Some common fixed-point and fixed-figure results with a function family on $S_{b}$-metric spaces

1.   Introduction and motivation

Fixed-point theory has been extensively researched in various areas, such as mathematics, engineering, and physics. Of particular importance is metric fixed-point theory, which is used in various branches of mathematics like topology, analysis, and applied mathematics. This theory was initiated with the famous Banach fixed-point theorem ^[1]. This theorem ensures that a self-mapping will have a unique fixed point. Despite this, there remain instances of self-mapping that have a fixed point but do not meet the criteria set by the Banach fixed point theorem, such as:

Let $\mathcal{X} = \mathbb{R}$, and $\left(\mathcal{X}, \zeta\right)$ be the usual metric space. Consider a self-mapping $g: \mathbb{R} \rightarrow \mathbb{R}$ defined as

for all $\mu\in \mathbb{R}$. Then $g$ has a unique fixed point $\mu = 4$, but $g$ does not meet the criteria of Banach contraction principle.

There are two popular methods used by researchers to generalize the Banach contraction principle. The first entails extending the utilized contractive condition while the second centers around generalizing the underlying metric space. For example, $G_{b}$-metric spaces, $G$-metric spaces ^[2], complex valued $G_{b}$-metric spaces ^[3,4], $S$-metric spaces, $A$-metric spaces ^[5], $S_{b}$-metric spaces, fuzzy cone metric spaces ^[6], modular metric spaces ^[7,8,9] et al. were defined for this purpose (for more details, see ^{[10,11,12,13]} and the references therein). Especially, we focus on the notion of $S_{b}$-metric spaces. To do this, we recall the following basic concepts:

Definition 1.1. ^[14] Let $\mathcal{X}$ be a nonempty set and $s\geq 1$ a given real number. A function $S_{b}:\mathcal{X}\times \mathcal{X} \times \mathcal{X}\rightarrow \lbrack 0, \infty)$ is said to be $S_{b}$ -metric if and only if for all $\mu, \tau, \hbar, \rho\in \mathcal{X}$ the following conditions are satisfied$:$

$(S_{b}1)$ $S_{b}(\mu, \tau, \hbar) = 0$ if and only if $\mu = \tau = \hbar$,

$(S_{b}2)$ $S_{b}(\mu, \tau, \hbar)\leq s[S_{b}(\mu, \mu, \rho)+S_{b}(\tau, \tau, \rho)+S_{b}(\hbar, \hbar, \rho)]$.

The pair $(\mathcal{X}, S_{b})$ is called an $S_{b}$-metric space.

As every $S$-metric is a $S_b$-metric with $s = 1$, we observe that $S_b$ -metric spaces are extensions of $S$-metric spaces. However, the converse statement is not always true $($see ^[14] and ^[15] for more details$)$.

Definition 1.2. ^[15] Let $(\mathcal{X}, S_{b})$ be an $S_{b}$-metric space and $s > 1$. An $S_{b}$-metric $S_{b}$ is called symmetric if

for all $\mu, \tau\in \mathcal{X}$.

Definition 1.3. ^[14] Let $(\mathcal{X}, S_{b})$ be an $S_{b}$ -metric space, and $\{\hbar_{n}\}$ be a sequence in $\mathcal{X}$.

1) Then the sequence $\{\hbar_{n}\}$ converges to $\hbar\in\mathcal{X}$ if and only if $S_{b}(\hbar_{n}, \hbar_{n}, \hbar)\rightarrow 0$ as $n\rightarrow \infty$, that is, for each $\varepsilon > 0$ there exists $n_{0}\in \mathbb{N}$ such that for all $n\geq n_{0}$, $S_{b}(\hbar_{n}, \hbar_{n}, \hbar) < \varepsilon$. It is denoted by

2) Then the sequence $\{\hbar_{n}\}$ is called a Cauchy sequence if for each $\varepsilon > 0$ there exists $n_{0}\in \mathbb{N}$ such that $S_{b}(\hbar_{n}, \hbar_{n}, \hbar_{m}) < \varepsilon$ for each $n, m\geq n_{0}$.

3) The $S_{b}$-metric space $(\mathcal{X}, S_{b})$ is called complete if every Cauchy sequence is convergent.

On the other hand, in the literature, besides various fixed point theorems there are also common fixed point theorems on metric and some extended metric spaces (for example, see ^{[14,16,17,18,19]} and the references therein).

