Some Fixed-Disc Results in Double Controlled Quasi-Metric Type Spaces

Abstract

In this paper, we introduce new types of general contractions for self mapping on double controlled quasi-metric type spaces, where we prove the existence and uniqueness of fixed disc and circle for such mappings.

1. Introduction

Lately, proving the existence of a fixed circle or a fixed disk on metric spaces or on a generalized form of a metric space has been the focus of many researchers (see [1,2,3,4,5]). For instance, in [1], they proved the existence of a fixed circle for the Caristi-type contraction on a regular metric spaces. Further, adopting the techniques of the Wardowski contractive mapping, the authors in [5] were able to prove some interesting fixed-circle theorems. In [2,3], the existence of a fixed-circle problem was investigated on the three dimensions of space, the so-called S-metric space. In [6], the authors also proved some new fixed-circle results for mappings that satisfies the modified Khan-type contraction on S-metric spaces. Some generalized fixed-circle results with geometric viewpoint were obtained on $S_b$-metric spaces and parametric $N_b$-metric spaces (see [7,8]). Moreover, it was an open question to study the existence of a fixed circle on extended $M_b$-metric spaces [9]. Further, an application of the obtained fixed-circle results was given to discontinuous activation functions on metric spaces (see [1,4,10]).

One of the most useful tools to show that a fractional differential equation has a solution is a fixed-point theory, see [11,12]; however, the existence of a fixed point leads us to conclude that such types of equations have a solution, but in some cases a map has a fixed point but it is not necessarily unique. So, in a case where we have more than one fixed point, what can we say about the set of a fixed point? In this manuscript, we are interested in the type of mapping that the set of fixed points is a disc or a circle. Of course the shape of a circle or a disc varies from one metric space to another. For our purposes, we study the existence and uniqueness of fixed circle and disc in double controlled quasi-metric type spaces. In the next section, we present some preliminaries.

2. Preliminaries

Definition 1

([13]). Consider the set $\mathbb{B}\ne \varnothing .$ Given non-comparable functions $\mathbb{K},\mathbb{L}:\mathbb{B}\times \mathbb{B}\to [1,\infty )$. If $\mathbb{M}:\mathbb{B}\times \mathbb{B}\to [0,\infty )$ satisfies

$$\begin{array}{c}\left(\mathbb{M}1\right) \mathbb{M}(x,\upsilon )=0\iff x=\upsilon \hfill \\ \left(\mathbb{M}2\right) \mathbb{M}(x,\upsilon )\le \mathbb{K}(x,\beta )\mathbb{M}(x,\beta )+\mathbb{L}(\beta ,\upsilon )\mathbb{M}(\beta ,\upsilon ),\hfill \end{array}$$

for all $x,\upsilon ,\beta \in \mathbb{B}$. Then $\mathbb{M}$ is called a double controlled quasi-metric type with the functions $\mathbb{K}$, $\mathbb{L}$ and $\left(\mathbb{B},\mathbb{M}\right)$ is a double controlled quasi-metric type space.

Throughout the rest of this manuscript we denote double controlled quasi-metric type space by $\left(DCQMS\right).$

Example 1

([13]).Let $\mathbb{B}=\{0,1,2\}$. Define $\mathbb{M}:\mathbb{B}\times \mathbb{B}\to [0,\infty )$ by

$$\begin{array}{ccc}\hfill \mathbb{M}(0,1)& =& 4,\mathbb{M}(0,2)=1,\hfill \\ \hfill \mathbb{M}(1,0)& =& \mathbb{M}(1,2)=3,\hfill \\ \hfill \mathbb{M}(2,0)& =& 0,\mathbb{M}(2,1)=2,\hfill \\ \hfill \mathbb{M}(0,0)& =& \mathbb{M}(1,1)=\mathbb{M}(2,2)=0.\hfill \end{array}$$

Then $\mathbb{M}$ is a double controlled quasi-metric type with the functions $\mathbb{K},\mathbb{L}:\mathbb{B}\times \mathbb{B}\to [1,\infty )$ defined as

$$\begin{array}{ccc}\hfill \mathbb{K}(0,1)& =& \mathbb{K}(1,0)=\mathbb{K}(1,2)=1,\hfill \\ \hfill \mathbb{K}(0,2)& =& \frac{5}{4},\mathbb{K}(2,0)=\frac{10}{9},\mathbb{K}(2,1)=\frac{20}{19},\hfill \\ \hfill \mathbb{K}(0,0)& =& \mathbb{K}(1,1)=\mathbb{K}(2,2)=1\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \mathbb{L}(0,1)& =& \mathbb{L}(1,0)=\mathbb{L}(0,2)=\mathbb{L}(1,2)=1,\hfill \\ \hfill \mathbb{L}(2,0)& =& \frac{3}{2},\mathbb{L}(2,1)=\frac{11}{8},\hfill \\ \hfill \mathbb{L}(0,0)& =& \mathbb{L}(1,1)=\mathbb{L}(2,2)=1.\hfill \end{array}$$

Throughout this paper, we denote by $\mathbb{R}$ the set of all real numbers, and $\mathbb{N}$ represents the set of all positive integers.

Example 2.

Let $\mathbb{B}=l_1$ be defined by

$$l_1=\left\{{\left\{{\varpi }_n\right\}}_{n\ge 1}\subset \mathbb{R}:\sum _{n=1}^{\infty }\left|{\varpi }_n\right|<\infty \right\}.$$

Consider $\mathbb{M}:\mathbb{B}\times \mathbb{B}\to [0,\infty )$ such that

$$\mathbb{M}(\eta ,\varpi )=\sum _{n=1}^{\infty }{({\varpi }_n-{\eta }_n)}^{+}$$

where ${\alpha }^{+}:=max\{\alpha ,0\}$ denotes the positive part of a number $\alpha \in \mathbb{R}$., and $\varpi =\left\{{\varpi }_n\right\}$ and $\eta =\left\{{\eta }_n\right\}$ are in $\mathbb{B}$. Further, let $\mathbb{K}(\varpi ,\eta )=max\{\varpi ,\eta \}+2$ and $\mathbb{L}(\varpi ,\eta )=max\{\varpi ,\eta \}+3.$

Note that $(\mathbb{B},\mathbb{M})$ is a $\left(DCQMS\right)$ with control functions $\mathbb{K},\mathbb{L}.$

Now, we remind the reader of the topological properties of $\left(DCQMS\right).$

Definition 2.

Let $(\mathbb{B},\mathbb{M})$ be a $\left(DCQMS\right),$$\left\{{\varpi }_n\right\}$ be a sequence in $\mathbb{B}$ and $\varpi \in \mathbb{B}$. The sequence $\left\{{\varpi }_n\right\}$ converges to ϖ⇔

$$\underset{n\to \infty }{lim}\mathbb{M}({\varpi }_n,\varpi )=\underset{n\to \infty }{lim}\mathbb{M}(\varpi ,{\varpi }_n)=0.$$

(1)

Remark 1.

In a $\left(DCQMS\right),$$(\mathbb{B},\mathbb{M})$, the limit for a convergent sequence is unique. Further, if ${\varpi }_n\to \varpi$, we have for all $\eta \in \mathbb{B}$

$$\underset{n\to \infty }{lim}\mathbb{M}({\varpi }_n,\eta )=\mathbb{M}(\varpi ,\eta )\mathrm{and}\underset{n\to \infty }{lim}\mathbb{M}(\eta ,{\varpi }_n)=\mathbb{M}(\eta ,\varpi ).$$

In fact, ${\varpi }_n\to \varpi \mathrm{and}{\eta }_n\to \eta \Rightarrow \mathbb{M}({\varpi }_n,{\eta }_n)\to \mathbb{M}(\varpi ,\eta ).$

Definition 3

([14]).  Let $(\mathbb{B},\mathbb{M})$ be a $\left(DCQMS\right),$ and $\left\{{\varpi }_n\right\}$ be a sequence in $\mathbb{B}$. We say that $\left\{{\varpi }_n\right\}$ is right DCQ-Cauchy ⇔ for every $\epsilon >0$, there exists a positive integer $N=N\left(\epsilon \right)$ such that $\mathbb{M}({\varpi }_n,{\varpi }_m)<\epsilon$ for all $n\ge m>N$.

