Some Fixed-Point Results in Extended S-Metric Space of Type (α,β)

Abstract

In this paper, we introduce the notion of extended S-metric space of type $\left(\alpha ,\beta \right)$. This extension is a generalization of S-metric space, defined by employing two functions instead of considering a constant in the second condition of the S-metric space definition. Accordingly, we prove some fixed-point results and give some examples to illustrate the validity of our work, along with giving an application of the Fredholm integral equation.

1. Introduction

The fixed-point concept has been a major area of research in mathematics. It is useful in various directions of mathematics, with many applications in various fields in science, like physics, engineering, economics and computer science. Fixed-point theory was established in 1911 by Brouwer [1], who stated that any continuous function from a closed unit ball in finite-dimensional space mapping to itself must have a fixed point. But this result is not constructive, because it does not give information on how to find the fixed point. Later, specifically in 1922, Banach pioneered his theory [2], which is well known as the Banach contraction principle and played an important role in solving many problems in many branches that were related to the fixed point and its existence. It states some criteria guaranteeing the existence and uniqueness of fixed points. This leads Banach theory to be both theoretical and practical at the same time. Banach investigated his theory with some conditions of a contraction mapping or a domain on a complete metric space and showed the underlying importance of completeness.

It is well known that there are many generalizations of metric spaces, such as G-metric space, b-metric space, S-metric space and more. This allows for more flexible frameworks for studying fixed-point results in more general settings, as mentioned in [3,4,5]. Some researchers generalize the Banach principle by changing the type of contraction and adding conditions that satisfy the existence of a fixed point, like Karman et al. [6], who established the notion of extended b-metric space and proved some fixed-point theorems in extended b-metric space. In recent years, there have been several important fixed-point results for some contractions in extended S-metric spaces and $S_b$-metric spaces; see [7,8,9,10,11]. This paper aims to investigate a generalization of S-metric space, called extended S-metric space of type $(\alpha ,\beta )$. As a result, several theorems and results in connection with the existence and uniqueness of fixed points are consequently proved.

2. Preliminaries

In this section, we recall the primary definitions and results related to S-metric space, $S_b$-metric space and extended $S_b$-metric space.

Definition 1

([12]). Let $\mathcal{W}$ be a nonempty set and $\phi :{\mathcal{W}}^3\to [0,\infty )$ be a given function. Let the following conditions hold:

i.

   $\phi (\rho ,\varrho ,\eta )=0$ if and only if $\rho =\varrho =\eta$;

ii.

  $\phi (\rho ,\varrho ,\eta )\le \phi (\rho ,\rho ,\zeta )+\phi (\varrho ,\varrho ,\zeta )+\phi (\eta ,\eta ,\zeta )$.

Then, the pair $(\mathcal{W},\phi )$ is called S-metric space.

In particular, if one assumes $\mathcal{W}={\mathbb{R}}^n$ and $\left|\right|.\left|\right|$ is a norm on $\mathcal{W}$, then $\phi (\rho ,\varrho ,\eta )=\left|\right|\varrho +\eta -2\rho \left|\right|+\left|\right|\varrho -\eta \left|\right|$ is an S-metric space on $\mathcal{W}$ [12].

Definition 2

([13]). Let $\mathcal{W}$ be a nonempty set, $\kappa \ge 1$ be a given real number and ${\phi }_b:{\mathcal{W}}^3\to [0,\infty )$ be a given function. Let the following conditions hold:

i.

   ${\phi }_b(\rho ,\varrho ,\eta )=0$ if and only if $\rho =\varrho =\eta$;

ii.

  ${\phi }_b(\rho ,\varrho ,\varrho )={\phi }_b(\varrho ,\varrho ,\rho )$, for all $\rho ,\varrho \in \mathcal{W}$;

iii.

 ${\phi }_b(\rho ,\varrho ,\eta )\le \kappa [{\phi }_b(\rho ,\rho ,\zeta )+{\phi }_b(\varrho ,\varrho ,\zeta )+{\phi }_b(\eta ,\eta ,\zeta )]$.

Then, the pair $(\mathcal{W},{\phi }_b)$ is called $S_b$-metric space.

Herein, if we let $\mathcal{W}\ne \varphi$, such that $card\left(\mathcal{W}\right)\ge 5$, and suppose that $\mathcal{W}={\mathcal{W}}_1\cup {\mathcal{W}}_2$ is a partition of $\mathcal{W}$, such that $card\left({\mathcal{W}}_1\right)\ge 4$, then for $\kappa \ge 1$, we can define ${\phi }_b:{\mathcal{W}}^3\to [0,+\infty )$ as

$${\phi }_b(\rho ,\varrho ,\eta )=\left\{\begin{array}{cc}0\hfill & \mathrm{if}\rho =\varrho =\eta =0\hfill \\ 3\kappa \hfill & \mathrm{if}\rho ,\varrho ,\eta \in {\mathcal{W}}_1^3\hfill \\ 1\hfill & \mathrm{if}\rho ,\varrho ,\eta \notin {\mathcal{W}}_1^3\hfill \end{array}\right.$$

for all $\rho ,\varrho ,\eta \in \mathcal{W}$. In this case, we notice that $(\mathcal{W},{\phi }_b)$ is an $S_b$-metric space [13].

Lemma 1

([13]). Any $S_b$-metric space is an S-metric space if $\kappa =1$.

In light of Lemma 1, one might observe that any S-metric space is not necessarily an $S_b$-metric space. This means that $S_b$-metric spaces are more general than S-metric spaces.