Recently, as a geometric approach to the fixed-point theory, the fixed-circle problem (see ^[20]) and the fixed-figure problem (see ^[21]) have been introduced. When there are more than one fixed points, it is interesting to investigate for the possible solutions as follows:

Let us define a self-mapping, $g: \mathbb{R} \rightarrow \mathbb{R}$, where $\mathbb{R}$ is with the usual metric

for all $\mu\in \mathbb{R}$. Then $g$ has two fixed points

$\mu_{1} = -1$ and $\mu_{2} = 1$. We consider these fixed points as a unit circle $C_{0, 1} = \left\{ -1, 1\right\}$.

For this reason, there exist some studies related to these recent aspects (for example, see ^{[22,23,24,25,26,27,28]} and the references therein).

By the above motivation, in this paper, we prove a common fixed-point theorem and some fixed-figure results on $S_{b}$-metric spaces.

2.   A common fixed-point result

In this section, we prove a common fixed-point result on $S_{b}$-metric spaces. To do this, we are inspired by the function family $\mathcal{F}_{6}$ introduced in ^[29] and the function family $\mathcal{M}$ defined in ^[30]. We modify these families as follows:

Let $\Psi$ be the family of all lower semi-continuous functions $\psi : \mathbb{R} _{+}^{6}\rightarrow \mathbb{R}$ that satisfy the following condition:

$\left(\psi ^{\ast }\right).$ For all $\mu, \tau, \hbar\geq 0$ and $s\geq 1$, there exists a $k\in \lbrack 0, 1)$ such that

with $\hbar\leq 2s\mu+s\tau$ then

If we define the function $\psi : \mathbb{R} _{+}^{6}\rightarrow \mathbb{R}$ such as

with $k\in \lbrack 0, 1)$. Then $\psi \in \Psi$.

Now, we give the following theorem.

Theorem 2.1. Let $\left(\mathcal{X}, S_{b}\right)$ be a complete continuous $S_{b}$ -metric space with the symmetric metric $S_{b}$. Let $g, h, G, H:\mathcal{X} \rightarrow \mathcal{X}$ be four self-mappings, where $g, G$ and $H$ are continuous and satisfying the following conditions:

$(i)$ $g(\mathcal{X})\subset H(\mathcal{X})$ and $h(\mathcal{X})\subset G(\mathcal{X})$,

$(ii)$ For all $\mu, \tau\in \mathcal{X}$ and $\psi \in \Psi$,

$(iii)$ For all $\mu, \tau\in \mathcal{X}$ and $\psi \in \Psi$,

holds, then $g$, $h$, $G$ and $H$ have a common fixed point in $\mathcal{X}$.

Proof. Let $\hbar_{0}\in \mathcal{X}$, $\hbar_{1} = g\hbar_{0}$ and $\hbar_{2} = h\hbar_{1}$. Using the condition $(ii)$, we get

By $(S_{b}2)$ and the symmetry property of $S_{b}$, we have

Using (2.1), (2.2) and $\left(\psi ^{\ast }\right)$, there exists a $k\in \lbrack 0, 1)$ such that

Continuing this process with induction with the condition $(i)$, we can define the sequence $\left\{ \hbar_{n}\right\}$ as follows:

and

Using $(ii)$, for $\mu = \hbar_{2n}$ and $\tau = \hbar_{2n+1}$, we find

By $(S_{b}2)$ and the symmetry property of $S_{b}$, we have

Using (2.3), (2.4) and $\left(\psi ^{\ast }\right)$, there exists a $k\in \lbrack 0, 1)$ such that

Using $(iii)$, for $\mu = \hbar_{2n-1}$ and $\tau = \hbar_{2n}$, we get

By $(S_{b}2)$ and the symmetry property of $S_{b}$, we find

Using (2.6), (2.7) and $\left(\psi ^{\ast }\right)$, there exists a $k\in \lbrack 0, 1)$ such that

Using the inequalities (2.5) and (2.8), we get

and so, using similar arguments, we have

Now we show that the sequence $\left\{ \hbar_{n}\right\}$ is Cauchy. Using $(S_{b}2)$, (2.9) and the symmetry property of $S_{b}$, for all $n, m\in \mathbb{N}$ with $m > n$, we get

Since $s\geq 1$ and $k\in \lbrack 0, 1)$, taking $n, m\rightarrow \infty$, we get

and so $\left\{ \hbar_{n}\right\}$ is Cauchy. Since $\left(\mathcal{X}, S_{b}\right)$ is a complete $S_{b}$-metric space, $\left\{ \hbar_{n}\right\}$ is convergent to a point $\hbar\in \mathcal{X}$, that is,

