Some Common Fixed-Circle Results on Metric Spaces

Abstract

Recently, the fixed-circle problems have been studied with different approaches as an interesting and geometric generalization. In this paper, we present some solutions to an open problem $CC$: what is (are) the condition(s) to make any circle $C_{{\varpi }_0,\sigma }$ as the common fixed circle for two (or more than two) self-mappings? To do this, we modify some known contractions which are used in fixed-point theorems such as the Hardy–Rogers-type contraction, Kannan-type contraction, etc.

1. Introduction

In the recent past, the fixed-circle problem has been introduced as a new geometric generalization of fixed-point theory. After that, some solutions to this problem have been investigated using various techniques (for example, see [1,2,3,4,5,6,7,8], and the references therein). In addition, in [1], the following open problem was given:

Let $\left(X,\mathfrak{D}\right)$ be a metric space and $C_{{\varpi }_0,\sigma }=\left\{\varpi \in X:\mathfrak{D}(\varpi ,{\varpi }_0)=\sigma \right\}$ be any circle on X.

Open Problem$CC$: What is (are) the condition(s) to make any circle $C_{{\varpi }_0,\sigma }$ as the common fixed circle for two (or more than two) self-mappings?

Let $\xi$ and g be two self-mappings on a set X. If $\xi \varpi =g\varpi =\varpi$ for all $\varpi \in C_{{\varpi }_0,\sigma }$, then $C_{{\varpi }_0,\sigma }$ is called a common fixed circle of the pair $\left(\xi ,g\right)$ (see [9] for more details).

Some solutions were given for this open problem (for example, see [8,9]). To obtain new solutions, in this paper, we define new contractions for the pair $\left(\xi ,g\right)$ and prove new common fixed-circle results on metric spaces. Before moving on to the main results, we recall the following.

Throughout this article, we denote by $\mathbb{R}$ the set of all real numbers and by ${\mathbb{R}}_{+}$ the set of all positive real numbers.

Let $\xi$ and g be self-mappings on a set X. If $\xi \varpi =g\varpi =w$ for some $\varpi$ in X, then $\varpi$ is called a coincidence point of $\xi$ and g, w is called a point of coincidence of $\xi$ and g.

Let $C\left(\xi ,g\right)=\left\{\varpi \in X:\xi \varpi =g\varpi =\varpi \right\}$ denote the set of all common fixed-points of self-mappings $\xi$ and g.

In [10], Wardowski introduced the following family of functions to obtain a new type of contraction called $\mathcal{F}$-contraction.

Let $\mathbb{F}$ be the family of all mappings $\mathcal{F}:{\mathbb{R}}_{+}\to \mathbb{R}$ that satisfy the following conditions:

$\left(\mathcal{F}1\right)$

$\mathcal{F}$ is strictly increasing, that is, for all $a,b\in$ ${\mathbb{R}}_{+}$ such that $a<b$ implies that $\mathcal{F}\left(a\right)<\mathcal{F}\left(b\right)$;

$\left(\mathcal{F}2\right)$

For every sequence ${\left\{a_n\right\}}_{n\in N}$ of positive real numbers, ${lim}_{n\to \infty }{a}_n=0$ and ${lim}_{n\to \infty }\mathcal{F}\left(a_n\right)=-\infty$ are equivalent;

$\left(\mathcal{F}3\right)$

There exists $k\in \left(0,1\right)$ such that ${lim}_{a\to 0^{+}}{a}^k\mathcal{F}\left(a\right)=0.$

Some examples of functions that confirm the conditions $\left(\mathcal{F}1\right)$, $\left(\mathcal{F}2\right)$, and $\left(\mathcal{F}3\right)$ are as follows:

• $\mathcal{F}\left(a\right)=ln\left(a\right)$;
• $\mathcal{F}\left(a\right)=ln\left(a\right)+a$;
• $\mathcal{F}\left(a\right)=ln\left(a^2+a\right)$;
• $\mathcal{F}\left(a\right)=-\frac{1}{\sqrt{a}}$ (see [10] for more details).

Definition 1.

[10] Let $\left(X,\mathfrak{D}\right)$ be a metric space, $\mathcal{F}\in \mathbb{F}$ and $\xi :X\to X.$ The mapping ξ is called an $\mathcal{F}$-contraction if there exists $\tau >0$ such that

$$\tau +\mathcal{F}\left(\mathfrak{D}\left(\xi \varpi ,\xi v\right)\right)\le \mathcal{F}\left(\mathfrak{D}\left(\varpi ,v\right)\right)$$

for all $\varpi ,v\in X$ satisfying $\mathfrak{D}\left(T\varpi ,Tv\right)>0.$

2. Main Results

In this section, we prove new common fixed-circle theorems on metric spaces. For this purpose, we modify some well-known contractions such as the Wardowski-type contraction [10], Nemytskii–Edelstein-type contraction [11,12], Banach-type contraction [13], Hardy–Rogers-type contraction [14], Reich-type contraction [15], Chatterjea-type contraction [16], and Kannan-type contraction [17].

At first, we introduce the following new contraction type for two mappings to obtain some common fixed-circle results on metric spaces.

Definition 2.

Let $\left(X,\mathfrak{D}\right)$ be a metric space and $\xi ,g$ be two self-mappings on X. If there exist $\tau >0,\mathcal{F}\in \mathbb{F}$ and ${\varpi }_0\in X$ such that

$$\tau +\mathcal{F}(\mathfrak{D}\left(\varpi ,\xi \varpi \right)+\mathfrak{D}\left(\varpi ,g\varpi \right))\le \mathcal{F}\left(\mathfrak{D}\left({\varpi }_0,\varpi \right)\right)$$

for all $\varpi \in X$ satisfying $min\left\{\mathfrak{D}\left(\varpi ,\xi \varpi \right),\mathfrak{D}\left(\varpi ,g\varpi \right)\right\}>0,$ then the pair $\left(\xi ,g\right)$ is called a Wardowski-type ${\mathcal{F}}_{\xi g}$-contraction.

Notice that the point ${\varpi }_0$ mentioned in Definition 2 must be a common fixed-point of the mappings $\xi$ and g. In fact, if ${\varpi }_0$ is not a common fixed-point of $\xi$ and g, then we have $\mathfrak{D}\left({\varpi }_0,\xi {\varpi }_0\right)>0$ and $\mathfrak{D}\left({\varpi }_0,g{\varpi }_0\right)>0.$ Hence, we obtain

$$min\left\{\mathfrak{D}\left({\varpi }_0,\xi {\varpi }_0\right),\mathfrak{D}\left({\varpi }_0,g{\varpi }_0\right)\right\}>0⟹\tau +\mathcal{F}(\mathfrak{D}\left({\varpi }_0,\xi {\varpi }_0\right)+\mathfrak{D}\left({\varpi }_0,g{\varpi }_0\right))\le \mathcal{F}\left(\mathfrak{D}\left({\varpi }_0,{\varpi }_0\right)\right).$$

This gives a contradiction since the domain of $\mathcal{F}$ is $\left(0,\infty \right).$ As a result, we receive the following proposition as a consequence of Definition 2.