Definition 3

([14]). Let $\mathcal{W}$ be a nonempty set, ${\phi }_{\theta }:{\mathcal{W}}^3\to [0,\infty )$ be a given function and $\theta :{\mathcal{W}}^3\to [1,\infty )$. Let the following conditions hold:

i.

   ${\phi }_{\theta }(\rho ,\varrho ,\eta )=0$ if and only if $\rho =\varrho =\eta$;

ii.

  ${\phi }_{\theta }(\rho ,\varrho ,\eta )\le \theta (\rho ,\varrho ,\eta )[{\phi }_{\theta }(\rho ,\rho ,\zeta )+{\phi }_{\theta }(\varrho ,\varrho ,\zeta )+{\phi }_{\theta }(\eta ,\eta ,\zeta )]$.

Then, the pair $(\mathcal{W},{\phi }_{\theta })$ is called extended $S_b$-metric space.

For instance, if one supposes that $\mathcal{W}=C[a,d]$, the set of all continuous real-valued functions on $[a,d]$, and ${\phi }_{\theta }:{\mathcal{W}}^3\to [0,\infty )$ and $\theta :{\mathcal{W}}^3\to [1,\infty )$ are two mappings such that

$$\phi (\rho \left(\zeta \right),\varrho \left(\zeta \right),\eta \left(\zeta \right))=\underset{\zeta \in [a,d]}{sup}{|max\{\rho \left(\zeta \right),\varrho \left(\zeta \right)\}-\eta \left(\zeta \right)|}^2,$$

and

$$\theta \left(\rho \right(\zeta ),\varrho (\zeta ),\eta (\zeta \left)\right)=max\left\{\right|\rho \left(\zeta \right)|,|\varrho \left(\zeta \right)\left|\right\}+\left|\eta \right(\zeta \left)\right|+1,$$

then the pair $(\mathcal{W},{\phi }_{\theta })$ is an extended $S_b$-metric space [14].

3. Main Results

The aim of this section is to present the main results of this work. In particular, we here intend to establish a novel notion of extended S-metric space of type $(\alpha ,\beta )$, and as a consequence, we provide several fixed-point theorems, definitions, examples and further results. To do so, we first introduce the following definition of extended S-metric space of type $(\alpha ,\beta )$, followed by an example on the proposed space for clarification purposes.

Definition 4.

Let $\mathcal{W}$ be a nonempty set. Suppose that $\alpha ,\beta :{\mathcal{W}}^3\to [1,\infty )$ and ${\phi }_{\alpha \beta }:{\mathcal{W}}^3\to [0,\infty )$ are given mappings. For all $\rho ,\varrho ,\eta \in \mathcal{W}$, let the following conditions hold:

i.

   ${\phi }_{\alpha \beta }(\rho ,\varrho ,\eta )=0$ if and only if $\rho =\varrho =\eta$;

ii.

  ${\phi }_{\alpha \beta }(\rho ,\varrho ,\eta )\le \alpha (\rho ,\varrho ,\eta ){\phi }_{\alpha \beta }(\rho ,\rho ,\zeta )+\beta (\rho ,\varrho ,\eta ){\phi }_{\alpha \beta }(\varrho ,\varrho ,\zeta )+{\phi }_{\alpha \beta }(\eta ,\eta ,\zeta )$.

Then, the pair $(\mathcal{W},{\phi }_{\alpha \beta })$ is called extended S-metric space of type $(\alpha ,\beta )$.

Example 1.

Let $\mathcal{W}=\{0,1,2\}$, and define

$${\phi }_{\alpha \beta }(\rho ,\varrho ,\eta )=\left\{\begin{array}{cc}0\hfill & ,\rho =\varrho =\eta \hfill \\ 1\hfill & ,\rho \ne \varrho \ne \eta \hfill \\ \frac{3}{2}\hfill & ,\rho =\varrho ,\varrho \ne \eta \hfill \end{array}\right..$$

Define $\alpha ,\beta :{\mathcal{W}}^3\to [1,\infty )$ as

$$\alpha (\rho ,\varrho ,\eta )=1+\rho +\varrho +\eta ,$$

and

$$\beta (\rho ,\varrho ,\eta )=1+\rho \varrho \eta .$$

Then, $(\mathcal{W},{\phi }_{\alpha \beta })$ is an extended S-metric space of type $(\alpha ,\beta )$.

Example 2.

Let $\mathcal{W}=R$, and define the mappings ${\phi }_{\alpha \beta }(\rho ,\varrho ,\eta ):{\mathcal{W}}^3\to [0,\infty )$ and $\alpha ,\beta :{\mathcal{W}}^3\to [1,\infty )$ as follows:

$${\phi }_{\alpha \beta }(\rho ,\varrho ,\eta )=max\{\rho ,\varrho ,\eta \},$$

$$\alpha (\rho ,\varrho ,\eta )=|\rho -\varrho |+1,$$

and

$$\beta (\rho ,\varrho ,\eta )=|\rho -\eta |+|\varrho -\eta |+2.$$

Then, $(\mathcal{W},{\phi }_{\alpha \beta })$ is an extended S-metric space of type $(\alpha ,\beta )$.

Lemma 2.

Every S-metric space is an extended S-metric space of type $(\alpha ,\beta )$.

Proof. 

This result can be obtained immediately by assuming $\alpha =\beta =1$. □

Lemma 3.

In an extended S-metric space of type $(\alpha ,\beta )$, we have

$${\phi }_{\alpha \beta }(\rho ,\rho ,\varrho )={\phi }_{\alpha \beta }(\varrho ,\varrho ,\rho ).$$

Proof. 