Definition 4

([14]).Let $(\mathbb{B},\mathbb{M})$ be a $\left(DCQMS\right),$ and $\left\{{\varpi }_n\right\}$ be a sequence in $\mathbb{B}$. We say that $\left\{{\varpi }_n\right\}$ is left DCQ-Cauchy ⇔ for every $\epsilon >0$, there exists a positive integer $N=N\left(\epsilon \right)$ such that $\mathbb{M}({\varpi }_n,{\varpi }_m)<\epsilon$ for all $m\ge n>N$.

Definition 5.

Let $(\mathbb{B},\mathbb{M})$ be a $\left(DCQMS\right),$ and $\left\{{\varpi }_n\right\}$ be a sequence in $\mathbb{B}$. We say that $\left\{{\varpi }_n\right\}$ is DCQ-Cauchy ⇔ for every $\epsilon >0$, there exists a positive integer $N=N\left(\epsilon \right)$ such that $\mathbb{M}({\varpi }_n,{\varpi }_m)<\epsilon$ for all $m,n>N$.

Remark 2.

A sequence $\left\{{\varpi }_n\right\}$ in a $\left(DCQMS\right),$ is DCQ-Cauchy ⇔ it is right DCQ-Cauchy and left DCQ-Cauchy.

Definition 6

([15]).Let $(\mathbb{B},\mathbb{M})$$DCQMS.$

$\left(1\right)$$(\mathbb{B},\mathbb{M})$ is said left-complete ⇔ each left DCQ-Cauchy sequence in $\mathbb{B}$ is convergent.

$\left(2\right)$$(\mathbb{B},\mathbb{M})$ is said right-complete ⇔ each right DCQ-Cauchy sequence in $\mathbb{B}$ is convergent.

$\left(3\right)$$(\mathbb{B},\mathbb{M})$ is said complete ⇔ each DCQ-Cauchy sequence in $\mathbb{B}$ is convergent.

Remark 3.

If $\mathbb{M}$ is a $DCQMS$ on $\mathbb{B},$ then $\overline{\mathbb{M}}(x,\upsilon )=\mathbb{M}(\upsilon ,x)$ for all $x,\upsilon \in \mathbb{B}$ is another quasi-metric, called the conjugate of $\mathbb{M}$ and ${\mathbb{M}}^s(x,\upsilon )=max\{\mathbb{M}(x,\upsilon ),\overline{\mathbb{M}}(x,\upsilon )\}$ for all $x,\upsilon \in \mathbb{B}$ is a metric on $\mathbb{B}.$ Moreover, we have

• ${\varpi }_n{\to }_{\mathbb{M}}\varpi \iff {lim}_{n\to \infty }\mathbb{M}(\varpi ,{\varpi }_n)=0;$
• ${\varpi }_n{\to }_{\overline{\mathbb{M}}}\varpi \iff {lim}_{n\to \infty }\overline{\mathbb{M}}(\varpi ,{\varpi }_n)=0\iff {lim}_{n\to \infty }\mathbb{M}({\varpi }_n,\varpi )=0.$

Further, note that

$$\begin{array}{cc}\hfill {\varpi }_n{\to }_{{\mathbb{M}}^s}\varpi & \iff \underset{n\to \infty }{lim}\mathbb{M}(\varpi ,{\varpi }_n)=0\mathrm{and}\underset{n\to \infty }{lim}\mathbb{M}({\varpi }_n,\varpi )=0\hfill \\ \hfill & \iff {\varpi }_n{\to }_{\mathbb{M}}\varpi \mathrm{and}{\varpi }_n{\to }_{\overline{\mathbb{M}}}\varpi .\hfill \end{array}$$

Hence, ${\varpi }_n{\to }_{\mathbb{M}}\varpi \mathrm{implies}{\varpi }_n{\to }_{{\mathbb{M}}^s}\varpi .$

Lemma 1.

Let $(\mathbb{B},\mathbb{M})$ be a $DCQMS$ and $\mathbb{W}:\mathbb{B}\to \mathbb{B}$ be a self-mapping. Suppose that $\mathbb{W}$ is continuous at $\varpi \in \mathbb{B}$. Then for each sequence $\left\{{\varpi }_n\right\}$ in $\mathbb{B}$ such that ${\varpi }_n\to \varpi$, we have $\mathbb{W}{\varpi }_n\to \mathbb{W}\varpi$, that is,

$$\underset{n\to \infty }{lim}\mathbb{M}(\mathbb{W}{\varpi }_n,\mathbb{W}\varpi )=\underset{n\to \infty }{lim}\mathbb{M}(\mathbb{W}\varpi ,\mathbb{W}{\varpi }_n)=0.$$

3. Main Results

One way to generalize the fixed-point results is to study the geometric properties of the set of fixed points when we do not have a unique fixed point. Let $(\mathbb{B},\mathbb{M})$ be a $DCQMS$, $x_0\in \mathbb{B}$ and $r>0$. The upper closed ball of radius r centered $x_0$ and the lower closed ball of radius r centered $x_0$ are defined by,

$$\overline{B^{+}}(x_0,r)=\left\{x\in \mathbb{B}:\mathbb{M}(x,x_0)\le r\right\}$$

and

$$\overline{B^{-}}(x_0,r)=\left\{x\in \mathbb{B}:\mathbb{M}(x_0,x)\le r\right\},$$

Next, we present the definitions of a circle and a disc on a $DCQMS$$(\mathbb{B},\mathbb{M})$: Let $r\ge 0$ and $x_0\in \mathbb{B}$. The circle $C_{x_0,r}^{\mathbb{M}}$ and the disc $D_{x_0,r}^{\mathbb{M}}$ are

$$C_{x_0,r}^{\mathbb{M}}=\left\{x\in \mathbb{B}:\mathbb{M}(x_0,x)=\mathbb{M}(x,x_0)=r\right\}$$

and

$$D_{x_0,r}^{\mathbb{M}}=\overline{B^{+}}(x_0,r)\cap \overline{B^{-}}(x_0,r)=\left\{x\in \mathbb{B}:\mathbb{M}(x_0,x)\le r\mathrm{and}\mathbb{M}(x,x_0)\le r\right\}.$$

Notice that the disc $D_{x_0,r}^{\mathbb{M}}$ form a closed ball with respect to the associated metric ${\mathbb{M}}^s$. That is,

$$\mathbb{M}(x_0,x)\le r\mathrm{and}\mathbb{M}(x,x_0)\le r\iff max\left\{\mathbb{M}(x_0,x),\mathbb{M}(x,x_0)\right\}\le r\iff {\mathbb{M}}^s(x,x_0)\le r.$$

Let $(\mathbb{B},\mathbb{M})$ be a $DCQMS$ and $\mathbb{W}$ be a self-mapping on $\mathbb{B}$. Further, let

$$r=inf\left\{\mathbb{M}\right(x,\mathbb{W}x)\mid x\in \mathbb{B},\mathbb{W}x\ne x\}.$$

(2)

3.1. DCQMS-$F_{\mathbb{M}}$-Contractions

In [16], Wardowski defined a new class of functions as follows.

Definition 7

([16]).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

$$\underset{n\to \infty }{lim}{\alpha }_n=0\iff \underset{n\to \infty }{lim}F\left({\alpha }_n\right)=-\infty ;$$

$\left(F_3\right)$ There exists $k\in (0,1)$ such that $\underset{\alpha \to 0^{+}}{lim}{\alpha }^{k}F\left(\alpha \right)=0$.

Next, we presnt the following contractive type of mappings.

Definition 8.

Let $(\mathbb{B},\mathbb{M})$ be a $DCQMS$, $\mathbb{W}$ a self-mapping on $\mathbb{B}$ and $F\in \mathbb{F}$. Then $\mathbb{W}$ is said to be a $DCQMS$-$F_{\mathbb{M}}$-contraction if there exist $t>0$ and $x_0\in \mathbb{B}$ such that

$$\mathbb{M}(x,\mathbb{W}x)>0\Rightarrow t+F\left(\mathbb{M}(x,\mathbb{W}x)\right)\le F\left(\mathbb{M}(x_0,x)\right),$$

(3)

for each $x\in \mathbb{B}$.