Proposition 1.

Let $\left(X,\mathfrak{D}\right)$ be a metric space. If the pair $\left(\xi ,g\right)$ is a Wardowski-type ${\mathcal{F}}_{\xi g}$-contraction with ${\varpi }_0\in X,$ then we have $\xi {\varpi }_0=g{\varpi }_0={\varpi }_0.$

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

Theorem 1.

Let $\left(X,\mathfrak{D}\right)$ be a metric space and the pair $\left(\xi ,g\right)$ be a Wardowski-type ${\mathcal{F}}_{\xi g}$-contraction with ${\varpi }_0\in X.$ Define the number σ by

$$\sigma =inf\left\{\mathfrak{D}\left(\varpi ,\xi \varpi \right)+\mathfrak{D}\left(\varpi ,g\varpi \right):\varpi \ne \xi \varpi ,\varpi \ne g\varpi ,\varpi \in X\right\}.$$

(1)

Then, $C_{{\varpi }_0,\sigma }$ is a common fixed circle of the pair $\left(\xi ,g\right)$. Especially, ξ and g fix every circle $C_{{\varpi }_0,r}$ where $r<\sigma .$

Proof. 

We distinguish two cases.

Case 1: Let $\sigma =0.$ Clearly, $C_{{\varpi }_0,\sigma }=\left\{{\varpi }_0\right\}$ and by Proposition 1, we see that $C_{{\varpi }_0,\sigma }$ is a common fixed circle of the pair $\left(\xi ,g\right)$.

Case 2: Let $\sigma >0$ and $\varpi \in C_{{\varpi }_0,\sigma }.$ If $\xi \varpi \ne \varpi$ and $g\varpi \ne \varpi ,$ then by (1), we have $\mathfrak{D}\left(\varpi ,\xi \varpi \right)+\mathfrak{D}\left(\varpi ,g\varpi \right)\ge \sigma .$ Hence, using the Wardowski-type ${\mathcal{F}}_{\xi g}$-contraction property and the fact that $\mathcal{F}$ is increasing, we obtain

$$\begin{array}{ccc}\hfill \mathcal{F}\left(\sigma \right)& \le & \mathcal{F}\left(\mathfrak{D}\left(\varpi ,\xi \varpi \right)+\mathfrak{D}\left(\varpi ,g\varpi \right)\right)\hfill \\ & \le & \mathcal{F}\left(\mathfrak{D}\left({\varpi }_0,\varpi \right)\right)-\tau \hfill \\ & <& \mathcal{F}\left(\mathfrak{D}\left({\varpi }_0,\varpi \right)\right)\hfill \\ & =& \mathcal{F}\left(\sigma \right)\hfill \end{array}$$

This gives a contradiction. Therefore, we have $\mathfrak{D}\left(\varpi ,\xi \varpi \right)+\mathfrak{D}\left(\varpi ,g\varpi \right)=0$, that is, $\varpi =\xi \varpi$ and $\varpi =g\varpi$. As a consequence, $C_{{\varpi }_0,\sigma }$ is a common fixed circle of the pair $\left(\xi ,g\right)$.

Now, we show that $\xi$ and g also fix any circle $C_{{\varpi }_0,r}$ with $r<\sigma .$ Let $\varpi \in C_{{\varpi }_0,r}$ and suppose that $\mathfrak{D}\left(\varpi ,\xi \varpi \right)+\mathfrak{D}\left(\varpi ,g\varpi \right)>0.$ With the Wardowski-type ${\mathcal{F}}_{\xi g}$-contraction property, we have

$$\begin{array}{ccc}\hfill \mathcal{F}\left(\mathfrak{D}\left(\varpi ,\xi \varpi \right)+\mathfrak{D}\left(\varpi ,g\varpi \right)\right)& \le & \mathcal{F}\left(\mathfrak{D}\left({\varpi }_0,\varpi \right)\right)-\tau \hfill \\ & <& \mathcal{F}\left(\mathfrak{D}\left({\varpi }_0,\varpi \right)\right)\hfill \\ & =& \mathcal{F}\left(r\right).\hfill \end{array}$$

Since $\mathcal{F}$ is increasing, then we find

$$\mathfrak{D}\left(\varpi ,\xi \varpi \right)+\mathfrak{D}\left(\varpi ,g\varpi \right)<\mathfrak{D}\left({\varpi }_0,\varpi \right)<r<\sigma .$$

However, $\sigma =inf\left\{\mathfrak{D}\left(\varpi ,\xi \varpi \right)+\mathfrak{D}\left(\varpi ,g\varpi \right):\varpi \ne \xi \varpi ,\varpi \ne g\varpi ,\varpi \in X\right\},$ so this gives a contradiction. Thus, $\mathfrak{D}\left(\varpi ,\xi \varpi \right)+\mathfrak{D}\left(\varpi ,g\varpi \right)=0$ and $\varpi =\xi \varpi =g\varpi$. Hence, $C_{{\varpi }_0,r}$ is a common fixed circle of the pair $\left(\xi ,g\right).$ □

Example 1.

Let $X=\left\{0,1,-e,e,e-1,e+1,-e^2,e^2,e^2-1,e^2+1,e^2-e,e^2+e\right\}$ with usual metric. Define $\xi ,g:X\to X$ by

$$\xi \varpi =\left\{\begin{array}{cc}1,& \varpi =0\\ \varpi ,& otherwise\end{array}\right.$$

and

$$g\varpi =\left\{\begin{array}{cc}e-1,& \varpi =0\\ \varpi ,& otherwise\end{array}\right..$$

Take $\mathcal{F}\left(a\right)=ln\left(a\right)+a,$ $a>0,\tau =e$ and ${\varpi }_0=e^2.$ Thus, the pair $\left(\xi ,g\right)$ is a Wardowski-type ${\mathcal{F}}_{\xi g}$-contraction. For $\varpi =0$, we have

$$\begin{array}{ccc}\hfill min\left\{\mathfrak{D}\left(\varpi ,\xi \varpi \right),\mathfrak{D}\left(\varpi ,g\varpi \right)\right\}& =& min\left\{\mathfrak{D}\left(0,1\right),\mathfrak{D}\left(0,e-1\right)\right\}\hfill \\ & =& min\left\{1,e-1\right\}\hfill \\ & =& 1>0\hfill \end{array}$$

In addition, we can easily see that the following inequality is satisfied:

$$\begin{array}{ccc}\hfill \tau +\mathcal{F}\left(\mathfrak{D}\left(\varpi ,\xi \varpi \right)+\mathfrak{D}\left(\varpi ,g\varpi \right)\right)& \le & \mathcal{F}\left(\mathfrak{D}\left({\varpi }_0,\varpi \right)\right)\hfill \\ \hfill e+\mathcal{F}\left(1+e-1\right)& \le & \mathcal{F}\left(e^2\right)\hfill \\ \hfill e+lne+e& \le & ln{e}^2+e^2\hfill \\ \hfill 2e+1& \le & 2+e^2\hfill \end{array}$$

With Theorem (1), we obtain

$$\sigma =inf\left\{\mathfrak{D}\left(\varpi ,\xi \varpi \right)+\mathfrak{D}\left(\varpi ,g\varpi \right):\varpi \ne \xi \varpi ,\varpi \ne g\varpi ,\varpi \in X\right\}=inf\left\{1+e-1\right\}=e$$

and $\xi ,g$ fix the circle $C_{e^2,e}=\left\{e^2-e,e^2+e\right\}.$ Notice that ξ and g fix also the circle $C_{e^2,1}=\left\{e^2-1,e^2+1\right\}.$

The converse of Theorem 1 fails. The following example confirms this statement.