For $\rho ,\varrho \in \mathcal{W}$, the second condition of Definition 4 implies that

$${\phi }_{\alpha \beta }(\rho ,\rho ,\varrho )\le \alpha (\rho ,\rho ,\varrho ){\phi }_{\alpha \beta }(\rho ,\rho ,\rho )+\beta (\rho ,\rho ,\varrho ){\phi }_{\alpha \beta }(\rho ,\rho ,\rho )+{\phi }_{\alpha \beta }(\varrho ,\varrho ,\rho ),$$

(1)

and

$${\phi }_{\alpha \beta }(\varrho ,\varrho ,\rho )\le \alpha (\varrho ,\varrho ,\rho ){\phi }_{\alpha \beta }(\varrho ,\varrho ,\varrho )+\beta (\varrho ,\varrho ,\rho ){\phi }_{\alpha \beta }(\varrho ,\varrho ,\varrho )+{\phi }_{\alpha \beta }(\rho ,\rho ,\varrho ).$$

(2)

Hence, we have

$${\phi }_{\alpha \beta }(\varrho ,\varrho ,\rho )={\phi }_{\alpha \beta }(\rho ,\rho ,\varrho ).$$

 □

In view of the previous discussion, we can also establish the following definition connected with convergence, followed by a proposition that demonstrates a needed condition for the convergence of a sequence in the proposed space.

Definition 5.

Let $(\mathcal{W},{\phi }_{\alpha \beta })$ be an extended S-metric space of type $(\alpha ,\beta )$ and $\left({\rho }_n\right)$ be a sequence in $\mathcal{W}$. Then, the following apply:

1.

 A sequence $\left\{{\rho }_n\right\}$ in $\mathcal{W}$ is said to be convergent to $\rho \in \mathcal{W}$ if for each $ϵ>0$, there exists $N=N\left(ϵ\right)\in \mathbb{N}$ such that ${\phi }_{\alpha \beta }({\rho }_n,{\rho }_n,\rho )<ϵ$, for all $n\ge N$.

2.

 A sequence $\left\{{\rho }_n\right\}$ in $\mathcal{W}$ is said to be Cauchy if for each $ϵ>0$, there exists $N=N\left(ϵ\right)\in \mathbb{N}$ such that ${\phi }_{\alpha \beta }({\rho }_n,{\rho }_n,{\rho }_m)<ϵ$, for all $n,m\ge N$.

3.

 An extended S-metric space of type $(\alpha ,\beta )$ is said to be complete if every Cauchy sequence is convergent.

Proposition 1.

A sequence $\left\{{\rho }_n\right\}$ in an extended S-metric space of type $(\alpha ,\beta )$ converges to some $u\in \mathcal{W}$ if ${\displaystyle \underset{n\to \infty }{lim}{\phi }_{\alpha \beta }(u,u,{\rho }_n)=0}$

Proof. 

This result is immediately followed by Lemma 3 and Definition 5. □

In the following content, we continue our investigation by proposing several definitions and establishing certain theorems and lemmas needed to clarify the proposed space, the extended S-metric space of type $(\alpha ,\beta )$.

Definition 6.

Let $(\mathcal{W},{\phi }_{\alpha \beta })$ be an extended S-metric space of type $(\alpha ,\beta )$. For $r>0$ and $x\in \mathcal{W}$, we define open ball $B(a,r)$ and closed ball $B[a,r]$ with radius r and center a as follows:

$$B(a,r)=\{b\in \mathcal{W}:{\phi }_{\alpha \beta }(b,b,a)<r\},$$

and

$$B[a,r]=\{b\in \mathcal{W}:{\phi }_{\alpha \beta }(b,b,a)\le r\}.$$

Lemma 4.

Let $(\mathcal{W},{\phi }_{\alpha \beta })$ be an extended S-metric space of type $(\alpha ,\beta )$. If sequence $\left\{{\rho }_n\right\}$ in $\mathcal{W}$ converges to ρ, then ρ is unique.

Proof. 

Let $\left\{{\rho }_n\right\}$ be a sequence in $\mathcal{W}$ that converges to $\rho$. To prove that $\rho$ is unique, we assume that there exists $\varrho \in \mathcal{W}$ with $\varrho \ne \rho$ such that ${\rho }_n\to \varrho$. Since $\mathcal{W}$ is an extended S-metric space of type $(\alpha ,\beta )$, then

$${\phi }_{\alpha \beta }(\rho ,\rho ,\varrho )\le \alpha (\rho ,\rho ,\varrho ){\phi }_{\alpha \beta }\left(\rho ,\rho ,{\rho }_n\right)+\beta (\rho ,\rho ,\varrho ){\phi }_{\alpha \beta }\left(\rho ,\rho ,{\rho }_n\right)+{\phi }_{\alpha \beta }({\rho }_n,{\rho }_n,\varrho )\to 0.$$

As a result, ${\phi }_{\alpha \beta }(\rho ,\rho ,\varrho )=0$; hence, $\rho =\varrho$, which is a contradiction. Therefore, $\rho$ is unique. □

Definition 7.

Let $(\mathcal{W},{\phi }_{\alpha \beta })$ be an extended S-metric space of type $(\alpha ,\beta )$. A self-mapping $T:\mathcal{W}\to \mathcal{W}$ is called contraction if there exists a constant $\lambda \in (0,1)$ such that

$${\phi }_{\alpha \beta }(T\rho ,T\varrho ,T\eta )\le \lambda {\phi }_{\alpha \beta }(\rho ,\varrho ,\eta ),\mathit{for}\mathit{all}\rho ,\varrho ,\eta \in \mathcal{W}.$$

Theorem 1.

