New Types of F_c-Contractions and the Fixed-Circle Problem

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 $\left(\mathbb{C},d\right)$ be the usual metric space and $C_{0,1}$ be the unit circle. Let us consider the self-mappings $T_1:\mathbb{C}\to \mathbb{C}$ and $T_2:\mathbb{C}\to \mathbb{C}$ defined by

$$T_{1}z=\left\{\begin{array}{ccc}\frac{1}{\overline{z}}& \mathrm{if}& z\ne 0\\ 0& \mathrm{if}& z=0\end{array}\right.$$

and

$$T_{2}z=\left\{\begin{array}{ccc}\frac{1}{z}& \mathrm{if}& z\ne 0\\ 0& \mathrm{if}& z=0\end{array}\right.,$$

for all $z\in \mathbb{C}$ where $\overline{z}$ is the complex conjugate of the complex number z. Then, we have $T_i\left(C_{0,1}\right)=C_{0,1}$ ($i=1,2$), 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 [5], fixed-circle results were proved using the Caristi’s inequality on metric spaces. In [8], it was given a fixed-circle theorem for a self-mapping that maps a given circle onto itself. In [9], 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 [10], it was proved some fixed-point results using an F-contraction of the Hardy-Rogers-type and in [11], 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 $T:X\to X$ 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:

$$C_{u_0,r}=\left\{u\in X:d(u,u_0)=r\right\}$$

and

$$D_{u_0,r}=\left\{u\in X:d(u,u_0)\le r\right\}.$$

Definition 1

([5]).Let $C_{u_0,r}$ be a circle on X. If $Tu=u$ for every $u\in C_{u_0,r}$ then the circle $C_{u_0,r}$ is said to be a fixed circle of T.

Definition 2

([13]).Let $\mathbb{F}$ be the family of all functions $F:(0,\infty )\to \mathbb{R}$ such that

• $\left(F_1\right)$ F is strictly increasing,
• $\left(F_2\right)$ For each sequence $\left\{{\alpha }_n\right\}$ in $\left(0,\infty \right)$ the following holds

  $$\underset{n\to \infty }{lim}{\alpha }_n=0\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}\underset{n\to \infty }{lim}F\left({\alpha }_n\right)=-\infty ,$$
• $\left(F_3\right)$ There exists $k\in (0,1)$ such that $\underset{\alpha \to 0^{+}}{lim}{\alpha }^{k}F\left(\alpha \right)=0$.

Definition 3

([9]).If there exist $t>0$, $F\in \mathbb{F}$ and $u_0\in X$ such that for all $u\in X$ the following holds:

$$d(u,Tu)>0\Rightarrow t+F\left(d(u,Tu)\right)\le F\left(d(u_0,u)\right),$$

then T is said to be an $F_c$-contraction on X.

Theorem 1

([9]).Let T be an $F_c$-contractive self-mapping with $u_0\in X$ and

$$r=min\left\{d(u,Tu):u\ne Tu\right\}.$$

(1)

Then $C_{u_0,r}$ is a fixed circle of T. Especially, T fixes every circle $C_{u_0,\rho }$ where $\rho <r$.

Now we define new contractive conditions and give some fixed-circle results.

Definition 4.

If there exist $t>0$, $F\in \mathbb{F}$ and $u_0\in X$ such that for all $u\in X$ the following holds:

$$d(u,Tu)>0⟹t+F\left(d(u,Tu)\right)\le F\left(m(u,u_0)\right),$$

(2)

where

$$m(u,v)=max\left\{d(u,v),d(u,Tu),d(v,Tv),\frac{1}{2}\left[d(u,Tv)+d(v,Tu)\right]\right\},$$

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 $u_0\in X$ then we have $T{u}_0=u_0$.

Proof. 

Assume that $T{u}_0\ne u_0$. From the definition of a Ćirić type $F_c$-contraction, we get

$$\begin{array}{ccc}\hfill d(u_0,T{u}_0)& >& 0⟹t+F\left(d(u_0,T{u}_0)\right)\le F\left(m(u_0,u_0)\right)\hfill \\ & =& F\left(max\left\{\begin{array}{c}d(u_0,u_0),d(u_0,T{u}_0),d(u_0,T{u}_0),\\ \frac{1}{2}\left[d(u_0,T{u}_0)+d(u_0,T{u}_0)\right]\end{array}\right\}\right)\hfill \\ & =& F\left(d(u_0,T{u}_0)\right),\hfill \end{array}$$

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 $u_0\in X$ and r be defined as in (1). If $d(u_0,Tu)=r$ for all $u\in C_{u_0,r}$ then $C_{u_0,r}$ is a fixed circle of T. Especially, T fixes every circle $C_{u_0,\rho }$ with $\rho <r$.