Next, we establish that $\hbar$ is a common fixed point of $g$, $h$, $G$ and $H$. Using $(ii)$, for $\mu = \hbar_{2n}$ and $\tau = \hbar$, we get

and taking $n\rightarrow \infty$, we obtain

and

Using (2.10), (2.11) and $\left(\psi ^{\ast }\right)$, there exists a $k\in \lbrack 0, 1)$ such that

that is,

Using the continuity hypothesis of $g$, we have

Hence $\hbar$ is a common fixed point $g$ and $h$. Using $(iii)$, for $\mu = \hbar_{2n}$ and $\tau = \hbar$, we get

and taking $n\rightarrow \infty$, we obtain

and

Using (2.12), (2.13) and $\left(\psi ^{\ast }\right)$, there exists a $k\in \lbrack 0, 1)$ such that

that is,

Using the continuity hypothesis of $G$, we have

Consequently, we obtain

that is, $\hbar$ is a common fixed point of four self-mappings $g$, $h$, $G$ and $H$.

3.   Some fixed-figure results

In this section, we investigate some fixed-figure results on $S_{b}$-metric spaces. At first, we recall the following notions:

Definition 3.1. ^[22,31] Let $\left(\mathcal{X}, S_{b}\right)$ be an $S_{b}$-metric space with $s\geq 1$ and $\hbar_{0}, \hbar_{1}, \hbar_{2}\in \mathcal{X}$, $r\in \lbrack 0, \infty)$.

● The circle is defined by

● The disc is defined by

● The ellipse is defined by

● The hyperbola is defined by

● The Cassini curve is defined by

● The Apollonious circle is defined by

Definition 3.2. ^[22] Let $g:\mathcal{X}\rightarrow \mathcal{X}$ be a self-mapping where $\left(\mathcal{X}, S_{b}\right)$ is a $S_{b}$-metric space with $s\geq 1$. Let $Fix(g)$ be set of all fixed points of $g$, then a geometric figure $\mathcal{F}$ is said to be a fixed figure of $g$ if $\mathcal{F}$ is contained in $Fix(g)$.

Let us define the number $r$ as

Theorem 3.1. Let $\left(\mathcal{X}, S_{b}\right)$ be an $S_{b}$-metric space with $s\geq 1$, $g:\mathcal{X}\rightarrow \mathcal{X}$ be a self-mapping, $S_{b}$ be symmetric and $r$ be defined as in $($3.1$)$. If there exist $\hbar_{0}\in \mathcal{X}$ and $\psi \in \Psi$ for all $\mu\in \mathcal{X}-\{\hbar_{0}\}$ such that

and $g\hbar_{0} = \hbar_{0}$, then $D_{\hbar_{0}, r}^{S_{b}}\subset Fix(g)$. Especially, we have $C_{\hbar_{0}, r}^{S_{b}}\subset Fix(g)$.

Proof. Let $r = 0$. Then we have $D_{\hbar_{0}, r}^{S_{b}} = \{\hbar_{0}\}$. By the hypothesis $g\hbar_{0} = \hbar_{0}$, we obtain

Let $r > 0$ and $\mu\in D_{\hbar_{0}, r}^{S_{b}}$ such that $\mu\notin Fix(g)$. Using the hypothesis, we get

By $(S_{b}2)$ and the symmetry property of $S_{b}$, we have

Using (3.2), (3.3) and $\left(\psi ^{\ast }\right)$, there exists a $k\in \lbrack 0, 1)$ such that

a contradiction. Hence it should be $\mu \in Fix(g)$. Consequently, we get

Using the similar arguments, it can be easily see that

Theorem 3.2. Let $\left(\mathcal{X}, S_{b}\right)$ be an $S_{b}$-metric space with $s\geq 1$, $g:\mathcal{X}\rightarrow \mathcal{X}$ be self-mapping, $S_{b}$ be a symmetric and $r$ be defined as in $($3.1$)$. If there exist $\hbar_{1}, \hbar_{2}\in \mathcal{X}$ and $\psi \in \Psi$ for all $\mu\in \mathcal{X}-\{\hbar_{1}, \hbar_{2}\}$ such that

and $g\hbar_{1} = \hbar_{1}$, $g\hbar_{2} = \hbar_{2}$ with $g(\mu)\in E_{r}^{S_{b}}(\hbar_{1}, \hbar_{2})$, then $E_{r}^{S_{b}}(\hbar_{1}, \hbar_{2})\subset Fix(g)$.