Let $(\mathcal{W},{\phi }_{\alpha \beta })$ be a complete, extended S-metric space of type $(\alpha ,\beta )$ and T be a self-mapping on $\mathcal{W}$ satisfying the following condition:

$${\phi }_{\alpha \beta }(T\rho ,T\varrho ,T\eta )\le \lambda {\phi }_{\alpha \beta }(\rho ,\varrho ,\eta ),\mathit{for\; all}\rho ,\varrho ,\eta \in \mathcal{W},$$

(3)

where $\lambda \in (0,1)$. For every ${\rho }_{\circ }\in \mathcal{W}$, suppose that ${\rho }_n=T^n{\rho }_{\circ }$, and for $m>i$, we have

$$\underset{m,i\to \infty }{lim}\frac{\alpha ({\rho }_{i+1},{\rho }_{i+1},{\rho }_m)+\beta ({\rho }_{i+1},{\rho }_{i+1},{\rho }_m)}{\alpha ({\rho }_i,{\rho }_i,{\rho }_m)+\beta ({\rho }_i,{\rho }_i,{\rho }_m)}<\frac{1}{\lambda }.$$

(4)

Additionally, assume that for every $\rho \in \mathcal{W}$, we have

$$\underset{n\to \infty }{lim}[\alpha ({\rho }_n,{\rho }_n,\rho )+\beta ({\rho }_n,{\rho }_n,\rho )]<\infty .$$

(5)

Then, T has a unique fixed point, say, $u\in \mathcal{W}$.

Proof. 

To prove this result, we consider sequence ${\rho }_n=T^n{\rho }_{\circ }$. Now, by condition (3), we have

$${\phi }_{\alpha \beta }({\rho }_n,{\rho }_n,{\rho }_{n+1})={\phi }_{\alpha \beta }(T^n{\rho }_{\circ },T^n{\rho }_{\circ },T^n{\rho }_1)\le {\lambda }^n{\phi }_{\alpha \beta }({\rho }_{\circ },{\rho }_{\circ },{\rho }_1),\mathit{for\; all} n\ge 0.$$

In this regard, for all natural numbers $n<m$, we have

$$\begin{array}{cc}\hfill {\phi }_{\alpha \beta }({\rho }_n,{\rho }_n,{\rho }_m)& \le \alpha ({\rho }_n,{\rho }_n,{\rho }_m){\phi }_{\alpha \beta }({\rho }_n,{\rho }_n,{\rho }_{n+1})+\beta ({\rho }_n,{\rho }_n,{\rho }_m){\phi }_{\alpha \beta }({\rho }_n,{\rho }_n,{\rho }_{n+1})\hfill \\ \hfill & +{\phi }_{\alpha \beta }({\rho }_{n+1},{\rho }_{n+1},{\rho }_m)\hfill \\ \hfill & \le \alpha ({\rho }_n,{\rho }_n,{\rho }_m){\phi }_{\alpha \beta }({\rho }_n,{\rho }_n,{\rho }_{n+1})+\beta ({\rho }_n,{\rho }_n,{\rho }_m){\phi }_{\alpha \beta }({\rho }_n,{\rho }_n,{\rho }_{n+1})\hfill \\ \hfill & +\alpha ({\rho }_{n+1},{\rho }_{n+1},{\rho }_m){\phi }_{\alpha \beta }({\rho }_{n+1},{\rho }_{n+1},{\rho }_{n+2})+\beta ({\rho }_{n+1},{\rho }_{n+1},{\rho }_m)\hfill \\ \hfill & {\phi }_{\alpha \beta }({\rho }_{n+1},{\rho }_{n+1},{\rho }_{n+2})+{\phi }_{\alpha \beta }({\rho }_{n+2},{\rho }_{n+2},{\rho }_m)\hfill \\ \hfill & \le [\alpha ({\rho }_n,{\rho }_n,{\rho }_m)+\beta ({\rho }_n,{\rho }_n,{\rho }_m)]{\phi }_{\alpha \beta }({\rho }_n,{\rho }_n,{\rho }_{n+1})\hfill \\ \hfill & +[\alpha ({\rho }_{n+1},{\rho }_{n+1},{\rho }_m)+\beta ({\rho }_{n+1},{\rho }_{n+1},{\rho }_m)]{\phi }_{\alpha \beta }({\rho }_{n+1},{\rho }_{n+1},{\rho }_{n+2})+\cdots +\hfill \\ \hfill & [\alpha ({\rho }_{m-2},{\rho }_{m-2},{\rho }_m)+\beta ({\rho }_{m-2},{\rho }_{m-2},{\rho }_m)]{\phi }_{\alpha \beta }({\rho }_{m-2},{\rho }_{m-2},{\rho }_{m-1})\hfill \\ \hfill & +{\phi }_{\alpha \beta }({\rho }_{m-1},{\rho }_{m-1},{\rho }_m).\hfill \end{array}$$

Consequently, due to $[\alpha ({\rho }_{m-1},{\rho }_{m-1},{\rho }_m)+\beta ({\rho }_{m-1},{\rho }_{m-1},{\rho }_m)]\ge 1$, we obtain