Denote the set of fixed-points of a map $\mathbb{W}$ by $Fix\left(\mathbb{W}\right)$.

Theorem 1.

Let $(\mathbb{B},\mathbb{M})$ be a $DCQMS$, $\mathbb{W}$ be a $DCQMS$-$F_{\mathbb{M}}$-contraction with $x_0\in \mathbb{B}$ on $\mathbb{B}$ and r defined as in (2). Then we have $\overline{B^{-}}(x_0,r)\subseteq Fix\left(\mathbb{W}\right)$, in particular $\mathbb{W}$ fixes the disc $D_{x_0,r}^{\mathbb{M}}$.

Proof. 

First of all, we show that $x_0$ is a fixed point of $\mathbb{W}$. Assume that $\mathbb{M}(x_0,\mathbb{W}{x}_0)>0$. By the quasi-$F_{\mathbb{M}}$-contractive property of $\mathbb{W}$ we deduce that

$$t+F\left(\mathbb{M}(x_0,\mathbb{W}{x}_0)\right)\le F\left(\mathbb{M}(x_0,x_0)\right).$$

Thus, $F\left(\mathbb{M}(x_0,\mathbb{W}{x}_0)\right)<F\left(0\right),$ which leads to a contradiction and that is due to the fact that F is strictly increasing. Thus, we arrive at $\mathbb{W}{x}_0=x_0$.

If $r=0$ then we obtain $\overline{B^{-}}(x_0,r)=D_{x_0,r}^{\mathbb{M}}=\left\{x_0\right\}$ and clearly, $\mathbb{W}$ fixes the center of the disc $D_{x_0,r}^{\mathbb{M}}$ and the whole disc $D_{x_0,r}^{\mathbb{M}}$.

Let $r>0$ and $x\in \overline{B^{-}}(x_0,r)$ with $\mathbb{W}x\ne x$. By the definition of r, we have $r\le \mathbb{M}(x,\mathbb{W}x)$. Hence, by the $DCQMS$-$F_{\mathbb{M}}$-contractive property, there exist $F\in \mathbb{F}$, $t>0$ and $x_0\in \mathbb{B}$ such that

$$t+F\left(\mathbb{M}(x,\mathbb{W}x)\right)\le F\left(\mathbb{M}(x_0,x)\right)\le F\left(r\right)\le F\left(\mathbb{M}(x,\mathbb{W}x)\right),$$

for all $x\in \mathbb{B}$ which leads us to a contradiction; therefore, we deduce that $\mathbb{W}x=x$, hence $\overline{B^{-}}(x_0,r)\subseteq Fix\left(\mathbb{W}\right),$ in particular $\mathbb{W}$ fixes the disc $D_{x_0,r}^{\mathbb{M}}$. □

Now, we introduce a new rational type contractive condition.

Definition 9.

Let $(\mathbb{B},\mathbb{M})$ be a $DCQMS$, $\mathbb{W}$ a self-mapping on $\mathbb{B}$ and $F\in \mathbb{F}.$ Then $\mathbb{W}$ is said to be $DCQMS$-$F_{\mathbb{M}}$-rational contraction if there exist $t>0$ and $x_0\in \mathbb{B}$ such that

$$\mathbb{M}(x,\mathbb{W}x)>0\Rightarrow t+F\left(\mathbb{M}(x,\mathbb{W}x)\right)\le F\left(M_R^{\mathbb{M}}(x_0,x)\right),$$

(4)

for all $x\in \mathbb{B}$, where

$$M_R^{\mathbb{M}}(x,\upsilon )=max\left\{\begin{array}{c}\mathbb{M}(x,\upsilon ),\mathbb{M}(x,\mathbb{W}x),\mathbb{M}(y,\mathbb{W}y),\\ \frac{\mathbb{M}(x,\mathbb{W}x)\mathbb{M}(y,\mathbb{W}y)}{1+\mathbb{M}(x,\upsilon )},\frac{\mathbb{M}(x,\mathbb{W}x)\mathbb{M}(y,\mathbb{W}y)}{1+\mathbb{M}(\mathbb{W}x,\mathbb{W}y)}\end{array}\right\}.$$

Theorem 2.

Let $(\mathbb{B},\mathbb{M})$ be a $DCQMS$, $\mathbb{W}$ a $DCQMS$-$F_{\mathbb{M}}$-rational contraction self-mapping with $x_0\in \mathbb{B}$ on $\mathbb{B}$, $\mathbb{W}{x}_0=x_0$ and r defined as in (2). Then we have $\overline{B^{-}}(x_0,r)\subseteq Fix\left(\mathbb{W}\right)$, in particular $\mathbb{W}$ fixes the disc $D_{x_0,r}^{\mathbb{M}}$.

Proof. 

Suppose that $r=0$. So we have $\overline{B^{-}}(x_0,r)=D_{x_0,r}^{\mathbb{M}}=\left\{x_0\right\}$. Using the hypothesis $\mathbb{W}{x}_0=x_0$, $\mathbb{W}$ fixes the disc $D_{x_0,r}^{\mathbb{M}}$.

Let $r>0$ and $x\in \overline{B^{-}}(x_0,r)$ with $\mathbb{W}x\ne x$. By the definition of r, we have $r\le \mathbb{M}(x,\mathbb{W}x)$. Because of the $DCQMS$-$F_{\mathbb{M}}$-rational contractive property, there exist $F\in \mathbb{F}$, $t>0$ and $x_0\in \mathbb{B}$ such that

$$t+F\left(\mathbb{M}(x,\mathbb{W}x)\right)\le F\left(M_R^{\mathbb{M}}(x_0,x)\right),$$

for all $x\in \mathbb{B}$. Then we obtain

$$\begin{array}{ccc}\hfill t+F\left(\mathbb{M}\right(x,\mathbb{W}x\left)\right)& \le & F\left(M_R^{\mathbb{M}}(x_0,x)\right)\hfill \\ & =& F\left(max\left\{\begin{array}{c}\mathbb{M}(x_0,x),\mathbb{M}(x_0,\mathbb{W}{x}_0),\mathbb{M}(x,\mathbb{W}x),\\ \frac{\mathbb{M}(x_0,\mathbb{W}{x}_0)\mathbb{M}(x,\mathbb{W}x)}{1+\mathbb{M}(x_0,x)},\frac{\mathbb{M}(x_0,\mathbb{W}{x}_0)\mathbb{M}(x,\mathbb{W}x)}{1+\mathbb{M}(\mathbb{W}{x}_0,\mathbb{W}x)}\end{array}\right\}\right)\hfill \\ & \le & F(max\{r,\mathbb{M}(x,\mathbb{W}x)\left\}\right)=F\left(\mathbb{M}\right(x,\mathbb{W}x\left)\right),\hfill \end{array}$$

a contradiction. Hence it should be $\mathbb{W}x=x$. Consequently, $\overline{B^{-}}(x_0,r)\subseteq Fix\left(\mathbb{W}\right),$ in particular $\mathbb{W}$ fixes the disc $D_{x_0,r}^{\mathbb{M}}$. □

3.2. DCQMS-$\alpha$-$x_0$-Contractive Type Mappings

First, we present the definition of an $x_0$-contractive mapping in $DCQMS$.

Definition 10.

Let $(\mathbb{B},\mathbb{M})$ be a $DCQMS$, $\mathbb{W}$ a self-mapping on $\mathbb{B}$ and $0<k<1$. Then $\mathbb{W}$ is said to be a $DCQMS$-$x_0$-contractive mapping if there exist $x_0\in \mathbb{B}$ such that

$$\mathbb{M}(x,\mathbb{W}x)\le k\mathbb{M}(x_0,x),$$

(5)

for every $x\in \mathbb{B}$.

Clearly, $x_0$ is always a fixed point of $\mathbb{W}$ in Definition 10. Now, we show that if $\mathbb{W}$ is a $DCQMS$-$x_0$-contractive mapping, then $Fix\left(\mathbb{W}\right)$ contains a disc.

Theorem 3.