Example 2.

Let $\left(X,\mathfrak{D}\right)$ be a metric space with any point ${\varpi }_0\in X.$ Define the self-mappings ξ and g as follows:

$$\xi \varpi =\left\{\begin{array}{cc}\varpi ,& \mathfrak{D}\left(\varpi ,{\varpi }_0\right)\le \mu \\ {\varpi }_0,& \mathfrak{D}\left(\varpi ,{\varpi }_0\right)>\mu \end{array}\right.$$

and

$$g\varpi =\left\{\begin{array}{cc}\varpi ,& \mathfrak{D}\left(\varpi ,{\varpi }_0\right)\le \mu \\ {\varpi }_0,& \mathfrak{D}\left(\varpi ,{\varpi }_0\right)>\mu \end{array}\right.,$$

for all $\varpi \in X$ with any $\mu >0.$ Then, it can be easily checked that the pair $\left(\xi ,g\right)$ is not a Wardowski-type ${\mathcal{F}}_{\xi g}$-contraction for the point ${\varpi }_0$ but ξ and g fix every circle $C_{{\varpi }_0,r}$ where $r\le \mu .$

Example 3.

Let $\mathbb{C}$ be the set of complex numbers, $\left(\mathbb{C},\mathfrak{D}\right)$ be the usual metric space, and define the self-mappings $\xi ,g:\mathbb{C}\to \mathbb{C}$ as follows:

$$\xi \varpi =\left\{\begin{array}{cc}\varpi ,& \left|\varpi -2\right|<e\\ \varpi +\frac{1}{2},& \left|\varpi -2\right|\ge e\end{array}\right.$$

and

$$g\varpi =\left\{\begin{array}{cc}\varpi ,& \left|\varpi -2\right|<e\\ \varpi -\frac{1}{2},& \left|\varpi -2\right|\ge e\end{array}\right.,$$

for all $\varpi \in C.$ We have $\sigma =inf\left\{\mathfrak{D}\left(\varpi ,\xi \varpi \right)+\mathfrak{D}\left(\varpi ,g\varpi \right):\varpi \ne \xi \varpi ,\varpi \ne g\varpi ,\varpi \in C\right\}.$ Thus, the pair $\left(\xi ,g\right)$ is a Wardowski-type ${\mathcal{F}}_{\xi g}$-contraction with $\mathcal{F}=\ln \left(a\right),\tau =lne$ and ${\varpi }_0=2\in C.$ Obviously, the number of common fixed circles of ξ and g is infinite.

Definition 3.

If there exist $\tau >0,$ $\mathcal{F}\in \mathcal{F}$ and ${\varpi }_0\in X$ such that for all $\varpi \in X$ the following holds:

$$\tau +\mathcal{F}\left(\mathfrak{D}\left(\xi \varpi ,\varpi \right)+\mathfrak{D}\left(g\varpi ,\varpi \right)\right)<\mathcal{F}\left(\mathfrak{D}\left(\varpi ,{\varpi }_0\right)\right)$$

with $min\left\{\mathfrak{D}\left(\xi \varpi ,\varpi \right),\mathfrak{D}\left(g\varpi ,\varpi \right)\right\}>0,$ then the pair $\left(\xi ,g\right)$ is called a Nemytskii–Edelstein-type ${\mathcal{F}}_{\xi g}$-contraction.

Proposition 2.

Let $\left(X,\mathfrak{D}\right)$ be a metric space. If the pair $\left(\xi ,g\right)$ is a Nemytskii-Edelstein-type ${\mathcal{F}}_{\xi g}$-contraction with ${\varpi }_0\in X,$ then we have $\xi {\varpi }_0=g{\varpi }_0={\varpi }_0.$

Proof. 

It can be easily proved from the similar arguments used in Proposition 1. □

Theorem 2.

Let the pair $\left(\xi ,g\right)$ be a Nemytskii–Edelstein-type ${\mathcal{F}}_{\xi g}$-contraction with ${\varpi }_0\in X$ and σ be defined as in (1). Then, $C_{{\varpi }_0,\sigma }$ is a common fixed circle of the pair $\left(\xi ,g\right)$. Especially, ξ and g fix every circle $C_{{\varpi }_0,r}$ where $r<\sigma .$

Proof. 

It can be easily seen from the proof of Theorem 1. □

In addition, we inspire the classical Banach contraction principle to give the following definition:

Definition 4.

If there exist $\tau >0,$ $\mathcal{F}\in \mathcal{F}$ and ${\varpi }_0\in X$ such that for all $\varpi \in X$, the following holds:

$$\tau +\mathcal{F}\left(\mathfrak{D}\left(\xi \varpi ,\varpi \right)+\mathfrak{D}\left(g\varpi ,\varpi \right)\right)\le \mathcal{F}\left(\eta \mathfrak{D}\left(\varpi ,{\varpi }_0\right)\right)$$

with $min\left\{\mathfrak{D}\left(\xi \varpi ,\varpi \right),\mathfrak{D}\left(g\varpi ,\varpi \right)\right\}>0$ where $\eta \in [0,1),$ then the pair $\left(\xi ,g\right)$ is called a Banach-type ${\mathcal{F}}_{\xi g}$-contraction.

Proposition 3.

Let $\left(X,\mathfrak{D}\right)$ be a metric space. If the pair $\left(\xi ,g\right)$ is a Banach-type ${\mathcal{F}}_{\xi g}$-contraction with ${\varpi }_0\in X,$ then we have $\xi {\varpi }_0=g{\varpi }_0={\varpi }_0.$

Proof. 

It can be easily proved from the similar arguments used in Proposition 1. □

Theorem 3.

Let the pair $\left(\xi ,g\right)$ be a Banach-type ${\mathcal{F}}_{\xi g}$-contraction with ${\varpi }_0\in X$ and σ be defined as in (1). Then $C_{{\varpi }_0,\sigma }$ is a common fixed circle of the pair $\left(\xi ,g\right)$. Especially, ξ and g fix every circle $C_{{\varpi }_0,r}$ where $r<\sigma .$

Proof. 