$$\begin{array}{cc}\hfill {\phi }_{\alpha \beta }({\rho }_n,{\rho }_n,{\rho }_m)& \le [\alpha ({\rho }_n,{\rho }_n,{\rho }_m)+\beta ({\rho }_n,{\rho }_n,{\rho }_m)]{\phi }_{\alpha \beta }({\rho }_n,{\rho }_n,{\rho }_{n+1})\hfill \\ \hfill & +[\alpha ({\rho }_{n+1},{\rho }_{n+1},{\rho }_m)+\beta ({\rho }_{n+1},{\rho }_{n+1},{\rho }_m)]{\phi }_{\alpha \beta }({\rho }_{n+1},{\rho }_{n+1},{\rho }_{n+2})\hfill \\ \hfill & +\cdots +[\alpha ({\rho }_{m-2},{\rho }_{m-2},{\rho }_m)+\beta ({\rho }_{m-2},{\rho }_{m-2},{\rho }_m)]{\phi }_{\alpha \beta }({\rho }_{m-2},{\rho }_{m-2},{\rho }_{m-1})\hfill \\ \hfill & +[\alpha ({\rho }_{m-1},{\rho }_{m-1},{\rho }_m)+\beta ({\rho }_{m-1},{\rho }_{m-1},{\rho }_m)]{\phi }_{\alpha \beta }({\rho }_{m-1},{\rho }_{m-1},{\rho }_m)\hfill \\ \hfill & =\sum _{i=n}^{m-1}[\alpha ({\rho }_i,{\rho }_i,{\rho }_m)+\beta ({\rho }_i,{\rho }_i,{\rho }_m)]{\phi }_{\alpha \beta }({\rho }_i,{\rho }_i,{\rho }_{i+1})\hfill \\ \hfill & \le \sum _{i=n}^{m-1}[\alpha ({\rho }_i,{\rho }_i,{\rho }_m)+\beta ({\rho }_i,{\rho }_i,{\rho }_m)]{\lambda }^i{\phi }_{\alpha \beta }({\rho }_{\circ },{\rho }_{\circ },{\rho }_1).\hfill \end{array}$$

Hence, letting ${\displaystyle L_p=\sum _{i=1}^{p-1}[\alpha ({\rho }_i,{\rho }_i,{\rho }_m)+\beta ({\rho }_i,{\rho }_i,{\rho }_m)]{\lambda }^i}$ yields

$${\phi }_{\alpha \beta }({\rho }_n,{\rho }_n,{\rho }_m)\le [L_{m-1}-L_n]{\phi }_{\alpha \beta }({\rho }_{\circ },{\rho }_{\circ },{\rho }_1).$$

(6)

Accordingly, by hypothesis (4), which is concluded by using the ratio test, we obtain that ${lim}_{p\to \infty }{L}_p<\infty$, and sequence $\left\{L_p\right\}$ is Cauchy. Also, if one takes the limit in Equation (6) as $n,m\to \infty$, we deduce

$$\underset{n,m\to \infty }{lim}{\phi }_{\alpha \beta }({\rho }_n,{\rho }_n,{\rho }_m)=0.$$

Hence, $\left\{{\rho }_n\right\}$ is a Cauchy sequence. Thus, since $(\mathcal{W},{\phi }_{\alpha \beta })$ is a complete, extended S-metric space of type $(\alpha ,\beta )$, there exists $u\in \mathcal{W}$ such that ${\displaystyle \underset{n\to \infty }{lim}{\phi }_{\alpha \beta }({\rho }_n,{\rho }_n,u)=\underset{n\to \infty }{lim}{\phi }_{\alpha \beta }(u,u,{\rho }_n)=0}$. Now, it remains to show that u is a fixed point of T. To do so, we obtain, with the use of the definition of ${\phi }_{\alpha \beta }$, the following assertions:

$$\begin{array}{cc}\hfill {\phi }_{\alpha \beta }(Tu,Tu,u)& ={\phi }_{\alpha \beta }(u,u,Tu)\hfill \\ \hfill & \le \alpha (u,u,Tu){\phi }_{\alpha \beta }(u,u,{\rho }_{n+1})+\beta (u,u,Tu){\phi }_{\alpha \beta }(u,u,{\rho }_{n+1})\hfill \\ \hfill & +{\phi }_{\alpha \beta }(Tu,Tu,{\rho }_{n+1})\hfill \\ \hfill & \le [\alpha (u,u,Tu)+\beta (u,u,Tu)]{\phi }_{\alpha \beta }(u,u,{\rho }_{n+1})+\lambda {\phi }_{\alpha \beta }(u,u,{\rho }_n).\hfill \end{array}$$

Taking the limit in the above inequality as $n\to \infty$ yields

$${\phi }_{\alpha \beta }(Tu,Tu,u)=0,$$

i.e., $Tu=u$. Hence, u is a fixed point of T; therefore, the uniqueness of u follows from Lemma 4. □

Definition 8.

Let T be a self-mapping on extended S-metric space $(\mathcal{W},{\phi }_{\alpha \beta })$ of type $(\alpha ,\beta )$. For ${\rho }_{\circ }\in \mathcal{W}$, the set

$$\mathcal{O}({\rho }_{\circ },T)=\left\{{\rho }_{\circ },T{\rho }_{\circ },T^2{\rho }_{\circ },T^3{\rho }_{\circ },\cdots \right\}$$

is said to be an orbit of T at ${\rho }_{\circ }$.

Definition 9.

Let T be a self-mapping on extended S-metric space $(\mathcal{W},{\phi }_{\alpha \beta })$ of type $(\alpha ,\beta )$. A function $\mathcal{Q}:\mathcal{W}\to \mathcal{R}$ is said to be T-orbitally lower semi-continuous at $\ell \in \mathcal{W}$ if

$$\left\{{\rho }_n\right\}\subset \mathcal{O}({\rho }_{\circ },T)\mathrm{and}{\rho }_n\to \ell \mathrm{as}n\to \infty \mathrm{implies}\mathcal{Q}(\ell )\le \underset{n\to \infty }{lim\; inf}\mathcal{Q}\left({\rho }_n\right).$$

Theorem 2.