Proof. 

Let $u\in C_{u_0,r}$. Since $d(u_0,Tu)=r$, the self-mapping T maps $C_{u_0,r}$ into (or onto) itself. If $Tu\ne u$, by the definition of r, we have $d(u,Tu)\ge r$. So using the Ćirić type $F_c$-contractive property, Proposition 1 and the fact that F is increasing, we get

$$\begin{array}{ccc}\hfill F\left(r\right)& \le & F\left(d(u,Tu)\right)\le F\left(m(u,u_0)\right)-t<F\left(m(u,u_0)\right)\hfill \\ & =& F\left(max\left\{d(u,u_0),d(u,Tu),d(u_0,T{u}_0),\frac{1}{2}\left[d(u,T{u}_0)+d(u_0,Tu)\right]\right\}\right)\hfill \\ & =& F\left(max\left\{r,d(u,Tu),0,r\right\}\right)=F\left(d(u,Tu)\right),\hfill \end{array}$$

a contradiction. Therefore, $d(u,Tu)=0$ and so $Tu=u$. Consequently, $C_{u_0,r}$ is a fixed circle of T.

Now we show that T also fixes any circle $C_{u_0,\rho }$ with $\rho <r$. Let $u\in C_{u_0,\rho }$ and assume that $d(u,Tu)>0$. By the Ćirić type $F_c$-contractive property, we have

$$F\left(d(u,Tu)\right)\le F\left(m(u,u_0)\right)-t<F\left(m(u,u_0)\right)=F\left(d(u,Tu)\right),$$

a contradiction. Thus we obtain $d(u,Tu)=0$ and $Tu=u$. So, $C_{u_0,\rho }$ is a fixed circle of T.☐

Corollary 1.

Let T be a Ćirić type $F_c$-contractive self-mapping with $u_0\in X$ and r be defined as in (1). If $d(u_0,Tu)=r$ for all $u\in C_{u_0,r}$ then T fixes the disc $D_{u_0,r}$.

Definition 5.

If there exist $t>0$, $F\in \mathbb{F}$ and $u_0\in X$ such that for all $u\in X$ the following holds:

$$d(u,Tu)>0⟹t+F\left(d(u,Tu)\right)\le F\left(\begin{array}{c}\alpha d(u,u_0)+\beta d(u,Tu)+\gamma d(u_0,T{u}_0)\\ +\delta d(u,T{u}_0)+\eta d(u_0,Tu)\end{array}\right),$$

(3)

where

$$\alpha +\beta +\gamma +\delta +\eta =1,\alpha ,\beta ,\gamma ,\delta ,\eta \ge 0\mathrm{and}\alpha \ne 0,$$

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 $u_0\in X$ then we have $T{u}_0=u_0$.

Proof. 

Assume that $T{u}_0\ne u_0$. From the definition of a Hardy-Rogers type $F_c$-contraction, we get

$$\begin{array}{ccc}\hfill d(u_0,T{u}_0)& >& 0⟹t+F\left(d(u_0,T{u}_0)\right)\hfill \\ & \le & F\left(\begin{array}{c}\alpha d(u_0,u_0)+\beta d(u_0,T{u}_0)+\gamma d(u_0,T{u}_0)\\ +\delta d(u_0,T{u}_0)+\eta d(u_0,T{u}_0)\end{array}\right)\hfill \\ & =& F\left(\left(\beta +\gamma +\delta +\eta \right)d(u_0,T{u}_0)\right)\hfill \\ & <& F\left(d(u_0,T{u}_0)\right),\hfill \end{array}$$

a contradiction because of $t>0$. Then we have $T{u}_0=u_0$.☐

Using Proposition 2, we rewrite the condition (3) as follows:

$$d(u,Tu)>0⟹t+F\left(d(u,Tu)\right)\le F\left(\begin{array}{c}\alpha d(u,u_0)+\beta d(u,Tu)\\ +\delta d(u,T{u}_0)+\eta d(u_0,Tu)\end{array}\right),$$

where

$$\alpha +\beta +\delta +\eta \le 1,\alpha ,\beta ,\delta ,\eta \ge 0\mathrm{and}\alpha \ne 0.$$

Using this inequality, we obtain the following fixed-circle result.