It can be easily seen from the proof of Theorem 1. □

If we consider Example 1, then the pair $\left(\xi ,g\right)$ is both a Nemytskii–Edelstein-type ${\mathcal{F}}_{\xi g}$-contraction and a Banach-type ${\mathcal{F}}_{\xi g}$-contraction with $\mathcal{F}\left(a\right)=ln\left(a\right)+a,$ $a>0,\tau =e$, ${\varpi }_0=e^2$ and so $\xi$, g have two common fixed circles $C_{e^2,e}$ and $C_{e^2,1}$.

We introduce the notion of Hardy–Rogers-type ${\mathcal{F}}_{\xi g}$-contraction.

Definition 5.

Let $\left(X,\mathfrak{D}\right)$ be a metric space and $\xi ,g$ be two self-mappings on X. The pair $\left(\xi ,g\right)$ is called a Hardy–Rogers-type ${\mathcal{F}}_{\xi g}$-contraction if there exist $\tau >0$ and $\mathcal{F}\in \mathcal{F}$ such that

$$\tau +\mathcal{F}\left(\mathfrak{D}\left(\varpi ,\xi \varpi \right)+\mathfrak{D}\left(\varpi ,g\varpi \right)\right)\le \mathcal{F}\left(\begin{array}{c}\alpha \mathfrak{D}\left(\varpi ,{\varpi }_0\right)+\beta \mathfrak{D}\left(\varpi ,\xi \varpi \right)\\ +\gamma \mathfrak{D}\left(\varpi ,g\varpi \right)+\delta \mathfrak{D}\left({\varpi }_0,\xi {\varpi }_0\right)+\eta \mathfrak{D}\left({\varpi }_0,g{\varpi }_0\right)\end{array}\right)$$

(2)

holds for any $\varpi ,{\varpi }_0\in X$ with $min\left\{\mathfrak{D}\left(\varpi ,\xi \varpi \right),\mathfrak{D}\left(\varpi ,g\varpi \right)\right\}>0$, where $\alpha ,\beta ,\gamma ,\delta ,\eta$ are nonnegative numbers, $\alpha \ne 0$ and $\alpha +\beta +\gamma +\delta +\eta \le 1.$

Proposition 4.

If the pair $\left(\xi ,g\right)$ is a Hardy–Rogers-type ${\mathcal{F}}_{\xi g}$-contraction with ${\varpi }_0\in X,$ then we have $\xi {\varpi }_0=g{\varpi }_0={\varpi }_0.$

Proof. 

Suppose that $\xi {\varpi }_0\ne {\varpi }_0$ and $g{\varpi }_0\ne {\varpi }_0.$ From the definition of the Hardy–Rogers-type ${\mathcal{F}}_{\xi g}$-contraction with $min\left\{\mathfrak{D}\left({\varpi }_0,\xi {\varpi }_0\right),\mathfrak{D}\left({\varpi }_0,g{\varpi }_0\right)\right\}>0,$ we obtain

$$\begin{array}{ccc}\hfill \tau +\mathcal{F}\left(\mathfrak{D}\left({\varpi }_0,\xi {\varpi }_0\right)+\mathfrak{D}\left({\varpi }_0,g{\varpi }_0\right)\right)& \le & \mathcal{F}\left(\begin{array}{c}\alpha \mathfrak{D}\left({\varpi }_0,{\varpi }_0\right)+\beta \mathfrak{D}\left({\varpi }_0,\xi {\varpi }_0\right)\\ +\gamma \mathfrak{D}\left({\varpi }_0,g{\varpi }_0\right)+\delta \mathfrak{D}\left({\varpi }_0,\xi {\varpi }_0\right)+\eta \mathfrak{D}\left({\varpi }_0,g{\varpi }_0\right)\end{array}\right)\hfill \\ & =& \mathcal{F}\left(\left(\beta +\delta \right)\mathfrak{D}\left({\varpi }_0,\xi {\varpi }_0\right)+\left(\gamma +\eta \right)\mathfrak{D}\left({\varpi }_0,g{\varpi }_0\right)\right)\hfill \\ & <& \mathcal{F}\left(\mathfrak{D}\left({\varpi }_0,\xi {\varpi }_0\right)+\mathfrak{D}\left({\varpi }_0,g{\varpi }_0\right)\right)\hfill \end{array}$$

a contradiction because of $\tau >0.$ Thus, we have $\xi {\varpi }_0=g{\varpi }_0={\varpi }_0.$ □

Using Proposition 4, we rewrite the condition (2) as follows:

$$\tau +\mathcal{F}\left(\mathfrak{D}\left(\varpi ,\xi \varpi \right),\mathfrak{D}\left(\varpi ,g\varpi \right)\right)\le \mathcal{F}\left(\alpha \mathfrak{D}\left(\varpi ,{\varpi }_0\right)+\beta \mathfrak{D}\left(\varpi ,\xi \varpi \right)+\gamma \mathfrak{D}\left(\varpi ,g\varpi \right)\right)$$

with $min\left\{\mathfrak{D}\left(\varpi ,\xi \varpi \right),\mathfrak{D}\left(\varpi ,g\varpi \right)\right\}>0$ where $\alpha ,\beta ,\gamma$ are nonnegative numbers, $\alpha \ne 0$ and $\alpha +\beta +\gamma \le 1.$

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

Theorem 4.

Let the pair $\left(\xi ,g\right)$ be a Hardy–Rogers-type ${\mathcal{F}}_{\xi g}$-contraction with ${\varpi }_0\in X$ and σ be defined as in (1). If $\beta =\gamma ,$ then $C_{{\varpi }_0,\sigma }$ is a common fixed circle of the pair $\left(\xi ,g\right)$. In addition, ξ and g fix every circle $C_{{\varpi }_0,r}$ with $r<\sigma .$

Proof. 

We distinguish two cases.

Case 1: Let $\sigma =0.$ Clearly, $C_{{\varpi }_0,\sigma }=\left\{{\varpi }_0\right\}$ and by Proposition 4, we see that $C_{{\varpi }_0,\sigma }$ is a common fixed circle of the pair $\left(\xi ,g\right)$.

Case 2: Let $\sigma >0$ and $\varpi \in C_{{\varpi }_0,\sigma .}$ Using the Hardy–Rogers-type ${\mathcal{F}}_{\xi g}$-contractive property and the fact that $\mathcal{F}$ is increasing, we have

$$\begin{array}{ccc}\hfill \mathcal{F}\left(\sigma \right)& \le & \mathcal{F}\left(\mathfrak{D}\left(\varpi ,\xi \varpi \right)+\mathfrak{D}\left(\varpi ,g\varpi \right)\right)\hfill \\ & \le & \mathcal{F}\left(\alpha \mathfrak{D}\left(\varpi ,{\varpi }_0\right)+\beta \mathfrak{D}\left(\varpi ,\xi \varpi \right)+\gamma \mathfrak{D}\left(\varpi ,g\varpi \right)\right)-\tau \hfill \\ & <& \mathcal{F}\left(\alpha \sigma +\beta \left(\mathfrak{D}\left(\varpi ,\xi \varpi \right)+\mathfrak{D}\left(\varpi ,g\varpi \right)\right)\right)\hfill \\ & <& \mathcal{F}\left(\left(\alpha +\beta \right)\left(\mathfrak{D}\left(\varpi ,\xi \varpi \right)+\mathfrak{D}\left(\varpi ,g\varpi \right)\right)\right)\hfill \\ & <& \mathcal{F}\left(\mathfrak{D}\left(\varpi ,\xi \varpi \right)+\mathfrak{D}\left(\varpi ,g\varpi \right)\right).\hfill \end{array}$$

This gives a contradiction. Therefore, $\mathfrak{D}\left(\varpi ,\xi \varpi \right)+\mathfrak{D}\left(\varpi ,g\varpi \right)=0$ and so $\xi \varpi =\varpi =g\varpi$. As a result, $C_{{\varpi }_0,\sigma }$ is a common fixed circle of the pair $\left(\xi ,g\right)$.