Let $(\mathcal{W},{\phi }_{\alpha \beta })$ be a complete, extended S-metric space of type $(\alpha ,\beta )$, and let T be a self-mapping on $\mathcal{W}$ satisfying the following condition:

$$\begin{array}{c}\hfill {\phi }_{\alpha \beta }(T\rho ,T\rho ,T^2\rho )\le \lambda {\phi }_{\alpha \beta }(\rho ,\rho ,T\rho ),\mathit{for}\mathit{all}\rho \in \mathcal{W},\end{array}$$

(7)

where $0<\lambda <1$. Suppose that for every ${\rho }_{\circ }\in \mathcal{W}$ and for $m>i$, we have

$$\underset{m,i\to \infty }{lim}\frac{\alpha ({\rho }_{i+1},{\rho }_{i+1},{\rho }_m)+\beta ({\rho }_{i+1},{\rho }_{i+1},{\rho }_m)}{\alpha ({\rho }_i,{\rho }_i,{\rho }_m)+\beta ({\rho }_i,{\rho }_i,{\rho }_m)}<\frac{1}{\lambda }.$$

Then, sequence $\left\{T^n{\rho }_{\circ }\right\}$ is convergent to some $u\in \mathcal{W}$. Moreover, u is a fixed point of T if and only if $\mathcal{Q}\left(\rho \right)={\phi }_{\alpha \beta }(\rho ,\rho ,T\rho )$ is T-orbitally lower semi-continuous at u.

Proof. 

To prove this result, we consider ${\rho }_{\circ }\in \mathcal{W}$ and define sequence $\left\{{\rho }_n\right\}$ as follows:

$${\rho }_1=T{\rho }_{\circ },{\rho }_2=T{\rho }_1=T^2{\rho }_{\circ },\cdots ,{\rho }_n=T^n{\rho }_{\circ }.$$

By condition (7), we obtain

$$\begin{array}{cc}\hfill {\phi }_{\alpha \beta }({\rho }_n,{\rho }_n,{\rho }_{n+1})& \le \lambda {\phi }_{\alpha \beta }({\rho }_{n-1},{\rho }_{n-1},{\rho }_n)\hfill \\ \hfill & \le {\lambda }^n{\phi }_{\alpha \beta }({\rho }_{\circ },{\rho }_{\circ },{\rho }_1).\hfill \end{array}$$

Similarly to the proof of Theorem 1, we conclude that $\left\{{\rho }_n\right\}$ is a Cauchy sequence, and so, it converges, say, to $u\in \mathcal{W}$. Now, suppose that $\mathcal{Q}$ is T-orbitally lower semi-continuous at u.

So, we have

$$\begin{array}{cc}\hfill {\phi }_{\alpha \beta }(u,u,Tu)& \le \underset{n\to \infty }{lim\; inf}{\phi }_{\alpha \beta }({\rho }_n,{\rho }_n,{\rho }_{n+1})\hfill \\ \hfill & \le \underset{n\to \infty }{lim\; inf}{\lambda }^n{\phi }_{\alpha \beta }({\rho }_{\circ },{\rho }_{\circ },{\rho }_1)\hfill \\ \hfill & =0,\hfill \end{array}$$

which consequently implies that $Tu=u$.

Conversely, suppose that $Tu=u$ and $\left\{{\rho }_n\right\}\subset \mathcal{O}({\rho }_{\circ },T)$ with $\left\{{\rho }_n\right\}\to u$ as $n\to \infty$.

Hence, we have

$$\mathcal{Q}\left(u\right)={\phi }_{\alpha \beta }(u,u,Tu)=0\le \underset{n\to \infty }{lim\; inf}{\phi }_{\alpha \beta }({\rho }_n,{\rho }_n,{\rho }_{n+1})=\underset{n\to \infty }{lim\; inf}\mathcal{Q}\left({\rho }_n\right),$$

which completes the proof. □

Example 3.

Let $\mathcal{W}=R$. Define mappings ${\phi }_{\alpha \beta }:{\mathcal{W}}^3\to [0,\infty )$ and $\alpha ,\beta :{\mathcal{W}}^3\to [1,\infty )$ as follows:

$${\phi }_{\alpha \beta }(\rho ,\varrho ,\eta )=|\rho -\eta |+|\varrho -\eta |,$$

$$\alpha (\rho ,\varrho ,\eta )=max\{\rho ,\varrho \}+\eta +1,$$

and

$$\beta (\rho ,\varrho ,\eta )=|\rho +\varrho -\eta |+1.$$

Suppose that T is a self-mapping on $\mathcal{W}$ defined by

$$T\left(\rho \right)=\frac{\rho }{2}.$$

It should be noted, based on the above assumptions, that

$$\begin{array}{cc}\hfill {\phi }_{\alpha \beta }(T\rho ,T\varrho ,T\eta )& =\left|\frac{T\rho }{2}-\frac{T\eta }{2}\right|+\left|\frac{T\varrho }{2}-\frac{T\eta }{2}\right|\hfill \\ \hfill & =\left|\frac{\rho }{4}-\frac{\eta }{4}\right|+\left|\frac{\varrho }{4}-\frac{\eta }{4}\right|\hfill \\ \hfill & =\frac{1}{4}\left(|\rho -\eta |+|\varrho -\eta |\right)\hfill \\ \hfill & =\frac{1}{4}{\phi }_{\alpha \beta }(\rho ,\varrho ,\eta ).\hfill \end{array}$$