Let $(\mathbb{B},\mathbb{M})$ be a $DCQMS$, $\mathbb{W}$ a $DCQMS$-$x_0$-contractive self-mapping with $x_0\in \mathbb{B}$ on $\mathbb{B}$ and r defined as in (2). Then we have $\overline{B^{-}}(x_0,r)\subseteq Fix\left(\mathbb{W}\right),$ in particular $\mathbb{W}$ fixes the disc $D_{x_0,r}^{\mathbb{M}}$.

Proof. 

In the case $r=0$, it is clear that $\overline{B^{-}}(x_0,r)=D_{x_0,r}^{\mathbb{M}}=\left\{x_0\right\}$ is a fixed disc of $\mathbb{W}$.

Suppose that $r>0.$ Let $x\in \overline{B^{-}}(x_0,r)$ be such that $\mathbb{W}x\ne x$. By the definition of r, we have $r\le \mathbb{M}(x,\mathbb{W}x).$ On the other hand, using the $DCQMS$-$x_0$-contractive property of $\mathbb{W}$, we obtain

$$0<\mathbb{M}(x,\mathbb{W}x)\le k\mathbb{M}(x_0,x)\le kr<r,$$

which leads us to a contradiction. Thus, $\mathbb{W}x=x$ for every $x\in \overline{B^{-}}(x_0,r)$, that is, $\overline{B^{-}}(x_0,r)\subseteq Fix\left(\mathbb{W}\right).$ In particular, $\mathbb{W}$ fixes the disc $D_{x_0,r}^{\mathbb{M}}$. □

Now, we define the concept of $DCQMS$-$\alpha$-$x_0$-contractive self-mappings in quasi-metric spaces.

Definition 11.

Let $\mathbb{W}$ be a self mapping on a $DCQMS$$(\mathbb{B},\mathbb{M})$. Then $\mathbb{W}$ is said to be a $DCQMS$-α-$x_0$-contractive self-mapping if there exist a function $\alpha :\mathbb{B}\times \mathbb{B}\to (0,\infty ),$$0<k<1$ and $x_0\in \mathbb{B}$ such that

$$\alpha (x_0,\mathbb{W}x)\mathbb{M}(x,\mathbb{W}x)\le k\mathbb{M}(x_0,x),$$

(6)

for all $x\in \mathbb{B}$.

We recall $\alpha$-$x_0$-admissible maps as follows:

Definition 12.

Let $\mathbb{B}$ be a non-empty set. Given a function $\alpha :\mathbb{B}\times \mathbb{B}\to (0,\infty )$ and $x_0\in \mathbb{B}.$ Then $\mathbb{W}$ is said to be an α-$x_0$-admissible if for every $x\in \mathbb{B},$

$$\alpha (x_0,x)\ge 1\Rightarrow \alpha (x_0,\mathbb{W}x)\ge 1.$$

Theorem 4.

Let $(\mathbb{B},\mathbb{M})$ be a $DCQMS$, $\mathbb{W}$ a $DCQMS$-α-$x_0$-contractive self-mapping with $x_0\in \mathbb{B}$ on $\mathbb{B}$ and r defined as in (2). Assume that $\mathbb{W}$ is α-$x_0$-admissible and $\alpha (x_0,x)\ge 1$ for all $x\in \overline{B^{-}}(x_0,r)$. Then we have $\overline{B^{-}}(x_0,r)\subseteq Fix\left(\mathbb{W}\right),$ in particular $\mathbb{W}$ fixes the disc $D_{x_0,r}^{\mathbb{M}}$.

Proof. 

By the definition of a $DCQMS$-$\alpha$-$x_0$-contractive self-mapping, it is easy to see that $x_0$ is always a fixed point of $\mathbb{W}$; therefore, if $r=0$ then we have $\overline{B^{-}}(x_0,r)=D_{x_0,r}^{\mathbb{M}}=\left\{x_0\right\}$ and the proof follows.

Suppose that $r>0.$ Let $x\in \overline{B^{-}}(x_0,r)$ such that $\mathbb{W}x\ne x.$ By the definition of r, we have $r\le \mathbb{M}(x,\mathbb{W}x).$ On the other hand, we have $\alpha (x_0,x)\ge 1$. Using the $\alpha$-$x_0$-admissible property and the $DCQMS$-$\alpha$-$x_0$-contractive property of $\mathbb{W}$, we find

$$0<\mathbb{M}(x,\mathbb{W}x)\le \alpha (x_0,\mathbb{W}x)\mathbb{M}(x,\mathbb{W}x)\le k\mathbb{M}(x_0,x)\le kr<r,$$

which leads us to a contradiction. Thus, $\overline{B^{-}}(x_0,r)\subseteq Fix\left(\mathbb{W}\right),$ in particular $\mathbb{W}$ fixes the disc $D_{x_0,r}^{\mathbb{M}}$. □

The concept of a $DCQMS$-$F_{\mathbb{M}}^{\alpha }$-contractive mapping is defined as follows.

Definition 13.

Let $(\mathbb{B},\mathbb{M})$ be a $DCQMS$, $\mathbb{W}$ a self-mapping on $\mathbb{B}$ and $F\in \mathbb{F}$. Then $\mathbb{W}$ is called a $DCQMS$-$F_{\mathbb{M}}^{\alpha }$-contraction if there exist $t>0,$ a function $\alpha :\mathbb{B}\times \mathbb{B}\to (0,\infty )$ and $x_0\in \mathbb{B}$ such that

$$\mathbb{M}(x,\mathbb{W}x)>0\Rightarrow t+\alpha (x_0,\mathbb{W}x)F\left(\mathbb{M}(x,\mathbb{W}x)\right)\le F\left(\mathbb{M}(x_0,x)\right),$$

(7)

for all $x\in \mathbb{B}$.

Theorem 5.

Let $(\mathbb{B},\mathbb{M})$ be a $DCQMS$, $\mathbb{W}$ a $DCQMS$-$F_{\mathbb{M}}^{\alpha }$-contractive self-mapping with $x_0\in \mathbb{B}$ and r defined as in (2). Suppose that $\mathbb{W}$ is α-$x_0$-admissible and $\alpha (x_0,x)\ge 1$ for all $x\in \overline{B^{-}}(x_0,r)$. Then we have $\overline{B^{-}}(x_0,r)\subseteq Fix\left(\mathbb{W}\right),$ in particular $\mathbb{W}$ fixes the disc $D_{x_0,r}^{\mathbb{M}}$.

Proof. 

At first, using the $DCQMS$-$F_{\mathbb{M}}^{\alpha }$-contractive property, one can easily deduce that $\mathbb{W}{x}_0=x_0$. Hence we have $\overline{B^{-}}(x_0,r)=D_{x_0,r}^{\mathbb{M}}=\left\{x_0\right\}$ if $r=0$. Clearly, $\mathbb{W}$ fixes the disc $D_{x_0,r}^{\mathbb{M}}$.

Assume that $r>0.$ Let $x\in \overline{B^{-}}(x_0,r)$ where $\mathbb{W}x\ne x$; therefore, by the definition of r, we have $r\le \mathbb{M}(x,\mathbb{W}x).$ On the other hand, we have $\alpha (x_0,x)\ge 1$ and $\mathbb{W}$ is $\alpha$-$x_0$-admissible. So, using the $DCQMS$-$F_{\mathbb{M}}^{\alpha }$-contractive property of $\mathbb{W}$, we deduce

$$F\left(\mathbb{M}(x,\mathbb{W}x)\right)<t+\alpha (x_0,\mathbb{W}x)F\left(\mathbb{M}(x,\mathbb{W}x)\right)\le F\left(\mathbb{M}(x_0,x)\right)\le F\left(r\right)\le F\left(\mathbb{M}(x,\mathbb{W}x)\right).$$

Thus, by the fact that F is strictly increasing and $t>0$ we come to a contradiction. Hence, we have $\overline{B^{-}}(x_0,r)\subseteq Fix\left(T\right),$ in particular $\mathbb{W}$ fixes the disc $D_{x_0,r}^{\mathbb{M}}$. □

Definition 14.