Now, we show that $\xi$ and g also fix any circle $C_{{\varpi }_0,r}$ with $r<\sigma .$ Let $\varpi \in C_{{\varpi }_0,r}$ and suppose that $\mathfrak{D}\left(\varpi ,\xi \varpi \right)+\mathfrak{D}\left(\varpi ,g\varpi \right)>0.$ By the Hardy–Rogers-type ${\mathcal{F}}_{\xi g}$-contraction, we have

$$\begin{array}{ccc}\hfill \mathcal{F}\left(\mathfrak{D}\left(\varpi ,\xi \varpi \right)+\mathfrak{D}\left(\varpi ,g\varpi \right)\right)& \le & \mathcal{F}\left(\alpha \mathfrak{D}\left(\varpi ,{\varpi }_0\right)+\beta \mathfrak{D}\left(\varpi ,\xi \varpi \right)+\gamma \mathfrak{D}\left(\varpi ,g\varpi \right)\right)-\tau \hfill \\ & <& \mathcal{F}\left(\alpha \mathfrak{D}\left(\varpi ,{\varpi }_0\right)+\beta \mathfrak{D}\left(\varpi ,\xi \varpi \right)+\gamma \mathfrak{D}\left(\varpi ,g\varpi \right)\right)\hfill \\ & <& \mathcal{F}\left(\mathfrak{D}\left(\varpi ,\xi \varpi \right)+\mathfrak{D}\left(\varpi ,g\varpi \right)\right)\hfill \end{array}$$

a contradiction. So, we obtain $\mathfrak{D}\left(\varpi ,\xi \varpi \right)+\mathfrak{D}\left(\varpi ,g\varpi \right)=0$ and $\xi \varpi =\varpi =g\varpi$. Thus, $C_{{\varpi }_0,r}$ is a common fixed circle of the pair $\left(\xi ,g\right)$. □

Remark 1.

If we take $\alpha =1$ and $\beta =\gamma =\delta =\eta =0$ in Definition 5, then we obtain the concept of a Wardowski-type ${\mathcal{F}}_{\xi g}$-contractive mapping.

Now, we give the concept of a Reich-type ${\mathcal{F}}_{\xi g}$-contraction as follows.

Definition 6.

If there exist $\tau >0,$ $\mathcal{F}\in \mathcal{F}$ and ${\varpi }_0\in X$ such that for all $\varpi \in X$, the following holds:

$$\tau +\mathcal{F}\left(\mathfrak{D}\left(\xi \varpi ,\varpi \right)+\mathfrak{D}\left(g\varpi ,\varpi \right)\right)\le \mathcal{F}\left(\begin{array}{c}\alpha \mathfrak{D}\left(\varpi ,{\varpi }_0\right)+\beta \left[\mathfrak{D}\left(\varpi ,\xi \varpi \right)+\mathfrak{D}\left(\varpi ,g\varpi \right)\right]\\ +\gamma \left[\mathfrak{D}\left({\varpi }_0,\xi {\varpi }_0\right)+\mathfrak{D}\left({\varpi }_0,g{\varpi }_0\right)\right]\end{array}\right)$$

(3)

with $min\left\{\mathfrak{D}\left(\xi \varpi ,\varpi \right),\mathfrak{D}\left(g\varpi ,\varpi \right)\right\}>0$, where $\alpha +\beta +\gamma <1,$ $\alpha \ne 0$ and $\alpha ,\beta ,\gamma \in \left[0,\infty \right)$. Then, the pair $\left(\xi ,g\right)$ is called a Reich-type ${\mathcal{F}}_{\xi g}$-contraction on X.

Proposition 5.

If the pair $\left(\xi ,g\right)$ is a Reich-type ${\mathcal{F}}_{\xi g}$-contraction with ${\varpi }_0\in X,$ then we have $\xi {\varpi }_0={\varpi }_0=g{\varpi }_0.$

Proof. 

Assume that $\xi {\varpi }_0\ne {\varpi }_0$ and $g{\varpi }_0\ne {\varpi }_0.$ From the definition of the Reich-type ${\mathcal{F}}_{\xi g}$-contraction with $min\left\{\mathfrak{D}\left({\varpi }_0,\xi {\varpi }_0\right),\mathfrak{D}\left({\varpi }_0,g{\varpi }_0\right)\right\}>0,$ we get

$$\begin{array}{ccc}\hfill \tau +\mathcal{F}\left(\mathfrak{D}\left({\varpi }_0,\xi {\varpi }_0\right)+\mathfrak{D}\left({\varpi }_0,g{\varpi }_0\right)\right)& \le & \mathcal{F}\left(\begin{array}{c}\alpha \mathfrak{D}\left({\varpi }_0,{\varpi }_0\right)+\beta \left[\mathfrak{D}\left({\varpi }_0,\xi {\varpi }_0\right)+\mathfrak{D}\left({\varpi }_0,g{\varpi }_0\right)\right]\\ +\gamma \left[\mathfrak{D}\left({\varpi }_0,\xi {\varpi }_0\right)+\mathfrak{D}\left({\varpi }_0,g{\varpi }_0\right)\right]\end{array}\right)\hfill \\ & =& \mathcal{F}\left(\left(\beta +\gamma \right)\left[\mathfrak{D}\left({\varpi }_0,\xi {\varpi }_0\right)+\mathfrak{D}\left({\varpi }_0,g{\varpi }_0\right)\right]\right)\hfill \\ & <& \mathcal{F}\left(\mathfrak{D}\left({\varpi }_0,\xi {\varpi }_0\right)+\mathfrak{D}\left({\varpi }_0,g{\varpi }_0\right)\right)\hfill \end{array}$$

a contradiction because of $\tau >0.$ Then, we have $\xi {\varpi }_0={\varpi }_0=g{\varpi }_0.$ □

Using Proposition 5, we rewrite the condition (3) as follows:

$$\tau +\mathcal{F}\left(\mathfrak{D}\left(\xi \varpi ,\varpi \right)+\mathfrak{D}\left(g\varpi ,\varpi \right)\right)\le \mathcal{F}\left(\alpha \mathfrak{D}\left(\varpi ,{\varpi }_0\right)+\beta \left[\mathfrak{D}\left(\varpi ,\xi \varpi \right)+\mathfrak{D}\left(\varpi ,g\varpi \right)\right]\right)$$

with $min\left\{\mathfrak{D}\left(\xi \varpi ,\varpi \right),\mathfrak{D}\left(g\varpi ,\varpi \right)\right\}>0$ where $\alpha +\beta <1,$ $\alpha \ne 0$ and $\alpha ,\beta \in \left[0,\infty \right).$

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

Theorem 5.