Theorem 3.

Let T be a Hardy-Rogers type $F_c$-contraction with $u_0\in X$ and r be defined as in (1). If $d(u_0,Tu)=r$ for all $u\in C_{u_0,r}$ then $C_{u_0,r}$ is a fixed circle of T. Especially, T fixes every circle $C_{u_0,\rho }$ with $\rho <r$.

Proof. 

Let $u\in C_{u_0,r}$. Using the Hardy-Rogers type $F_c$-contractive property, Proposition 2 and the fact that F is increasing, we get

$$\begin{array}{ccc}\hfill F\left(r\right)& \le & F\left(d\right(u,Tu\left)\right)\hfill \\ & \le & F\left(\alpha d(u,u_0)+\beta d(u,Tu)+\delta d(u,T{u}_0)+\eta d(u_0,Tu)\right)-t\hfill \\ & <& F(\alpha r+\beta d(u,Tu)+\delta r+\eta r)\hfill \\ & \le & F\left(\right(\alpha +\beta +\delta +\eta \left)d\right(u,Tu\left)\right)\le F\left(d\right(u,Tu\left)\right),\hfill \end{array}$$

a contradiction. Therefore, $d(u,Tu)=0$ and so $Tu=u$. 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 $C_{u_0,\rho }$ with $\rho <r$.☐

Corollary 2.

Let T be a Hardy-Rogers type $F_c$-contractive self-mapping with $u_0\in X$ and r be defined as in (1). If $d(u_0,Tu)=r$ for all $u\in C_{u_0,r}$ then T fixes the disc $D_{u_0,r}$.

Remark 1.

If we consider $\alpha =1$ and $\beta =\gamma =\delta =\eta =0$ in Definition 5, then we get the notion of an $F_c$-contractive mapping.

In Definition 5, if we choose $\delta =\eta =0$, then we obtain the following definition.

Definition 6.

If there exist $t>0$, $F\in \mathbb{F}$ and $u_0\in X$ such that for all $u\in X$ the following holds:

$$d(u,Tu)>0⟹t+F\left(d(u,Tu)\right)\le F\left(\alpha d(u,u_0)+\beta d(u,Tu)+\gamma d(u_0,T{u}_0)\right),$$

(4)

where

$$\alpha +\beta +\gamma <1\mathrm{and}\alpha ,\beta ,\gamma \ge 0,$$

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 $u_0\in X$ 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 $\beta +\gamma <1$.☐

Using Proposition 3, we rewrite the condition (4) as follows:

$$d(u,Tu)>0⟹t+F\left(d(u,Tu)\right)\le F\left(\alpha d(u,u_0)+\beta d(u,Tu)\right),$$

where

$$\alpha +\beta <1\mathrm{and}\alpha ,\beta \ge 0.$$

Using this inequality, we obtain the following fixed-circle result.

Theorem 4.

Let T be a Reich type $F_c$-contraction with $u_0\in X$ and r be defined as in (1). Then $C_{u_0,r}$ is a fixed circle of T. Especially, T fixes every circle $C_{u_0,\rho }$ with $\rho <r$.

Proof. 

It can be easily seen since

$$F\left(r\right)\le F\left(d\right(u,Tu\left)\right)\le F\left(\right(\alpha +\beta \left)d\right(u,Tu\left)\right)<F\left(d\right(u,Tu\left)\right).$$

☐

Corollary 3.

Let T be a Reich type $F_c$-contractive self-mapping with $u_0\in X$ and r be defined as in (1). Then T fixes the disc $D_{u_0,r}$.

In Definition 5, if we choose $\alpha =\beta =\gamma =0$ and $\delta =\eta$, then we obtain the following definition.

Definition 7.