Proof. Let $r = 0$. Then we have $E_{r}^{S_{b}}(\hbar_{1}, \hbar_{2}) = \{\hbar_{1}\} = \{ \hbar_{2}\}$. By the hypothesis $g\hbar_{1} = \hbar_{1}$ and $g\hbar_{2} = \hbar_{2}$, we obtain

Let $r > 0$ and $\mu\in E_{r}^{S_{b}}(\hbar_{1}, \hbar_{2})$ such that $\mu\notin Fix(g)$. Using the hypothesis, we get

Since

using $\left(\psi ^{\ast }\right)$, there exists a $k\in \lbrack 0, 1)$ such that

a contradiction. Hence it should be $\mu\in Fix(g)$. Consequently, we get

Theorem 3.3. Let $\left(\mathcal{X}, S_{b}\right)$ be an $S_{b}$-metric space with $s\geq 1$, $g:\mathcal{X}\rightarrow \mathcal{X}$ be self-mapping, $S_{b}$ be a symmetric and $r$ be defined as in $($3.1$)$. If $r > 0$ and there exist $\hbar_{1}, \hbar_{2}\in \mathcal{X}$, $\psi \in \Psi$ for all $x\in \mathcal{X}-\{\hbar_{1}, \hbar_{2}\}$ such that

and $g\hbar_{1} = \hbar_{1}$, $g\hbar_{2} = \hbar_{2}$ with $g(\mu)\in H_{r}^{S_{b}}(\hbar_{1}, \hbar_{2})$, then $H_{r}^{S_{b}}(\hbar_{1}, \hbar_{2})\subset Fix(g)$.

Proof. Let $r > 0$ and $\mu\in H_{r}^{S_{b}}(\hbar_{1}, \hbar_{2})$ such that $\mu\notin Fix(g)$. Using the hypothesis, we get

Since

using $\left(\psi ^{\ast }\right)$, there exists a $k\in \lbrack 0, 1)$ such that

a contradiction. Hence it should be $\mu\in Fix(g)$. Consequently, we get

Theorem 3.4. Let $\left(\mathcal{X}, S_{b}\right)$ be an $S_{b}$-metric space with $s\geq 1$, $g:\mathcal{X}\rightarrow \mathcal{X}$ be a self-mapping, $S_{b}$ be symmetric and $r$ be defined as in $($3.1$)$. If there exist $\hbar_{1}, \hbar_{2}\in \mathcal{X}$ and $\psi \in \Psi$ for all $\mu\in \mathcal{X}-\{\hbar_{1}, \hbar_{2}\}$ such that

and $g\hbar_{1} = \hbar_{1}$, $g\hbar_{2} = \hbar_{2}$ with $g(\mu)\in C_{r}^{S_{b}}(\hbar_{1}, \hbar_{2})$, then $C_{r}^{S_{b}}(\hbar_{1}, \hbar_{2})\subset Fix(g)$.

Proof. Let $r = 0$. Then we have $C_{r}^{S_{b}}(\hbar_{1}, \hbar_{2}) = \{\hbar_{1}\}$ or $\{ \hbar_{2}\}$. By the hypothesis $g\hbar_{1} = \hbar_{1}$ and $g\hbar_{2} = \hbar_{2}$, we obtain

Let $r > 0$ and $\mu\in C_{r}^{S_{b}}(\hbar_{1}, \hbar_{2})$ such that $\mu\notin Fix(g)$. Using the hypothesis, we get

Since

using $\left(\psi ^{\ast }\right)$, there exists a $k\in \lbrack 0, 1)$ such that

a contradiction. Hence it should be $\mu\in Fix(g)$. Consequently, we get

Theorem 3.5. Let $\left(\mathcal{X}, S_{b}\right)$ be an $S_{b}$-metric space with $s\geq 1$, $g:\mathcal{X}\rightarrow \mathcal{X}$ be a self-mapping, $S_{b}$ be symmetric and $r$ be defined as in $($3.1$)$. If there exist $\hbar_{1}, \hbar_{2}\in \mathcal{X}$ and $\psi \in \Psi$ for all $\mu\in \mathcal{X}-\{\hbar_{1}, \hbar_{2}\}$ such that

and $g\hbar_{1} = \hbar_{1}$, $g\hbar_{2} = \hbar_{2}$ with $g(\mu)\in A_{r}^{S_{b}}(\hbar_{1}, \hbar_{2})$, then $A_{r}^{S_{b}}(\hbar_{1}, \hbar_{2})\subset Fix(g)$.

