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New Types of F_{c}-Contractions and the Fixed-Circle Problem

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Department of Mathematics, Balıkesir University, 10145 Balıkesir, Turkey

Department of Mathematical Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia

Author to whom correspondence should be addressed.

Received: 2 September 2018 / Revised: 26 September 2018 / Accepted: 27 September 2018 / Published: 2 October 2018

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.

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
and
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:

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

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

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.

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:
and

$${C}_{{u}_{0},r}=\left\{u\in X:d(u,{u}_{0})=r\right\}$$

$${D}_{{u}_{0},r}=\left\{u\in X:d(u,{u}_{0})\le r\right\}.$$

- $\left({F}_{1}\right)$ F is strictly increasing,
- $\left({F}_{2}\right)$ For each sequence $\left\{{\alpha}_{n}\right\}$ in $\left(0,\infty \right)$ the following holds$$\underset{n\to \infty}{lim}{\alpha}_{n}=0\phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\mathrm{only}\phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}\underset{n\to \infty}{lim}F\left({\alpha}_{n}\right)=-\infty ,$$
- $\left({F}_{3}\right)$ There exists $k\in (0,1)$ such that $\underset{\alpha \to {0}^{+}}{lim}{\alpha}^{k}F\left(\alpha \right)=0$.

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

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

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.

If there exist $t>0$, $F\in \mathbb{F}$ and ${u}_{0}\in X$ such that for all $u\in X$ the following holds:
where
then T is said to be a Ćirić type ${F}_{c}$-contraction on X.

$$d(u,Tu)>0\u27f9t+F\left(d(u,Tu)\right)\le F\left(m(u,{u}_{0})\right),$$

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

If T is a Ćirić type ${F}_{c}$-contraction with ${u}_{0}\in X$ then we have $T{u}_{0}={u}_{0}$.

Assume that $T{u}_{0}\ne {u}_{0}$. From the definition of a Ćirić type ${F}_{c}$-contraction, we get
a contradiction because of $t>0$. Then we have $T{u}_{0}={u}_{0}$.☐

$$\begin{array}{ccc}\hfill d({u}_{0},T{u}_{0})& >& 0\u27f9t+F\left(d({u}_{0},T{u}_{0})\right)\le F\left(m({u}_{0},{u}_{0})\right)\hfill \\ & =& F\left(max\left\{\begin{array}{c}d({u}_{0},{u}_{0}),d({u}_{0},T{u}_{0}),d({u}_{0},T{u}_{0}),\\ \frac{1}{2}\left[d({u}_{0},T{u}_{0})+d({u}_{0},T{u}_{0})\right]\end{array}\right\}\right)\hfill \\ & =& F\left(d({u}_{0},T{u}_{0})\right),\hfill \end{array}$$

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 $\rho <r$.

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

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

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

$$F\left(d(u,Tu)\right)\le F\left(m(u,{u}_{0})\right)-t<F\left(m(u,{u}_{0})\right)=F\left(d(u,Tu)\right),$$

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 ${D}_{{u}_{0},r}$.

If there exist $t>0$, $F\in \mathbb{F}$ and ${u}_{0}\in X$ such that for all $u\in X$ the following holds:
where
then T is said to be a Hardy-Rogers type ${F}_{c}$-contraction on X.

$$d(u,Tu)>0\u27f9t+F\left(d(u,Tu)\right)\le F\left(\begin{array}{c}\alpha d(u,{u}_{0})+\beta d(u,Tu)+\gamma d({u}_{0},T{u}_{0})\\ +\delta d(u,T{u}_{0})+\eta d({u}_{0},Tu)\end{array}\right),$$

$$\alpha +\beta +\gamma +\delta +\eta =1,\phantom{\rule{4.pt}{0ex}}\alpha ,\beta ,\gamma ,\delta ,\eta \ge 0\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\alpha \ne 0,$$

If T is a Hardy-Rogers type ${F}_{c}$-contraction with ${u}_{0}\in X$ then we have $T{u}_{0}={u}_{0}$.

Assume that $T{u}_{0}\ne {u}_{0}$. From the definition of a Hardy-Rogers type ${F}_{c}$-contraction, we get
a contradiction because of $t>0$. Then we have $T{u}_{0}={u}_{0}$.☐

$$\begin{array}{ccc}\hfill d({u}_{0},T{u}_{0})& >& 0\u27f9t+F\left(d({u}_{0},T{u}_{0})\right)\hfill \\ & \le & F\left(\begin{array}{c}\alpha d({u}_{0},{u}_{0})+\beta d({u}_{0},T{u}_{0})+\gamma d({u}_{0},T{u}_{0})\\ +\delta d({u}_{0},T{u}_{0})+\eta d({u}_{0},T{u}_{0})\end{array}\right)\hfill \\ & =& F\left(\left(\beta +\gamma +\delta +\eta \right)d({u}_{0},T{u}_{0})\right)\hfill \\ & <& F\left(d({u}_{0},T{u}_{0})\right),\hfill \end{array}$$

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

$$d(u,Tu)>0\u27f9t+F\left(d(u,Tu)\right)\le F\left(\begin{array}{c}\alpha d(u,{u}_{0})+\beta d(u,Tu)\\ +\delta d(u,T{u}_{0})+\eta d({u}_{0},Tu)\end{array}\right),$$

$$\alpha +\beta +\delta +\eta \le 1,\phantom{\rule{4.pt}{0ex}}\alpha ,\beta ,\delta ,\eta \ge 0\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\alpha \ne 0.$$

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

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 $\rho <r$.