Let $(\mathbb{B},\mathbb{M})$ be a $DCQMS$, $\mathbb{W}$ a self-mapping on $\mathbb{B}$ and $F\in \mathbb{F}$. Then $\mathbb{W}$ is called a Ćirić-type $DCQMS$-$F_{\mathbb{M}}$-contraction if there exist $t>0$ and $x_0\in \mathbb{B}$ such that

$$\mathbb{M}(x,\mathbb{W}x)>0⟹t+\alpha (x_0,\mathbb{W}x)F\left(\mathbb{M}(x,\mathbb{W}x)\right)\le F\left(M_C^{\mathbb{M}}(x_0,x)\right),$$

(8)

for all $x\in \mathbb{B}$, where

$$M_C^{\mathbb{M}}(x,\upsilon )=max\left\{\mathbb{M}(x,\upsilon ),\mathbb{M}(x,\mathbb{W}x),\mathbb{M}(y,\mathbb{W}y),\frac{\mathbb{M}(x,\mathbb{W}y)+\mathbb{M}(y,\mathbb{W}x)}{2}\right\}.$$

(9)

Proposition 1.

Let $(\mathbb{B},\mathbb{M})$ be a $DCQMS$. If $\mathbb{W}$ is a Ćirić-type $DCQMS$-$F_{\mathbb{M}}$-contraction with $x_0\in \mathbb{B}$ such that $\alpha (x_0,\mathbb{W}{x}_0)\ge 1$, then we have $\mathbb{W}{x}_0=x_0$.

Proof. 

Assume that $\mathbb{W}{x}_0\ne x_0$. From the definition of a Ćirić-type $DCQMS$-$F_{\mathbb{M}}$-contraction, we obtain

$$\begin{array}{ccc}\hfill \mathbb{M}(x_0,\mathbb{W}{x}_0)& >& 0⟹t+\alpha (x_0,\mathbb{W}{x}_0)F\left(\mathbb{M}(x_0,\mathbb{W}{x}_0)\right)\le F\left(M_C^{\mathbb{M}}(x_0,x_0)\right)\hfill \\ & =& F\left(max\left\{\begin{array}{c}\mathbb{M}(x_0,x_0),\mathbb{M}(x_0,\mathbb{W}{x}_0),\mathbb{M}(x_0,\mathbb{W}{x}_0),\\ \frac{\mathbb{M}(x_0,\mathbb{W}{x}_0)+\mathbb{M}(x_0,\mathbb{W}{x}_0)}{2}\end{array}\right\}\right)\hfill \\ & =& F\left(\mathbb{M}(x_0,\mathbb{W}{x}_0)\right),\hfill \end{array}$$

which is a contradiction since $t>0$. Then we have $\mathbb{W}{x}_0=x_0$. □

Theorem 6.

Let $(\mathbb{B},\mathbb{M})$ be a $DCQMS$, $\mathbb{W}$ a Ćirić-type $DCQMS$-$F_{\mathbb{M}}$-contraction with $x_0\in \mathbb{B}$ and r defined as in (2). Assume that $\mathbb{W}$ is α-$x_0$-admissible and if for every $x\in D_{x_0,r}^{\mathbb{M}}$, we have $\mathbb{M}(x_0,\mathbb{W}x)\le r$. Then $\mathbb{W}$ fixes the disc $D_{x_0,r}^{\mathbb{M}}.$

Proof. 

If $r=0$, clearly $D_{x_0,r}^{\mathbb{M}}=\left\{x_0\right\}$ is a fixed-disc (point) by Proposition 1.

Assume that $r>0.$ Let $x\in D_{x_0,r}^{\mathbb{M}}$. By the definition of r, we have $\mathbb{M}(x,\mathbb{W}x)\ge r$. So using the Ćirić-type $DCQMS$-$F_{\mathbb{M}}$-contractive property and the fact that $\mathbb{W}$ is $\alpha$-$x_0$-admissible and F is increasing, we obtain

$$\begin{array}{ccc}\hfill F\left(\mathbb{M}\right(x,\mathbb{W}x\left)\right)& \le & \alpha (x_0,\mathbb{W}x)F\left(\mathbb{M}(x,\mathbb{W}x)\right)+t\le F\left(M_C^{\mathbb{M}}(x_0,x)\right)\hfill \\ & =& F\left(max\left\{\mathbb{M}(x_0,x),\mathbb{M}(x_0,\mathbb{W}{x}_0),\mathbb{M}(x,\mathbb{W}x),\frac{\mathbb{M}(x_0,\mathbb{W}x)+\mathbb{M}(x,\mathbb{W}{x}_0)}{2}\right\}\right)\hfill \\ & \le & F\left(max\left\{r,\mathbb{M}(x,\mathbb{W}x),0,r\right\}\right)\le F\left(\mathbb{M}(x,\mathbb{W}x)\right),\hfill \end{array}$$

which leads to a contradiction. Therefore, $\mathbb{M}(x,\mathbb{W}x)=0$ and so $\mathbb{W}x=x$. Hence, $\mathbb{W}$ fixes the disc $D_{x_0,r}^{\mathbb{M}}$. □

3.3. DCQMS-$\alpha$-$\phi$-$x_0$-Contractive Type Mappings

At first, we recall the notion of $\left(c\right)$-comparison functions [17] (see also [18]).

Definition 15

([17]).A function $\phi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is called a $\left(c\right)$-comparison function if

${\left(i\right)}_{\phi }$ φ is increasing;

${\left(ii\right)}_{\phi }$ There exist $k_0\in \mathbb{N}$, $a\in (0,1)$ and a convergent series of nonnegative terms $\sum _{k=1}^{\infty }{\upsilon }_k$ such that

$${\phi }^{k+1}\left(t\right)\le a{\phi }^k\left(t\right)+{\upsilon }_k,$$

for $k\ge k_0$ and any $t\in {\mathbb{R}}_{+}$.

The class of $\left(c\right)$-comparison functions will be denoted by ${\mathsf{\Psi }}_c$.

Lemma 2

([17]). If $\phi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is a $\left(c\right)$-comparison function, then the followings hold:

$\left(i\right)$ φ is a comparison function;

$\left(ii\right)$ $\phi \left(t\right)<t$ for any $t\in {\mathbb{R}}_{+};$

$\left(iii\right)$ φ is continuous at $0;$

$\left(iv\right)$ the series $\sum _{k=0}^{\infty }{\phi }^k\left(t\right)$ converges for any $t\in {\mathbb{R}}_{+}$.

Next, we introduce two new contractions and obtain two new fixed-disc theorems as follows:

Definition 16.

Let $(\mathbb{B},\mathbb{M})$ be a $DCQMS$ and $\mathbb{W}$ a self-mapping on $\mathbb{B}$. Then $\mathbb{W}$ is said to be a $DCQMS$-α-φ-$x_0$-contraction if there exist $\alpha :\mathbb{B}\times \mathbb{B}\to (0,\infty )$, $\phi \in {\mathsf{\Psi }}_c$ and $x_0\in \mathbb{B}$ such that

$$\mathbb{M}(x,\mathbb{W}x)>0⟹\alpha (x_0,\mathbb{W}x)\mathbb{M}(x,\mathbb{W}x)\le \phi \left(\mathbb{M}(x_0,\mathbb{W}x)\right),$$

for each $x\in \mathbb{B}$.

Theorem 7.

Let $(\mathbb{B},\mathbb{M})$ be a $DCQMS$, $\mathbb{W}$ a $DCQMS$-α-φ-$x_0$-contractive self-mapping with $x_0\in \mathbb{B}$ and r defined as in (2). Assume that $\mathbb{W}$ is α-$x_0$-admissible. If $\alpha (x_0,x)\ge 1$ for $x\in \overline{B^{-}}(x_0,r)$ and $0<\mathbb{M}(x_0,\mathbb{W}x)\le r$ for $x\in \overline{B^{-}}(x_0,r)-\left\{x_0\right\}$, then we have $\overline{B^{-}}(x_0,r)\subseteq Fix\left(\mathbb{W}\right),$ in particular $\mathbb{W}$ fixes the disc $D_{x_0,r}^{\mathbb{M}}$.

Proof. 