Let the pair $\left(\xi ,g\right)$ be a Reich-type ${\mathcal{F}}_{\xi g}$-contraction with ${\varpi }_0\in X$ and σ be defined as in (1). Then, $C_{{\varpi }_0,\sigma }$ is a common fixed circle of the pair $\left(\xi ,g\right)$. Especially, ξ and g fix every circle $C_{{\varpi }_0,\rho }$ with $\rho <\sigma .$

Proof. 

We distinguish two cases.

Case 1: Let $\sigma =0.$ Clearly, $C_{{\varpi }_0,\sigma }=\left\{{\varpi }_0\right\}$ and by Proposition 5, we see that $C_{{\varpi }_0,\sigma }$ is a common fixed circle of the pair $\left(\xi ,g\right)$.

Case 2: Let $\sigma >0$ and $\varpi \in C_{{\varpi }_0,\sigma .}$ This case can be easily seen since

$$\begin{array}{ccc}\hfill \mathcal{F}\left(\sigma \right)& \le & \mathcal{F}\left(\mathfrak{D}\left(\xi \varpi ,\varpi \right)+\mathfrak{D}\left(g\varpi ,\varpi \right)\right)\hfill \\ & \le & \mathcal{F}\left(\left(\alpha +\beta \right)\left[\mathfrak{D}\left(\xi \varpi ,\varpi \right)+\mathfrak{D}\left(g\varpi ,\varpi \right)\right]\right)\hfill \\ & <& \mathcal{F}\left(\mathfrak{D}\left(\xi \varpi ,\varpi \right)+\mathfrak{D}\left(g\varpi ,\varpi \right)\right).\hfill \end{array}$$

Consequently, $C_{{\varpi }_0,\sigma }$ is a common fixed circle of the pair $\left(\xi ,g\right)$. Especially, $\xi$ and g fix every circle $C_{{\varpi }_0,\rho }$ with $\rho <\sigma .$ □

To obtain, some new common fixed-circle results, we define the following contractions.

Definition 7.

If there exist $\tau >0,$ $\mathcal{F}\in \mathcal{F}$ and ${\varpi }_0\in X$ such that for all $\varpi \in X$, the following holds:

$$\tau +\mathcal{F}\left(\mathfrak{D}\left(\xi \varpi ,\varpi \right)+\mathfrak{D}\left(g\varpi ,\varpi \right)\right)\le \mathcal{F}\left(\eta \left[\mathfrak{D}\left(\xi \varpi ,{\varpi }_0\right)+\mathfrak{D}\left(g\varpi ,{\varpi }_0\right)\right]\right)$$

with $min\left\{\mathfrak{D}\left(\xi \varpi ,\varpi \right),\mathfrak{D}\left(g\varpi ,\varpi \right)\right\}>0$ where $\eta \in \left(0,\frac{1}{3}\right),$ then the pair $\left(\xi ,g\right)$ is called a Chatterjea-type ${\mathcal{F}}_{\xi g}$-contraction.

Proposition 6.

If the pair $\left(\xi ,g\right)$ is a Chattereja-type ${\mathcal{F}}_{\xi g}$-contraction with ${\varpi }_0\in X,$ then we have $\xi {\varpi }_0={\varpi }_0=g{\varpi }_0.$

Proof. 

From the similar arguments used in Proposition 4, it can be easily proved. □

Theorem 6.

Let the pair $\left(\xi ,g\right)$ be a Chatterjea-type ${\mathcal{F}}_{\xi g}$-contraction with ${\varpi }_0\in X$ and σ be defined as in (1). Then, $C_{{\varpi }_0,\sigma }$ is a common fixed circle of the pair $\left(\xi ,g\right)$. Especially, ξ and g fix every circle $C_{{\varpi }_0,\rho }$ with $\rho <\sigma .$

Proof. 

We distinguish two cases.

Case 1: Let $\sigma =0.$ Clearly, $C_{{\varpi }_0,\sigma }=\left\{{\varpi }_0\right\}$ and by Proposition 6, we see that $C_{{\varpi }_0,\sigma }$ is a common fixed circle of the pair $\left(\xi ,g\right)$.

Case 2: Let $\sigma >0$ and $\varpi \in C_{{\varpi }_0,\sigma .}$ Using the Chatterjea-type ${\mathcal{F}}_{\xi g}$-contractive property, the fact that $\mathcal{F}$ is increasing, and the triangle inequality property of metric function d, we have

$$\begin{array}{ccc}\hfill \mathcal{F}\left(\sigma \right)& \le & \mathcal{F}\left(\mathfrak{D}\left(\xi \varpi ,\varpi \right)+\mathfrak{D}\left(g\varpi ,\varpi \right)\right)\hfill \\ & \le & \mathcal{F}\left(\eta \left[\mathfrak{D}\left(\xi \varpi ,{\varpi }_0\right)+\mathfrak{D}\left(g\varpi ,{\varpi }_0\right)\right]\right)-\tau \hfill \\ & \le & \mathcal{F}\left(\eta \left[\mathfrak{D}\left(\xi \varpi ,\varpi \right)+\mathfrak{D}\left(\varpi ,{\varpi }_0\right)+\mathfrak{D}\left(g\varpi ,\varpi \right)+\mathfrak{D}\left(\varpi ,{\varpi }_0\right)\right]\right)\hfill \\ & =& \mathcal{F}\left(\eta \left[2\mathfrak{D}\left(\varpi ,{\varpi }_0\right)+\left[\mathfrak{D}\left(\xi \varpi ,\varpi \right)+\mathfrak{D}\left(g\varpi ,\varpi \right)\right]\right]\right)\hfill \\ & =& \mathcal{F}\left(3\eta \left[\mathfrak{D}\left(\xi \varpi ,\varpi \right)+\mathfrak{D}\left(g\varpi ,\varpi \right)\right]\right)\hfill \\ & <& \mathcal{F}\left(\mathfrak{D}\left(\xi \varpi ,\varpi \right)+\mathfrak{D}\left(g\varpi ,\varpi \right)\right).\hfill \end{array}$$

This gives a contradiction. Thus, $\mathfrak{D}\left(\xi \varpi ,\varpi \right)+\mathfrak{D}\left(g\varpi ,\varpi \right)=0,$ that is, $\xi \varpi =\varpi =g\varpi$. As a result, $C_{{\varpi }_0,\sigma }$ is a common fixed circle of the pair $\left(\xi ,g\right)$. By the similar arguments used in the proof of Theorem 1, $\xi$ and g also fix any circle $C_{{\varpi }_0,\rho }$ with $\rho <\sigma .$ □

Definition 8.