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
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 $\rho <r$.☐

$$\begin{array}{ccc}\hfill F\left(r\right)& \le & F\left(d\right(u,Tu\left)\right)\hfill \\ & \le & F\left(\alpha d(u,{u}_{0})+\beta d(u,Tu)+\delta d(u,T{u}_{0})+\eta d({u}_{0},Tu)\right)-t\hfill \\ & <& F(\alpha r+\beta d(u,Tu)+\delta r+\eta r)\hfill \\ & \le & F\left(\right(\alpha +\beta +\delta +\eta \left)d\right(u,Tu\left)\right)\le F\left(d\right(u,Tu\left)\right),\hfill \end{array}$$

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 ${D}_{{u}_{0},r}$.

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.

If there exist $t>0$, $F\in \mathbb{F}$ and ${u}_{0}\in X$ such that for all $u\in X$ the following holds:
where
then T is said to be a Reich type ${F}_{c}$-contraction on X.

$$d(u,Tu)>0\u27f9t+F\left(d(u,Tu)\right)\le F\left(\alpha d(u,{u}_{0})+\beta d(u,Tu)+\gamma d({u}_{0},T{u}_{0})\right),$$

$$\alpha +\beta +\gamma <1\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\alpha ,\beta ,\gamma \ge 0,$$

If a self-mapping T on X is a Reich type ${F}_{c}$-contraction with ${u}_{0}\in X$ then we have $T{u}_{0}={u}_{0}$.

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:
where

$$d(u,Tu)>0\u27f9t+F\left(d(u,Tu)\right)\le F\left(\alpha d(u,{u}_{0})+\beta d(u,Tu)\right),$$

$$\alpha +\beta <1\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\alpha ,\beta \ge 0.$$

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

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 $\rho <r$.

It can be easily seen since
☐

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

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.

If there exist $t>0$, $F\in \mathbb{F}$ and ${u}_{0}\in X$ such that for all $u\in X$ the following holds:
where
then T is said to be a Chatterjea type ${F}_{c}$-contraction on X.

$$d(u,Tu)>0\u27f9t+F\left(d(u,Tu)\right)\le F\left(\eta (d(u,T{u}_{0})+d({u}_{0},Tu))\right),$$

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

If a self-mapping T on X is a Chatterjea type ${F}_{c}$-contraction with ${u}_{0}\in X$ then we have $T{u}_{0}={u}_{0}$.

From the similar arguments used in the proof of Proposition 2, it can be easily proved.☐

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 $\rho <r$.

By the similar arguments used in the proof of Theorem 3 and Definition 7, it can be easily checked.☐

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.

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
for all $u\in X$.

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

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
for $u=1$ and

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

$$\begin{array}{ccc}\hfill m(u,{u}_{0})& =& m(1,{e}^{3})=max\left\{d(1,{e}^{3}),d(1,2),\frac{1}{2}\left[d(1,{e}^{3})+d({e}^{3},2)\right]\right\}\hfill \\ & =& max\left\{{e}^{3}-1,1,{e}^{3}-\frac{3}{2}\right\}={e}^{3}-1.\hfill \end{array}$$

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
for $u=1$ and

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

$$\begin{array}{ccc}\hfill \alpha d(u,{u}_{0})+\beta d(u,Tu)+\delta d(u,T{u}_{0})+\eta d({u}_{0},Tu)& =& \frac{1}{3}\left[d(1,{e}^{3})+d(1,2)\right]\hfill \\ & =& \frac{1}{3}\left[{e}^{3}-1+1\right]=\frac{{e}^{3}}{3}.\hfill \end{array}$$

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
for $u=1$ and

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

$$\alpha d(u,{u}_{0})+\beta d(u,Tu)=\frac{1}{4}\left[d(1,{e}^{3})+d(1,2)\right]=\frac{1}{4}\left[{e}^{3}-1+1\right]=\frac{{e}^{3}}{4}.$$

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
for $u=1$ and

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

$$\begin{array}{ccc}\hfill \eta (d(u,T{u}_{0})+d({u}_{0},Tu))& =& \frac{1}{3}\left[d(1,{e}^{3})+d({e}^{3},2)\right]\hfill \\ & =& \frac{1}{3}\left[{e}^{3}-1+{e}^{3}-2\right]=\frac{2{e}^{3}}{3}-1.\hfill \end{array}$$

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.

Let ${x}_{0}\in X$ be any point and the self-mapping $T:X\to X$ be defined as
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 $.

$$Tu=\left\{\begin{array}{ccc}u& \mathrm{if}& u\in {D}_{{u}_{0},\mu}\\ {u}_{0}& \mathrm{if}& u\notin {D}_{{u}_{0},\mu}\end{array}\right.,$$

In this section, we give some examples of discontinuous functions and obtain a discontinuity result related to fixed circle.

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
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 ${e}^{3}-1$.

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

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
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 ${C}_{{e}^{3},1}$.

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

Consider the above examples, we give the following theorem.

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 $\underset{v\to u}{lim}m(u,v)\ne 0$.

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 $\underset{v\to u}{lim}m(u,v)\ne 0$.☐

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.

All authors contributed equally in writing this article. All authors read and approved the final manuscript.

This research received no external funding.

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.

The authors declare no conflicts of interest.

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