Using the $DCQMS$-$\alpha$-$\phi$-$x_0$-contractive property, we have $\mathbb{W}{x}_0=x_0$. Indeed, we assume $\mathbb{W}{x}_0\ne x_0$, that is, $\mathbb{M}(x_0,\mathbb{W}{x}_0)>0$. Then using the condition $\left(ii\right)$ in Lemma 2 and $\alpha$-$x_0$-admissibility, we obtain

$$\alpha (x_0,\mathbb{W}{x}_0)\mathbb{M}(x_0,\mathbb{W}{x}_0)\le \phi \left(\mathbb{M}(x_0,\mathbb{W}{x}_0)\right)<\mathbb{M}(x_0,\mathbb{W}{x}_0),$$

a contradiction. Thus, $\mathbb{W}{x}_0=x_0$.

Suppose that $r=0$. In this case, $\overline{B^{-}}(x_0,r)=D_{x_0,r}^{\mathbb{M}}=\left\{x_0\right\}$ and the proof follows.

Now we suppose that $r>0$ and $x\in \overline{B^{-}}(x_0,r)-\left\{x_0\right\}$ such that $x\ne \mathbb{W}x$. Using the definition of r, we have $r\le \mathbb{M}(x,\mathbb{W}x)$. By the hypothesis, we known $\alpha (x_0,x)\ge 1$. From the $DCQMS$-$\alpha$-$\phi$-$x_0$-contractive property and $\alpha$-$x_0$-admissibility, we obtain

$$\alpha (x_0,\mathbb{W}x)\mathbb{M}(x,\mathbb{W}x)\le \phi \left(\mathbb{M}(x_0,\mathbb{W}x)\right)<\mathbb{M}(x_0,\mathbb{W}x)\le r,$$

a contradiction. Therefore, $\mathbb{W}x=x$, that is, $\overline{B^{-}}(x_0,r)\subseteq Fix\left(\mathbb{W}\right),$ in particular $\mathbb{W}$ fixes the disc $D_{x_0,r}^{\mathbb{M}}$. □

Next, we define the following new contraction.

Definition 17.

Let $(\mathbb{B},\mathbb{M})$ be a $DCQMS$ and $\mathbb{W}$ a self-mapping on $\mathbb{B}$. Then $\mathbb{W}$ is said to be a Ćirić-type $DCQMS$-α-φ-$x_0$-contraction if there exist $\alpha :\mathbb{B}\times \mathbb{B}\to (0,\infty )$, $\phi \in {\Psi }_c$ and $x_0\in \mathbb{B}$ such that

$$\mathbb{M}(x,\mathbb{W}x)>0⟹\alpha (x_0,\mathbb{W}x)\mathbb{M}(x,\mathbb{W}x)\le \phi \left(M_C^{\mathbb{M}}(x_0,x)\right),$$

for each $x\in \mathbb{B}$.

Theorem 8.

Let $(\mathbb{B},\mathbb{M})$ be a $DCQMS$, $\mathbb{W}$ a Ćirić-type $DCQMS$-α-φ-$x_0$-contractive self-mapping with $x_0\in \mathbb{B}$ and r defined as in (2). Assume that $\mathbb{W}$ is α-$x_0$-admissible. If $\alpha (x_0,x)\ge 1$ and $\mathbb{M}(x_0,\mathbb{W}x)\le r$ for $x\in D_{x_0,r}^{\mathbb{M}}$, then $\mathbb{W}$ fixes the disc $D_{x_0,r}^{\mathbb{M}}$.

Proof. 

Using the hypothesis, we have $\mathbb{W}{x}_0=x_0$. Indeed, we assume $\mathbb{W}{x}_0\ne x_0$, that is, $\mathbb{M}(x_0,\mathbb{W}{x}_0)>0$. Then using the condition $\left(ii\right)$ in Lemma 2 and $\alpha$-$x_0$-admissibility, we obtain

$$M_C^{\mathbb{M}}(x_0,x_0)=max\left\{\begin{array}{c}\mathbb{M}(x_0,x_0),\mathbb{M}(x_0,\mathbb{W}{x}_0),\mathbb{M}(x_0,\mathbb{W}{x}_0),\\ \frac{\mathbb{M}(x_0,\mathbb{W}{x}_0)+\mathbb{M}(x_0,\mathbb{W}{x}_0)}{2}\end{array}\right\}=\mathbb{M}(x_0,\mathbb{W}{x}_0)$$

and

$$\alpha (x_0,\mathbb{W}{x}_0)\mathbb{M}(x_0,\mathbb{W}{x}_0)\le \phi \left(M_C^{\mathbb{M}}(x_0,x_0)\right)<\mathbb{M}(x_0,\mathbb{W}{x}_0),$$

a contradiction. It should be $\mathbb{W}{x}_0=x_0$.

Let $r=0$. In this case, we have $D_{x_0,r}^{\mathbb{M}}=\left\{x_0\right\}$.

Now we suppose that $r>0$ and $x\in D_{x_0,r}^{\mathbb{M}}-\left\{x_0\right\}$ such that $x\ne \mathbb{W}x$. Using the definition of r, we have $r\le \mathbb{M}(x,\mathbb{W}x)$. By the hypothesis, we known $\alpha (x_0,x)\ge 1$. By the Ćirić-type $DCQMS$-$\alpha$-$\phi$-$x_0$-contractive property and $\alpha$-$x_0$-admissibility, we obtain

$$M_C^{\mathbb{M}}(x_0,x)=max\left\{\begin{array}{c}\mathbb{M}(x_0,x),\mathbb{M}(x_0,\mathbb{W}{x}_0),\mathbb{M}(x,\mathbb{W}x),\\ \frac{\mathbb{M}(x_0,\mathbb{W}x)+\mathbb{M}(x,\mathbb{W}{x}_0)}{2}\end{array}\right\}\le \mathbb{M}(x,\mathbb{W}x)$$

and

$$\alpha (x_0,\mathbb{W}x)\mathbb{M}(x,\mathbb{W}x)\le \phi \left(M_C^{\mathbb{M}}(x,x_0)\right)<\mathbb{M}(x,\mathbb{W}x),$$

a contradiction. Therefore, $\mathbb{W}x=x$, that is, $D_{x_0,r}^{\mathbb{M}}$ is a fixed disc of $\mathbb{W}$. □

3.4. DCQMS-$\alpha$-$\psi$-$\phi$-$x_0$-Contractive Type Mappings

We recall the notion of an altering distance function.

Definition 18

([19]).A function $\psi :\left[0,\infty \right)\to \left[0,\infty \right)$ is called an altering distance function if the followings hold:

$\left(i\right)$ ψ is continuous and nondecreasing;

$\left(ii\right)$ $\psi \left(t\right)=0$⇔$t=0$.

Using this definition, we present two new contractive conditions and two new fixed-disc results.

Definition 19.

Let $(\mathbb{B},\mathbb{M})$ be a $DCQMS$ and $\mathbb{W}$ a self-mapping on $\mathbb{B}$. Then $\mathbb{W}$ is said to be a $DCQMS$-α-ψ-φ-$x_0$-contraction if there exist $\alpha :\mathbb{B}\times \mathbb{B}\to (0,\infty )$, two altering distance functions ψ, φ and $x_0\in \mathbb{B}$ such that

$$\mathbb{M}(x,\mathbb{W}x)>0⟹\alpha (x_0,\mathbb{W}x)\psi \left(\mathbb{M}(x,\mathbb{W}x)\right)\le \psi \left(\mathbb{M}(x_0,x)\right)-\phi \left(\mathbb{M}(x_0,x)\right),$$

for each $x\in \mathbb{B}$.

Theorem 9.

Let $(\mathbb{B},\mathbb{M})$ be a $DCQMS$, $\mathbb{W}$ a $DCQMS$-α-ψ-φ-$x_0$-contractive self-mapping with $x_0\in \mathbb{B}$ and r defined as in (2). Assume that $\mathbb{W}$ is α-$x_0$-admissible. If $\alpha (x_0,x)\ge 1$ for $x\in \overline{B^{-}}(x_0,r)$, then we have $\overline{B^{-}}(x_0,r)\subseteq Fix\left(\mathbb{W}\right),$ in particular $\mathbb{W}$ fixes the disc $D_{x_0,r}^{\mathbb{M}}$.