Proof. Let $r = 0$. Then we have $A_{r}^{S_{b}}(\hbar_{1}, \hbar_{2}) = \{\hbar_{1}\}$. By the hypothesis $g\hbar_{1} = \hbar_{1}$, we obtain

Let $r > 0$ and $\mu \in A_{r}^{S_{b}}(\hbar _{1}, \hbar _{2})$ such that $\mu \notin Fix(g)$. Using the hypothesis, we get

Since

using $\left(\psi ^{\ast }\right)$, there exists a $k\in \lbrack 0, 1)$ such that

a contradiction. Hence it should be $\mu \in Fix(g)$. Consequently, we get

Now we give the following illustrative example of above proved geometric results.

Example 3.1. Let us consider Example 2.2 given in ^[22]. Let $\mathcal{X} = [-1, 1]\cup \left\{ -7, -\sqrt{2}, \sqrt{2}, \frac{7}{3}, 7, 8, 21\right\}$ and the $S$-metric defined as

for all $\mu, \tau, \hbar \in \mathcal{X}$ ^[32]. Then the function $S$ is also an $S_{b}$-metric with $s = 1$. Let us define the function $g: \mathcal{X}\rightarrow \mathcal{X}$ as

for all $\mu \in \mathcal{X}$ and the function $\psi : \mathbb{R} _{+}^{6}\rightarrow \mathbb{R}$ as

with $k\in \lbrack 0, 1)$. Under these assumptions, we get

$\bigstar$ If we take $\hbar _{0} = 0$ and $k = \frac{1}{2}$, then the function $g$ satisfies the conditions of Theorem 3.1. Therefore, we obtain

and

$\bigstar$ If we take $\hbar _{1} = -\frac{1}{2}$, $\hbar _{2} = \frac{1}{2}$ and $k = \frac{1}{2}$, then the function $g$ satisfies the conditions of Theorem 3.2. Therefore, we obtain

$\bigstar$ If we take $\hbar _{1} = -1$, $\hbar _{2} = 1$ and $k = \frac{3}{4}$, then the function $g$ satisfies the conditions of Theorem 3.3. Therefore, we obtain

$\bigstar$ If we take $\hbar _{1} = -1$, $\hbar _{2} = 1$ and $k = \frac{3}{4}$, then the function $g$ satisfies the conditions of Theorem 3.4. Therefore, we obtain

$\bigstar$ If we take $\hbar _{1} = -7$, $\hbar _{2} = 7$ and $k = \frac{1}{5}$, then the function $g$ satisfies the conditions of Theorem 3.5. Therefore, we obtain

4.   An application to parametric rectified linear unit activation functions

Recently, activation functions have been used in applicable areas. Especially, these functions are used in neural network. For example, for state-of-the-art neural networks, rectified activation units are essential. Therefore, in this section, we focus on the parametric rectified linear unit ($PReLU$) activation functions ^[33]. $PReLU$ is defined as

where $\alpha$ is a coefficient.

Let $\mathcal{X} = \mathbb{R} ^{+}\cup \left\{ -2, -1, 0\right\}$, the $S$-metric defined as in Example 3.1 and the function $\psi : \mathbb{R} _{+}^{6}\rightarrow \mathbb{R}$ defined as in Example 3.1. Let us consider $\alpha = 0.8$, then we obtain the function $PReLU$ as

for all $\mu \in \mathbb{R}$. If we take $\hbar _{0} = 0$ and $k = \frac{1}{2}$, then the function $PReLU$ satisfies the conditions of Theorem 3.1. Indeed, for $\mu \in \left(-\infty, 0\right)$, we get

Also, we obtain

Consequently, we have

and similarly

Finally, we say that the parametric rectified linear unit ($PReLU$) activation function fixes the disc $D_{0, 0.2}^{S_{b}}$ and $C_{0, 0.2}^{S_{b}}$, that is, $PReLU$ has at least two fixed figure. In this way, the learning capacity of the activation function $PReLU$ increases.

Acknowledgements

The authors I. Ayoob and N. Mlaiki would like to thank the Prince Sultan University for paying the publication fees for this work through TAS LAB.

Conflict of interest

The authors declare no conflicts of interest.