If there exist $t>0$, $F\in \mathbb{F}$ and $u_0\in X$ such that for all $u\in X$ the following holds:

$$d(u,Tu)>0⟹t+F\left(d(u,Tu)\right)\le F\left(\eta (d(u,T{u}_0)+d(u_0,Tu))\right),$$

(5)

where

$$\eta \in \left(0,\frac{1}{2}\right),$$

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 $u_0\in X$ 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 $u_0\in X$ and r be defined as in (1). If $d(u_0,Tu)=r$ for all $u\in C_{u_0,r}$ then $C_{u_0,r}$ is a fixed circle of T. Especially, T fixes every circle $C_{u_0,\rho }$ with $\rho <r$.

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 $u_0\in X$ and r be defined as in (1). If $d(u_0,Tu)=r$ for all $u\in C_{u_0,r}$ then T fixes the disc $D_{u_0,r}$.

Now we give two illustrative examples of our obtained results.

Example 1.

Let $X=\left\{1,2,e^3-1,e^3,e^3+1\right\}$ be the metric space with the usual metric. Let us define the self-mapping $T:X\to X$ as

$$Tu=\left\{\begin{array}{ccc}2& \mathrm{if}& u=1\\ u& & \mathrm{otherwise}\end{array}\right.,$$

for all $u\in X$.

The Ćirić type $F_c$-contractive self-mapping T: The self-mapping T is a Ćirić type $F_c$-contractive self-mapping with $F=lnu$, $t=ln(e^3-1)$ and $u_0=e^3$. Indeed, we get

$$d(u,Tu)=d(1,T1)=d(1,2)=1>0$$

for $u=1$ and

$$\begin{array}{ccc}\hfill m(u,u_0)& =& m(1,e^3)=max\left\{d(1,e^3),d(1,2),\frac{1}{2}\left[d(1,e^3)+d(e^3,2)\right]\right\}\hfill \\ & =& max\left\{e^3-1,1,e^3-\frac{3}{2}\right\}=e^3-1.\hfill \end{array}$$

Then, we have

$$\begin{array}{ccc}\hfill t+F\left(d\right(u,Tu\left)\right)& =& ln(e^3-1)+ln\left(d(1,2)\right)=ln(e^3-1)\hfill \\ & \le & ln\left(d\left(m(u,u_0)\right)\right)=ln(e^3-1).\hfill \end{array}$$

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 $F=lnu$, $t=ln\left(e^3\right)-ln3$, $\alpha =\beta =\frac{1}{3}$, $\delta =\eta =0$ and $u_0=e^3$. Indeed, we get

$$d(u,Tu)=d(1,T1)=d(1,2)=1>0$$

for $u=1$ and

$$\begin{array}{ccc}\hfill \alpha d(u,u_0)+\beta d(u,Tu)+\delta d(u,T{u}_0)+\eta d(u_0,Tu)& =& \frac{1}{3}\left[d(1,e^3)+d(1,2)\right]\hfill \\ & =& \frac{1}{3}\left[e^3-1+1\right]=\frac{e^3}{3}.\hfill \end{array}$$

Then, we have

$$\begin{array}{ccc}\hfill t+F\left(d\right(u,Tu\left)\right)& =& ln\left(e^3\right)-ln3+ln\left(d(1,2)\right)=ln\left(e^3\right)-ln3\hfill \\ & \le & ln\left(d(\alpha d(u,u_0)+\beta d(u,Tu)+\delta d(u,T{u}_0)+\eta d(u_0,Tu))\right)\hfill \\ & =& ln\left(e^3\right)-ln3.\hfill \end{array}$$

The Reich type $F_c$-contractive self-mapping T: The self-mapping T is a Reich type $F_c$-contractive self-mapping with $F=lnu$, $t=ln\left(e^3\right)-ln4$, $\alpha =\beta =\frac{1}{4}$ and $u_0=e^3$. Indeed, we get

$$d(u,Tu)=d(1,T1)=d(1,2)=1>0$$

for $u=1$ and

$$\alpha d(u,u_0)+\beta d(u,Tu)=\frac{1}{4}\left[d(1,e^3)+d(1,2)\right]=\frac{1}{4}\left[e^3-1+1\right]=\frac{e^3}{4}.$$