Proof. 

Using the $DCQMS$-$\alpha$-$\psi$-$\phi$-$x_0$-contractive property, we have $\mathbb{W}{x}_0=x_0$. Indeed, we assume $\mathbb{W}{x}_0\ne x_0$, that is, $\mathbb{M}(x_0,\mathbb{W}{x}_0)>0$. Then using the condition $\left(ii\right)$ in Definition 18 and $\alpha$-$x_0$-admissibility, we obtain

$$\begin{array}{ccc}\hfill \alpha (x_0,\mathbb{W}{x}_0)\psi \left(\mathbb{M}(x_0,\mathbb{W}{x}_0)\right)& \le & \psi \left(\mathbb{M}(x_0,x_0)\right)-\phi \left(\mathbb{M}(x_0,x_0)\right)\hfill \\ & =& \psi \left(0\right)-\phi \left(0\right)=0,\hfill \end{array}$$

a contradiction. It should be $\mathbb{W}{x}_0=x_0$.

Suppose that $r=0$. In this case, we obtain $\overline{B^{-}}(x_0,r)=D_{x_0,r}^{\mathbb{M}}=\left\{x_0\right\}$.

Now, we suppose that $r>0$ and $x\in \overline{B^{-}}(x_0,r)-\left\{x_0\right\}$ such that $x\ne \mathbb{W}x$. Using the definition of r, we have $r\le \mathbb{M}(x,\mathbb{W}x)$. By the hypothesis, we know $\alpha (x_0,x)\ge 1$. By the $DCQMS$-$\alpha$-$\psi$-$\phi$-$x_0$-contractive property and $\alpha$-$x_0$-admissibility, we obtain

$$\begin{array}{ccc}\hfill \alpha (x_0,\mathbb{W}x)\psi \left(\mathbb{M}(x,\mathbb{W}x)\right)& \le & \psi \left(\mathbb{M}(x_0,x)\right)-\phi \left(\mathbb{M}(x_0,x)\right)\hfill \\ & =& \psi \left(r\right)-\phi \left(r\right)<\psi \left(r\right),\hfill \end{array}$$

a contradiction. Therefore $\mathbb{W}x=x$, that is, $\overline{B^{-}}(x_0,r)\subseteq Fix\left(\mathbb{W}\right).$ In particular, $\mathbb{W}$ fixes the disc $D_{x_0,r}^{\mathbb{M}}$. □

Definition 20.

Let $(\mathbb{B},\mathbb{M})$ be a $DCQMS$ and $\mathbb{W}$ a self-mapping on $\mathbb{B}$. Then $\mathbb{W}$ is said to be a Ćirić-type $DCQMS$-α-ψ-φ-$x_0$-contraction if there exist $\alpha :\mathbb{B}\times \mathbb{B}\to (0,\infty )$, two altering distance functions ψ, φ and $x_0\in \mathbb{B}$ such that

$$\mathbb{M}(x,\mathbb{W}x)>0⟹\alpha (x_0,\mathbb{W}x)\psi \left(\mathbb{M}(x,\mathbb{W}x)\right)\le \psi \left(M_C^{\mathbb{M}}(x_0,x)\right)-\phi \left(M_C^{\mathbb{M}}(x_0,x)\right),$$

for each $x\in \mathbb{B}$.

Theorem 10.

Let $(\mathbb{B},\mathbb{M})$ be a $DCQMS$, $\mathbb{W}$ a Ćirić-type $DCQMS$-α-ψ-φ-$x_0$-contractive self-mapping with $x_0\in \mathbb{B}$ and r defined as in (2). Assume that $\mathbb{W}$ is α-$x_0$-admissible. If $\alpha (x_0,x)\ge 1$ and $\mathbb{M}(x_0,\mathbb{W}x)\le r$ for $x\in D_{x_0,r}^{\mathbb{M}}$, then $\mathbb{W}$ fixes the disc $D_{x_0,r}^{\mathbb{M}}$.

Proof. 

Using the hypothesis, we have $\mathbb{W}{x}_0=x_0$. Indeed, we assume $\mathbb{W}{x}_0\ne x_0$, that is, $\mathbb{M}(x_0,\mathbb{W}{x}_0)>0$ and we obtain

$$\begin{array}{ccc}\hfill \alpha (x_0,\mathbb{W}{x}_0)\psi \left(\mathbb{M}(x_0,\mathbb{W}{x}_0)\right)& \le & \psi \left(M_C^{\mathbb{M}}(x_0,x_0)\right)-\phi \left(M_C^{\mathbb{M}}(x_0,x_0)\right)\hfill \\ & =& \psi \left(\mathbb{M}(x_0,\mathbb{W}{x}_0)\right)-\phi \left(\mathbb{M}(x_0,\mathbb{W}{x}_0)\right)\hfill \\ & <& \psi \left(\mathbb{M}(x_0,\mathbb{W}{x}_0)\right),\hfill \end{array}$$

a contradiction. It should be $\mathbb{W}{x}_0=x_0$.

Let $r=0$. In this case, we have $D_{x_0,r}^{\mathbb{M}}=\left\{x_0\right\}$ and the proof follows.

Now, we suppose that $r>0$ and $x\in D_{x_0,r}^{\mathbb{M}}-\left\{x_0\right\}$ such that $x\ne \mathbb{W}x$. Using the definition of r, we have $r\le \mathbb{M}(x,\mathbb{W}x)$. By the hypothesis, we know that $\alpha (x_0,x)\ge 1$. From the Ćirić-type $DCQMS$-$\alpha$-$\psi$-$\phi$-$x_0$-contractive property and $\alpha$-$x_0$-admissibility, we obtain

$$\begin{array}{ccc}\hfill \alpha (x_0,\mathbb{W}x)\psi \left(\mathbb{M}(x,\mathbb{W}x)\right)& \le & \psi \left(M_C^{\mathbb{M}}(x_0,x)\right)-\phi \left(M_C^{\mathbb{M}}(x_0,x)\right)\hfill \\ & <& \psi \left(\mathbb{M}(x,\mathbb{W}x)\right),\hfill \end{array}$$

a contradiction; therefore, $\mathbb{W}x=x$, that is, $D_{x_0,r}^{\mathbb{M}}$ is a fixed disc of $\mathbb{W}$. □

4. An Application: A Common Fixed-Disc Theorem

Let $(\mathbb{B},\mathbb{M})$ be a $DCQMS$, $\mathbb{W},S:\mathbb{B}\to \mathbb{B}$ be two self-mappings and $D_{x_0,r}^{\mathbb{M}}$ be a disc on $\mathbb{B}$. If $\mathbb{W}x=Sx=x$ for all $x\in D_{x_0,r}^{\mathbb{M}}$, then the disc $D_{x_0,r}^{\mathbb{M}}$ is called the common fixed disc of the pair $(\mathbb{W},S)$.

Following [20,21], we present the following.

$$M_{\mathbb{W},S}^{\mathbb{M}}(x,\upsilon )=max\left\{\mathbb{M}(\mathbb{W}x,Sy),\mathbb{M}(\mathbb{W}x,Sx),\mathbb{M}(\mathbb{W}y,Sy),\frac{\mathbb{M}(\mathbb{W}x,Sy)+\mathbb{M}(\mathbb{W}y,Sx)}{2}\right\}.$$

To obtain a common fixed-disc theorem, we define the following number:

$${\mu }^{\mathbb{M}}=inf\left\{\mathbb{M}(\mathbb{W}x,Sx):x\in \mathbb{B},\mathbb{W}x\ne Sx\right\}.$$

In the following theorem, we use the numbers $M_{\mathbb{W},S}^{\mathbb{M}}(x,\upsilon )$, r which is defined in (2), ${\mu }^{\mathbb{M}}$ and $\rho$ defined by

$$\rho =min\{r,{\mu }^{\mathbb{M}}\}.$$

Theorem 11.