If there exist $\tau >0,$ $\mathcal{F}\in \mathcal{F}$ and ${\varpi }_0\in X$ such that for all $\varpi \in X$ the following holds:

$$\tau +\mathcal{F}\left(\mathfrak{D}\left(\xi \varpi ,\varpi \right)+\mathfrak{D}\left(g\varpi ,\varpi \right)\right)\le \mathcal{F}\left(\eta \left[\mathfrak{D}\left(\varpi ,\xi {\varpi }_0\right)+\mathfrak{D}\left(\varpi ,g{\varpi }_0\right)\right]\right)$$

(4)

with $min\left\{\mathfrak{D}\left(\xi \varpi ,\varpi \right),\mathfrak{D}\left(g\varpi ,\varpi \right)\right\}>0$ where $\eta \in \left(0,\frac{1}{2}\right),$ then the pair $\left(\xi ,g\right)$ is called a Kannan-type ${\mathcal{F}}_{\xi g}$-contraction.

Proposition 7.

If the pair $\left(\xi ,g\right)$ is a Kannan-type ${\mathcal{F}}_{\xi g}$-contraction with ${\varpi }_0\in X,$ then we have $\xi {\varpi }_0={\varpi }_0=g{\varpi }_0.$

Proof. 

From the similar arguments used in Proposition 4, it can be easily obtained. □

Theorem 7.

Let the pair $\left(\xi ,g\right)$ be a Kannan-type ${\mathcal{F}}_{\xi g}$-contraction with ${\varpi }_0\in X$ and σ be defined as in (1). Then, $C_{{\varpi }_0,\sigma }$ is a common fixed circle of the pair $\left(\xi ,g\right)$. Especially, ξ and g fix every circle $C_{{\varpi }_0,\rho }$ with $\rho <\sigma .$

Proof. 

We distinguish two cases.

Case 1: Let $\sigma =0.$ Clearly, $C_{{\varpi }_0,\sigma }=\left\{{\varpi }_0\right\}$ and by Proposition 7, we see that $C_{{\varpi }_0,\sigma }$ is a common fixed circle of the pair $\left(\xi ,g\right)$.

Case 2: Let $\sigma >0$ and $\varpi \in C_{{\varpi }_0,\sigma .}$ Using the Kannan-type ${\mathcal{F}}_{\xi g}$-contractive property, the fact that $\mathcal{F}$ is increasing, and the triangle inequality property of metric function d, we have

$$\begin{array}{ccc}\hfill \mathcal{F}\left(\sigma \right)& \le & \mathcal{F}\left(\mathfrak{D}\left(\xi \varpi ,\varpi \right)+\mathfrak{D}\left(g\varpi ,\varpi \right)\right)\hfill \\ & \le & \mathcal{F}\left(\eta \left[\mathfrak{D}\left(\varpi ,\xi {\varpi }_0\right)+\mathfrak{D}\left(\varpi ,g{\varpi }_0\right)\right]\right)-\tau \hfill \\ & \le & \mathcal{F}\left(\eta \left[\mathfrak{D}\left(\varpi ,{\varpi }_0\right)+\mathfrak{D}\left(\varpi ,{\varpi }_0\right)\right]\right)\hfill \\ & \le & \mathcal{F}\left(2\eta \sigma \right)\hfill \\ & <& \mathcal{F}\left(\mathfrak{D}\left(\xi \varpi ,\varpi \right)+\mathfrak{D}\left(g\varpi ,\varpi \right)\right).\hfill \end{array}$$

This gives a contradiction. Thus, $\mathfrak{D}\left(\xi \varpi ,\varpi \right)+\mathfrak{D}\left(g\varpi ,\varpi \right)=0,$ that is, $\xi \varpi =\varpi =g\varpi$. As a result, $C_{{\varpi }_0,\sigma }$ is a common fixed circle of the pair $\left(\xi ,g\right)$. By similar arguments used in the proof of Theorem 1, $\xi$ and g also fix any circle $C_{{\varpi }_0,\rho }$ with $\rho <\sigma .$ □

Now, we present an illustrative example of our obtained results.

Example 4.

Let $X=\left\{1,2,e^2,e^2-1,e^2+1\right\}$ be the metric space with the usual metric. Let us define the self-mappings $\xi ,g:X⟶X$ as

$$\xi \varpi =\left\{\begin{array}{cc}2,& \varpi =1\\ \varpi ,& otherwise\end{array}\right.$$

and

$$g\varpi =\left\{\begin{array}{cc}2,& \varpi =1\\ \varpi ,& otherwise\end{array}\right.,$$

for all $\varpi \in X.$

The pair $\left(\xi ,g\right)$ is a Hardy–Rogers-type ${\mathcal{F}}_{\xi g}$-contraction with $\mathcal{F}=\ln a+a,$ $\tau =0.01,$ $\alpha =\beta =\gamma =\frac{1}{4}$ and ${\varpi }_0=e^2.$ Indeed, we get

$$min\left\{\mathfrak{D}\left(\varpi ,\xi \varpi \right),\mathfrak{D}\left(\varpi ,g\varpi \right)\right\}=min\left\{\mathfrak{D}\left(1,2\right),\mathfrak{D}\left(1,2\right)\right\}=1>0$$

for $\varpi =1$ and we get

$$\begin{array}{ccc}\hfill \alpha \mathfrak{D}\left(\varpi ,{\varpi }_0\right)+\beta \mathfrak{D}\left(\varpi ,\xi \varpi \right)+\gamma \mathfrak{D}\left(\varpi ,g\varpi \right)& =& \frac{1}{4}\left[\mathfrak{D}\left(1,e^2\right)+\mathfrak{D}\left(1,2\right)+\mathfrak{D}\left(1,2\right)\right]\hfill \\ & =& \frac{1}{4}\left[e^2-1+1+1\right]\hfill \\ & =& \frac{e^2+1}{4}.\hfill \end{array}$$

Then, we have

$$\begin{array}{ccc}\hfill \tau +\mathcal{F}\left(\mathfrak{D}\left(\varpi ,\xi \varpi \right)+\mathfrak{D}\left(\varpi ,g\varpi \right)\right)& =& 0.01+ln2+2\hfill \\ & \le & \mathcal{F}\left(\frac{e^2+1}{4}\right)\hfill \\ & =& ln\left(e^2+1\right)-ln4+\frac{e^2+1}{4}.\hfill \end{array}$$