So, we may take $\lambda =\frac{1}{4}$. Furthermore, for any $c\in \mathcal{W}$, sequence ${\rho }_n=T^{n}c$ is ${\rho }_n=\frac{c}{2^n}$. So, for $m>i$, we have

$$\begin{array}{cc}\hfill \underset{i,m\to \infty }{lim}& \frac{\alpha ({\rho }_{i+1},{\rho }_{i+1},{\rho }_m)+\beta ({\rho }_{i+1},{\rho }_{i+1},{\rho }_m)}{\alpha ({\rho }_i,{\rho }_i,{\rho }_m)+\beta ({\rho }_i,{\rho }_i,{\rho }_m)}\hfill \\ \hfill & =\underset{i,m\to \infty }{lim}\frac{1+max\{\frac{c}{2^{i+1}},\frac{c}{2^{i+1}}\}+\frac{c}{2^m}+1+|\frac{c}{2^{i+1}}+\frac{c}{2^{i+1}}-\frac{c}{2^m}|}{1+max\{\frac{c}{2^i},\frac{c}{2^i}\}+\frac{c}{2^m}+1+|\frac{c}{2^i}+\frac{c}{2^i}-\frac{c}{2^m}|}\hfill \\ \hfill & =\underset{i,m\to \infty }{lim}\frac{\frac{c}{2^{i+1}}+\frac{c}{2^m}+\frac{2c}{2^{i+1}}-\frac{c}{2^m}+2}{\frac{c}{2^i}+\frac{c}{2^m}+\frac{2c}{2^i}-\frac{c}{2^m}+2}\hfill \\ \hfill & =\underset{i\to \infty }{lim}\frac{\frac{3c}{2^{i+1}}+2}{\frac{3c}{2^i}+2}\hfill \\ \hfill & =1\hfill \\ \hfill & <\frac{1}{\lambda }.\hfill \end{array}$$

Hence, all conditions in Theorem 1 are satisfied, so T has a unique fixed point equal to 0.

In what follows, we shall use some nonlinear functions to expand the prior theorem. This is also supported by a given example.

Theorem 3.

Let $(\mathcal{W},{\phi }_{\alpha \beta })$ be a complete, extended S-metric space of type $(\alpha ,\beta )$ such that T is a continuous self-mapping on $\mathcal{W}$ satisfying

$$\begin{array}{c}\hfill {\phi }_{\alpha \beta }(T\rho ,T\varrho ,T\eta )\le \psi \left[{\phi }_{\alpha \beta }(\rho ,\varrho ,\eta )\right],\mathit{for}\mathit{all}\rho ,\varrho ,\eta \in \mathcal{W},\end{array}$$

(8)

where $\psi :[0,\infty )\to [0,\infty )$ is an increasing function such that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{\psi }^n\left(\zeta \right)=0,\end{array}$$

(9)

for each fixed $\zeta >0$. Also, suppose that there exist $\delta >4$ and some $L\in \mathbb{N}$ such that for every $c\in \mathcal{W}$, we have

$$\alpha ({\rho }_{n+1},{\rho }_{n+1},c)+\beta ({\rho }_{n+1},{\rho }_{n+1},c)<\frac{\delta }{2},\forall n\ge L.$$

Then, T has a unique fixed point in $\mathcal{W}$.

Proof. 

Let $\rho \in \mathcal{W}$, and construct a sequence $\left\{{\rho }_n\right\}$ in $\mathcal{W}$ as follows: ${\rho }_0=\rho$, ${\rho }_1=T{\rho }_0$, ${\rho }_2=T{\rho }_1=T^2{\rho }_0$, …, ${\rho }_n=T{\rho }_{n-1}=T^{n-1}{\rho }_0$. For $ϵ>0$, then (9) implies that for large n, we have ${\psi }^n\left(ϵ\right)<\frac{ϵ}{{\delta }^2}$. Without losing generality, we may choose n to be greater than L. Let $G=T^n$ and $\gamma ={\psi }^n$. By using (8) and the increasing property of $\psi$, we have

$$\begin{array}{ccc}\hfill {\phi }_{\alpha \beta }(G\rho ,G\rho ,G\varrho )& =& {\phi }_{\alpha \beta }(T^n\rho ,T^n\rho ,T^n\varrho )\hfill \\ & \le & \psi \left({\phi }_{\alpha \beta }(T^{n-1}\rho ,T^{n-1}\rho ,T^{n-1}\varrho )\right)\hfill \\ & \le & {\psi }^2\left({\phi }_{\alpha \beta }(T^{n-2}\rho ,T^{n-2}\rho ,T^{n-2}\varrho )\right)\hfill \\ & ⋮& \\ & \le & {\psi }^n\left({\phi }_{\alpha \beta }(\rho ,\rho ,\varrho )\right)\hfill \\ & =& \gamma \left({\phi }_{\alpha \beta }(\rho ,\rho ,\varrho )\right).\hfill \end{array}$$

So, ${\phi }_{\alpha \beta }({\rho }_{n+1},{\rho }_{n+1},{\rho }_n)\to 0$ as $n\to \infty$. Hence, there is $k\in \mathbb{N}$ such that ${\displaystyle {\phi }_{\alpha \beta }({\rho }_{k+1},{\rho }_{k+1},{\rho }_k)<\frac{ϵ}{2\delta }}$. It should be noted here that ${\rho }_k\in B({\rho }_k,ϵ)$, so $B({\rho }_k,ϵ)\ne \varphi .$ Hence, for all $\eta \in B({\rho }_k,ϵ)$, we have

$$\begin{array}{cc}\hfill {\phi }_{\alpha \beta }(G\eta ,G\eta ,G{\rho }_k)={\phi }_{\alpha \beta }(G{\rho }_k,G{\rho }_k,G\eta )& \le \gamma \left({\phi }_{\alpha \beta }({\rho }_k,{\rho }_k,\eta )\right)\hfill \\ \hfill & \le \gamma \left(ϵ\right)\hfill \\ \hfill & ={\psi }^n\left(ϵ\right)\hfill \\ \hfill & <\frac{ϵ}{{\delta }^2}.\hfill \end{array}$$