Let $(\mathbb{B},\mathbb{M})$ be a $DCQMS$, $\mathbb{W},S:\mathbb{B}\to \mathbb{B}$ two self-mappings and $\mathbb{W}$ an α-$x_0$-admissible map. Assume that there exist $F\in \mathbb{F}$, $t>0$ and $x_0\in \mathbb{B}$ such that

$$\mathbb{M}(\mathbb{W}x,Sx)>0⟹t+\alpha (x_0,\mathbb{W}x)F\left(\mathbb{M}(\mathbb{W}x,Sx)\right)\le F\left(M_{\mathbb{W},S}^{\mathbb{M}}(x,x_0)\right),$$

for each $x\in \mathbb{B}$ and

$$\alpha (x_0,x)\ge 1,\mathbb{M}(\mathbb{W}x,x_0)\le \rho ,\mathbb{M}(x_0,Sx)\le \rho ,$$

for each $x\in D_{x_0,\rho }^{\mathbb{M}}$. If $\mathbb{W}$ is a $DCQMS$-$F_{\mathbb{M}}$-contraction with $x_0\in \mathbb{B}$ and r and S is an $\mathbb{M}-$$F_{\mathbb{M}}$-contraction with $x_0\in \mathbb{B}$ and $r)$, then $D_{x_0,\rho }^{\mathbb{M}}$ is a common fixed disc of the pair $(\mathbb{W},S)$ in $\mathbb{B}$.

Proof. 

Let $x=x_0$. If $\mathbb{M}(\mathbb{W}{x}_0,S{x}_0)>0$ then we have

$$M_{\mathbb{W},S}^{\mathbb{M}}(x_0,x_0)=max\left\{\begin{array}{c}\mathbb{M}(\mathbb{W}{x}_0,S{x}_0),\mathbb{M}(\mathbb{W}{x}_0,S{x}_0),\mathbb{M}(\mathbb{W}{x}_0,S{x}_0),\\ \frac{\mathbb{M}(\mathbb{W}{x}_0,S{x}_0)+\mathbb{M}(\mathbb{W}{x}_0,S{x}_0)}{2}\end{array}\right\}=\mathbb{M}(\mathbb{W}{x}_0,S{x}_0)$$

and

$$\begin{array}{ccc}\hfill t+\alpha (x_0,\mathbb{W}{x}_0)F\left(\mathbb{M}(\mathbb{W}{x}_0,S{x}_0)\right)& \le & F\left(M_{\mathbb{W},S}^{\mathbb{M}}(x_0,x_0)\right)=F\left(\mathbb{M}(\mathbb{W}{x}_0,S{x}_0)\right)\hfill \\ & ⟹& t\le \left(1-\alpha (x_0,\mathbb{W}{x}_0)\right)F\left(\mathbb{M}(\mathbb{W}{x}_0,S{x}_0)\right),\hfill \end{array}$$

a contradiction with $t>0$; therefore $\mathbb{W}{x}_0=S{x}_0$, that is, $x_0$ is a coincidence point of the pair $(\mathbb{W},S)$. If $\mathbb{W}$ is a $\mathbb{M}-$$F_{\mathbb{M}}$-contraction (or S is a $\mathbb{M}-$$F_{\mathbb{M}}$-contraction) then using Theorem 1, we have $\mathbb{W}{x}_0=x_0$ (or $S{x}_0=x_0$) and hence $\mathbb{W}{x}_0=S{x}_0=x_0$.

Now if $\rho =0$, then clearly $D_{x_0,\rho }^{\mathbb{M}}=\left\{x_0\right\}$ and this disc is a common fixed disc of the pair $(\mathbb{W},S)$.

Let $\rho >0$ and $x\in D_{x_0,\rho }^{\mathbb{M}}$. Assume that $\mathbb{W}x\ne Sx$, that is, $\mathbb{M}(\mathbb{W}x,Sx)>0$. Using the hypothesis, $\alpha$-$x_0$-admissibility of $\mathbb{W}$ and the definition of $\rho$, we obtain

$$\begin{array}{ccc}\hfill M_{\mathbb{W},S}^{\mathbb{M}}(x,x_0)& =& max\left\{\mathbb{M}(\mathbb{W}x,S{x}_0),\mathbb{M}(\mathbb{W}x,Sx),\mathbb{M}(\mathbb{W}{x}_0,S{x}_0),\frac{\mathbb{M}(\mathbb{W}x,S{x}_0)+\mathbb{M}(\mathbb{W}{x}_0,Sx)}{2}\right\}\hfill \\ & =& max\left\{\mathbb{M}(\mathbb{W}x,x_0),\mathbb{M}(\mathbb{W}x,Sx),\frac{\mathbb{M}(\mathbb{W}x,x_0)+\mathbb{M}(x_0,Sx)}{2}\right\}\hfill \\ & \le & \mathbb{M}(\mathbb{W}x,Sx)\hfill \end{array}$$

and so

$$\begin{array}{ccc}\hfill t+\alpha (x_0,\mathbb{W}x)F\left(\mathbb{M}(\mathbb{W}x,Sx)\right)& \le & F\left(M_{\mathbb{W},S}^{\mathbb{M}}(x,x_0)\right)\le F\left(\mathbb{M}(\mathbb{W}x,Sx)\right)\hfill \\ & ⟹& t\le (1-\alpha (x_0,\mathbb{W}x))F\left(\mathbb{M}(\mathbb{W}x,Sx)\right),\hfill \end{array}$$

a contradiction with $t>0$. We have found that x is a coincidence point of the pair $(\mathbb{W},S)$, that is, $\mathbb{W}x=Sx$. If $\mathbb{W}$ (or S) is a $\mathbb{M}-$$F_{\mathbb{M}}$-contraction, then by Theorem 1, we have $\mathbb{W}x=x$ (or $Sx=x$) andhence $\mathbb{W}x=Sx=x$. Consequently, $D_{x_0,\rho }^{\mathbb{M}}$ is a common fixed disc of the pair $(\mathbb{W},S)$. □

We give an illustrative example.

Example 3.

Let $\mathbb{B}=(-\infty ,\infty )$ and for all $x,y\in \mathbb{B}$ let $\mathbb{M}(x,y)=|x-y|\mathrm{if}x\in (0,1)$ and $\mathbb{M}(0,1)=1,\mathbb{M}(1,0)=2.$ Next, let $\mathbb{K}(x,y)=max\{x,y\}+2$ and $\mathbb{L}(x,y)=max\{x,y\}+3.$ It is not difficult to see that $(\mathbb{B},\mathbb{M})$ is a $DCQMS.$ Now, define the self-mappings $\mathbb{W}:\mathbb{B}\to \mathbb{B}$ and $S:\mathbb{B}\to \mathbb{B}$ as

$$\mathbb{W}x=\left\{\begin{array}{ccc}\frac{1}{x^2}& if& x\in \{-1,1\}\\ x& if& x\in (-1,1)\\ x+2& otherwise\end{array}\right.,$$

and

$$Sx=\left\{\begin{array}{ccc}\frac{1}{\left|x\right|}& if& x\in \{-1,1\}\\ x& if& x\in (-1,1)\\ x+1& otherwise\end{array}\right.,$$

for all $x\in \mathbb{B}$. Define $\alpha (x,y)=e^{|x-y|}.$ First of all, note that both mappings $\mathbb{W},S$ are α-0-admissible. Moreover, the pair of the self-mappings $(\mathbb{W},S)$ satisfy the following condition

$$\mathbb{M}(\mathbb{W}x,Sx)>0⟹t+\alpha (0,\mathbb{W}x)F\left(\mathbb{M}(\mathbb{W}x,Sx)\right)\le F\left(M_{\mathbb{W},S}^{\mathbb{M}}(x,0)\right),$$

with $F=lnx$, $t=ln\frac{3}{2}$ and $x_0=0$. Now, it is not difficult to see that all the hypothesis of Theorem 11, are satisfied. Hence, $D_{0,1}^{\mathbb{M}}$ is a common fixed disk of the pair of mappings $(\mathbb{W},S)$ as required.

5. Conclusions

We have proved the existence of a fixed disk for self mappings in $DCQMS$ that satisfy different types of contractions. We provided an application of our result on common fixed disk for two self mapping on $DCQMS.$ In closing, we would like to bring to the readers attention the following question:

Question 1. Under what conditions these types of mappings in $DCQMS$ have a unique fixed disk?