Then, we have

$$\begin{array}{ccc}\hfill t+F\left(d\right(u,Tu\left)\right)& =& ln\left(e^3\right)-ln4+ln\left(d(1,2)\right)=ln\left(e^3\right)-ln4\hfill \\ & \le & ln\left(d(\alpha d(u,u_0)+\beta d(u,Tu))\right)=ln\left(e^3\right)-ln4.\hfill \end{array}$$

The Chatterjea type $F_c$-contractive self-mapping T: The self-mapping T is a Chatterjea type $F_c$-contractive self-mapping with $F=lnu$, $t=ln\left(\frac{2}{3}{e}^3-1\right)$, $\eta =\frac{1}{3}$ and $u_0=e^3$. Indeed, we get

$$d(u,Tu)=d(1,T1)=d(1,2)=1>0$$

for $u=1$ and

$$\begin{array}{ccc}\hfill \eta (d(u,T{u}_0)+d(u_0,Tu))& =& \frac{1}{3}\left[d(1,e^3)+d(e^3,2)\right]\hfill \\ & =& \frac{1}{3}\left[e^3-1+e^3-2\right]=\frac{2e^3}{3}-1.\hfill \end{array}$$

Then, we have

$$\begin{array}{ccc}\hfill t+F\left(d\right(u,Tu\left)\right)& =& ln\left(\frac{2}{3}{e}^3-1\right)+ln\left(d(1,2)\right)=ln\left(\frac{2}{3}{e}^3-1\right)\hfill \\ & \le & ln\left(\eta (d(u,T{u}_0)+d(u_0,Tu))\right)=ln\left(\frac{2}{3}{e}^3-1\right).\hfill \end{array}$$

Also, we obtain

$$r=min\left\{d(u,Tu):u\ne Tu\right\}=\left\{d(1,2)\right\}=1.$$

Consequently, T fixes the circle $C_{e^3,1}=\left\{e^3-1,e^3+1\right\}$ and the disc $D_{e^3,1}=\left\{e^3-1,e^3,e^3+1\right\}$.

In the following example, we see that the converse statements of Theorems 2–5 are not always true.

Example 2.

Let $x_0\in X$ be any point and the self-mapping $T:X\to X$ be defined as

$$Tu=\left\{\begin{array}{ccc}u& \mathrm{if}& u\in D_{u_0,\mu }\\ u_0& \mathrm{if}& u\notin D_{u_0,\mu }\end{array}\right.,$$

for all $u\in X$ with $\mu >0$. 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 $C_{x_0,\rho }$ where $\rho \le \mu$.

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 $X=\left\{1,2,e^3-1,e^3,e^3+1\right\}$ be the metric space with the usual metric. Let us define the self-mapping $T:X\to X$ as

$$Tu=\left\{\begin{array}{ccc}2& \mathrm{if}& u<e^3-1\\ u& \mathrm{if}& u\ge e^3-1\end{array}\right.,$$

for all $u\in X$. As in Example 1, it is easily verified that the self-mapping T is a Ćirić type $F_c$-contractive self-mapping and $C_{e^3,1}=\left\{e^3-1,e^3+1\right\}$ 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 $X=\left\{1,2,e^3-1,e^3,e^3+1\right\}$ be the metric space with the usual metric. Let us define the self-mapping $T:X\to X$ as

$$Tu=\left\{\begin{array}{ccc}2& \mathrm{if}& u<e^3-1\\ e^3-1& \mathrm{if}& e^3-1\le u<e^3\\ u& \mathrm{if}& e^3\le u\le e^3+1\\ u-1& \mathrm{if}& u>e^3+1\end{array}\right.,$$

for all $u\in X$. As in Example 1, it is easily checked that the self-mapping T is a Ćirić type $F_c$-contractive self-mapping and $C_{e^3,1}=\left\{e^3-1,e^3+1\right\}$ 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 $u_0\in X$ and r be defined as in (1). If $d(u_0,Tu)=r$ for all $u\in C_{u_0,r}$ then $C_{u_0,r}$ is a fixed circle of T. Also T is discontinuous at $u\in C_{u_0,r}$ if and only if $\underset{v\to u}{lim}m(u,v)\ne 0$.

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 [14], we see that T is discontinuous at $u\in C_{u_0,r}$ if and only if $\underset{v\to u}{lim}m(u,v)\ne 0$.☐

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.