 0 = e 3$. Indeed, we get
$d ( u , T u ) = d ( 1 , T 1 ) = d ( 1 , 2 ) = 1 > 0$
for $u = 1$ and
$α d ( u , u 0 ) + β d ( u , T u ) = 1 4 d ( 1 , e 3 ) + d ( 1 , 2 ) = 1 4 e 3 − 1 + 1 = e 3 4 .$
Then, we have
$t + F ( d ( u , T u ) ) = ln ( e 3 ) − ln 4 + ln ( d ( 1 , 2 ) ) = ln ( e 3 ) − ln 4 ≤ ln ( d ( α d ( u , u 0 ) + β d ( u , T u ) ) ) = ln ( e 3 ) − ln 4 .$
The Chatterjea type $F c$-contractive self-mapping T: The self-mapping T is a Chatterjea type $F c$-contractive self-mapping with $F = ln u$, $t = ln 2 3 e 3 − 1$, $η = 1 3$ and $u 0 = e 3$. Indeed, we get
$d ( u , T u ) = d ( 1 , T 1 ) = d ( 1 , 2 ) = 1 > 0$
for $u = 1$ and
$η ( d ( u , T u 0 ) + d ( u 0 , T u ) ) = 1 3 d ( 1 , e 3 ) + d ( e 3 , 2 ) = 1 3 e 3 − 1 + e 3 − 2 = 2 e 3 3 − 1 .$
Then, we have
$t + F ( d ( u , T u ) ) = ln 2 3 e 3 − 1 + ln ( d ( 1 , 2 ) ) = ln 2 3 e 3 − 1 ≤ ln ( η ( d ( u , T u 0 ) + d ( u 0 , T u ) ) ) = ln 2 3 e 3 − 1 .$
Also, we obtain
$r = min d ( u , T u ) : u ≠ T u = d ( 1 , 2 ) = 1 .$
Consequently, T fixes the circle $C e 3 , 1 = e 3 − 1 , e 3 + 1$ and the disc $D e 3 , 1 = e 3 − 1 , e 3 , e 3 + 1$.
In the following example, we see that the converse statements of Theorems 2–5 are not always true.
Example 2.
Let $x 0 ∈ X$ be any point and the self-mapping $T : X → X$ be defined as
$T u = u if u ∈ D u 0 , μ u 0 if u ∉ D u 0 , μ ,$
for all $u ∈ X$ with $μ > 0$. Then T is not a Ćirić type $F c$-contractive self-mapping (resp. Hardy-Rogers type $F c$-contractive self-mapping, Reich type $F c$-contractive self-mapping and Chatterjea type $F c$-contractive self-mapping). But T fixes every circle $C x 0 , ρ$ where $ρ ≤ μ$.

## 3. An Application to Discontinuity Problem

In this section, we give some examples of discontinuous functions and obtain a discontinuity result related to fixed circle.
Example 3.
Let $X = 1 , 2 , e 3 − 1 , e 3 , e 3 + 1$ be the metric space with the usual metric. Let us define the self-mapping $T : X → X$ as
$T u = 2 if u < e 3 − 1 u if u ≥ e 3 − 1 ,$
for all $u ∈ X$. As in Example 1, it is easily verified that the self-mapping T is a Ćirić type $F c$-contractive self-mapping and $C e 3 , 1 = e 3 − 1 , e 3 + 1$ is a fixed circle of T. We note that the self-mapping T is continuous at the point $e 3 + 1$ while the self-mapping T is discontinuous at the point $e 3 − 1$.
Example 4.
Let $X = 1 , 2 , e 3 − 1 , e 3 , e 3 + 1$ be the metric space with the usual metric. Let us define the self-mapping $T : X → X$ as
$T u = 2 if u < e 3 − 1 e 3 − 1 if e 3 − 1 ≤ u < e 3 u if e 3 ≤ u ≤ e 3 + 1 u − 1 if u > e 3 + 1 ,$
for all $u ∈ X$. As in Example 1, it is easily checked that the self-mapping T is a Ćirić type $F c$-contractive self-mapping and $C e 3 , 1 = e 3 − 1 , e 3 + 1$ is a fixed circle of T. We note that the self-mapping T is discontinuous at the center $e 3$ and on the circle $C e 3 , 1$.
Consider the above examples, we give the following theorem.
Theorem 6.
Let T be a Ćirić type $F c$-contraction with $u 0 ∈ X$ and r be defined as in (1). If $d ( u 0 , T u ) = r$ for all $u ∈ C u 0 , r$ then $C u 0 , r$ is a fixed circle of T. Also T is discontinuous at $u ∈ C u 0 , r$ if and only if $lim v → u m ( u , v ) ≠ 0$.
Proof.
From Theorem 2, we see that $C u 0 , r$ is a fixed circle of T. Used the idea given in Theorem 2.1 on page 1240 in , we see that T is discontinuous at $u ∈ C u 0 , r$ if and only if $lim v → u m ( u , v ) ≠ 0$.☐

## 4. Conclusions

We have presented new generalized fixed-circle results using new types of contractive conditions on metric spaces. The obtained results can be also considered as fixed-disc results. By means of some known techniques which are used to obtain some fixed-point results, we have generated useful fixed-circle theorems. As we have seen in the last section, our main results can be applied to other research areas.

## Funding

This research received no external funding.

## Acknowledgments

The third author would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.

## Conflicts of Interest

The authors declare no conflicts of interest.

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