The pair $\left(\xi ,g\right)$ is a Reich-type ${\mathcal{F}}_{\xi g}$-contraction with $\mathcal{F}=lna,$ $\tau =ln\left(e^2+1\right)-ln6,$ $\alpha =\beta =\frac{1}{3}$ and ${\varpi }_0=e^2.$ Indeed, we get

$$min\left\{\mathfrak{D}\left(\varpi ,\xi \varpi \right),\mathfrak{D}\left(\varpi ,g\varpi \right)\right\}=min\left\{\mathfrak{D}\left(1,2\right),\mathfrak{D}\left(1,2\right)\right\}=1>0$$

for $\varpi =1$ and we have

$$\begin{array}{ccc}\hfill \alpha \mathfrak{D}\left(\varpi ,{\varpi }_0\right)+\beta \left[\mathfrak{D}\left(\varpi ,\xi \varpi \right)+\mathfrak{D}\left(\varpi ,g\varpi \right)\right]& =& \frac{1}{3}\mathfrak{D}\left(1,e^2\right)+\frac{1}{3}\left[\mathfrak{D}\left(1,2\right)+\mathfrak{D}\left(1,2\right)\right]\hfill \\ & =& \frac{e^2+1}{3}.\hfill \end{array}$$

Then, we obtain

$$\begin{array}{ccc}\hfill \tau +\mathcal{F}\left(\mathfrak{D}\left(\varpi ,\xi \varpi \right)+\mathfrak{D}\left(\varpi ,g\varpi \right)\right)& =& ln\left(e^2+1\right)-ln6+ln2\hfill \\ & \le & \mathcal{F}\left(\frac{e^2+1}{3}\right)\hfill \\ & =& ln(e^2+1)-ln3.\hfill \end{array}$$

The pair $\left(\xi ,g\right)$ is both a Chatterjea-type ${\mathcal{F}}_{\xi g}$-contractions and a Kannan-type ${\mathcal{F}}_{\xi g}$-contraction with $\mathcal{F}=lna,$ $\tau =ln\left(e^2-2\right)-ln4,$ $\eta =\frac{1}{4}$ and ${\varpi }_0=e^2.$ Indeed, for Chatterjea-type ${\mathcal{F}}_{\xi g}$-contractions, we get

$$min\left\{\mathfrak{D}\left(\varpi ,\xi \varpi \right),\mathfrak{D}\left(\varpi ,g\varpi \right)\right\}=min\left\{\mathfrak{D}\left(1,2\right),\mathfrak{D}\left(1,2\right)\right\}=1>0$$

for $\varpi =1$ and we have

$$\begin{array}{ccc}\hfill \eta \left[\mathfrak{D}\left({\varpi }_0,\xi \varpi \right)+\mathfrak{D}\left({\varpi }_0,g\varpi \right)\right]& =& \frac{1}{4}\left[\mathfrak{D}\left(e^2,2\right)+\mathfrak{D}\left(e^2,2\right)\right]\hfill \\ & \le & \frac{1}{4}\left[2(e^2-2)\right]\hfill \\ & =& \frac{e^2-2}{2}.\hfill \end{array}$$

Then, we obtain

$$\begin{array}{ccc}\hfill \tau +\mathcal{F}\left(\mathfrak{D}\left(\varpi ,\xi \varpi \right)+\mathfrak{D}\left(\varpi ,g\varpi \right)\right)& =& ln\left(e^2-2\right)-ln4+ln2\hfill \\ & \le & \mathcal{F}\left(\frac{e^2-2}{2}\right)\hfill \\ & =& ln\left(e^2-2\right)-ln2.\hfill \end{array}$$

For Kannan-type ${\mathcal{F}}_{\xi g}$-contractions, we have

$$min\left\{\mathfrak{D}\left(\varpi ,\xi \varpi \right),\mathfrak{D}\left(\varpi ,g\varpi \right)\right\}=min\left\{\mathfrak{D}\left(1,2\right),\mathfrak{D}\left(1,2\right)\right\}=1>0$$

for $\varpi =1$ and we have

$$\begin{array}{ccc}\hfill \eta \left[\mathfrak{D}\left(\varpi ,\xi {\varpi }_0\right)+\mathfrak{D}\left(\varpi ,g_0\right)\right]& =& \frac{1}{4}\left[\mathfrak{D}\left(1,e^2\right)+\mathfrak{D}\left(1,e^2\right)\right]\hfill \\ & \le & \frac{1}{4}\left[2(e^2-1)\right]\hfill \\ & =& \frac{e^2-1}{2}.\hfill \end{array}$$

Then, we obtain

$$\begin{array}{ccc}\hfill \tau +\mathcal{F}\left(\mathfrak{D}\left(\varpi ,\xi \varpi \right)+\mathfrak{D}\left(\varpi ,g\varpi \right)\right)& =& ln\left(e^2-2\right)-ln4+ln2\hfill \\ & \le & \mathcal{F}\left(\frac{e^2-1}{2}\right)\hfill \\ & =& ln\left(e^2-1\right)-ln2.\hfill \end{array}$$

Consequently, $\xi$ and g fix the circle $C_{e^2,1}=\{e^2-1,e^2+1\}.$

If we combine the notions of Banach-type ${\mathcal{F}}_{\xi g}$-contractions, Chatterjea-type ${\mathcal{F}}_{\xi g}$-contractions, and Kannan-type ${\mathcal{F}}_{\xi g}$-contractions, then we get the following corollary. This corollary can be considered as Zamfirescu-type common fixed-circle result [18].

Corollary 1.

Let $\left(X,\mathfrak{D}\right)$ be a metric space, $\xi ,g:X⟶X$ be two self-mappings and σ be defined as in (1). If there exist $\tau >0,$ $\mathcal{F}\in \mathcal{F}$ and ${\varpi }_0\in X$ such that for all $\varpi \in X$, at least one of the followings holds:

$\left(1\right)$$\tau +\mathcal{F}\left(\mathfrak{D}\left(\xi \varpi ,\varpi \right)+\mathfrak{D}\left(g\varpi ,\varpi \right)\right)\le \mathcal{F}\left(\alpha \mathfrak{D}\left(\varpi ,{\varpi }_0\right)\right)$,

$\left(2\right)$$\tau +\mathcal{F}\left(\mathfrak{D}\left(\xi \varpi ,\varpi \right)+\mathfrak{D}\left(g\varpi ,\varpi \right)\right)\le \mathcal{F}\left(\beta \left[\mathfrak{D}\left(\xi \varpi ,{\varpi }_0\right)+\mathfrak{D}\left(g\varpi ,{\varpi }_0\right)\right]\right)$,

$\left(3\right)$$\tau +\mathcal{F}\left(\mathfrak{D}\left(\xi \varpi ,\varpi \right)+\mathfrak{D}\left(g\varpi ,\varpi \right)\right)\le \mathcal{F}\left(\gamma \left[\mathfrak{D}\left(\varpi ,\xi {\varpi }_0\right)+\mathfrak{D}\left(\varpi ,g{\varpi }_0\right)\right]\right),$

with $min\left\{\mathfrak{D}\left(\xi \varpi ,\varpi \right),\mathfrak{D}\left(g\varpi ,\varpi \right)\right\}>0$ where $0\le \alpha <1$, $0\le \beta ,\gamma <\frac{1}{2},$ then $C_{{\varpi }_0,\sigma }$ is a common fixed circle of the pair $\left(\xi ,g\right)$. Especially, ξ and g fix every circle $C_{{\varpi }_0,\rho }$ with $\rho <\sigma .$

Proof. 

It is obvious. □