Also, we have

$$\begin{array}{cc}\hfill {\phi }_{\alpha \beta }(G\eta ,G\eta ,{\rho }_{k+1})& \le [\alpha \left({\phi }_{\alpha \beta }(G\eta ,G\eta ,{\rho }_k)\right)+\beta \left({\phi }_{\alpha \beta }(G\eta ,G\eta ,{\rho }_k)\right)]{\phi }_{\alpha \beta }(G\eta ,G\eta ,G{\rho }_k)\hfill \\ \hfill & +{\phi }_{\alpha \beta }({\rho }_{k+1},{\rho }_{k+1},{\rho }_k)\hfill \\ \hfill & {\displaystyle <\frac{\delta }{2}\frac{ϵ}{{\delta }^2}+\frac{ϵ}{2\delta }}\hfill \\ \hfill & {\displaystyle =\frac{ϵ}{\delta }.}\hfill \end{array}$$

Now, due to ${\phi }_{\alpha \beta }(T{\rho }_k,T{\rho }_k,T{\rho }_k)={\phi }_{\alpha \beta }({\rho }_{k+1},{\rho }_{k+1},{\rho }_k)<\frac{ϵ}{\delta }$, we obtain

$$\begin{array}{cc}\hfill {\phi }_{\alpha \beta }({\rho }_k,{\rho }_k,G\eta )& \le \alpha ({\rho }_k,{\rho }_k,G\eta ){\phi }_{\alpha \beta }({\rho }_k,{\rho }_k,{\rho }_{k+1})+\beta ({\rho }_k,{\rho }_k,G\eta ){\phi }_{\alpha \beta }({\rho }_k,{\rho }_k,{\rho }_{k+1})\hfill \\ \hfill & +{\phi }_{\alpha \beta }(G\eta ,G\eta ,{\rho }_{k+1})\hfill \\ \hfill & \le [\alpha ({\rho }_k,{\rho }_k,G\eta )+\beta ({\rho }_k,{\rho }_k,G\eta )]{\phi }_{\alpha \beta }({\rho }_k,{\rho }_k,{\rho }_{k+1})+{\phi }_{\alpha \beta }(G\eta ,G\eta ,{\rho }_{k+1})\hfill \\ \hfill & \le [\alpha ({\rho }_k,{\rho }_k,G\eta )+\beta ({\rho }_k,{\rho }_k,G\eta )]{\phi }_{\alpha \beta }({\rho }_k,{\rho }_k,{\rho }_{k+1})\hfill \\ \hfill & +[\alpha (G\eta ,G\eta ,{\rho }_{k+1})+\beta (G\eta ,G\eta ,{\rho }_{k+1})]{\phi }_{\alpha \beta }(G\eta ,G\eta ,{\rho }_{k+1})\hfill \\ \hfill & <\frac{\delta }{2}\left(\frac{ϵ}{2\delta }\right)+\frac{\delta }{2}\left(\frac{ϵ}{\delta }\right)\hfill \\ \hfill & <ϵ.\hfill \end{array}$$

Hence, G maps $B({\rho }_k,ϵ)$ to itself. In the same regard, since ${\rho }_k\in B({\rho }_k,ϵ)$, we have $G{\rho }_k\in B({\rho }_k,ϵ)$. Now, by repeating the same process as above, we obtain

$$G_{{\rho }_k}^m\in B({\rho }_k,ϵ),\mathrm{for}\mathrm{all}m\in \mathbb{N}.$$

In other words, we have

$${\rho }_l\in B({\rho }_k,ϵ),\mathrm{for}\mathrm{all}l\ge k.$$

Consequently, we obtain

$${\phi }_{\alpha \beta }({\rho }_m,{\rho }_m,{\rho }_l)<ϵ,\mathrm{for}\mathrm{all}m,l>k.$$

Therefore, $\left\{{\rho }_k\right\}$ is a Cauchy sequence. Now, since $\mathcal{W}$ is complete, there exists $c\in \mathcal{W}$ such that ${\rho }_k\to c$ as $k\to \infty$. Moreover, we have

$$c=\underset{k\to \infty }{lim}{\rho }_{k+1}=\underset{k\to \infty }{lim}{\rho }_k=G\left(c\right),$$

so G has a fixed point c. Thus, after we have finished proving the existence part, we intend to prove the uniqueness part. To do so, we let c and v be two fixed points of G. So, we have

$$\begin{array}{cc}\hfill {\phi }_{\alpha \beta }(c,c,v)& ={\phi }_{\alpha \beta }(Gc,Gc,Gv)\hfill \\ \hfill & \le {\psi }^n\left({\phi }_{\alpha \beta }(c,c,v)\right)\hfill \\ \hfill & =\gamma ({\phi }_{\alpha \beta }(c,c,v)\hfill \\ \hfill & <{\phi }_{\alpha \beta }(c,c,v).\hfill \end{array}$$

Thus, ${\phi }_{\alpha \beta }(c,c,v)=0$, so $c=v$. Hence, G has a unique fixed point. On the other hand, it should be noted that $T^{nk+r}\left(\rho \right)=G^k\left(T^r\left(\rho \right)\right)\to c$, as $k\to \infty$. Therefore, $T^m\rho \to c$, as $m\to \infty$, for all $\rho$. This implies that $c={lim}_{m\to \infty }T{\rho }_m=T\left(c\right)$; hence, T possesses a unique fixed point equal to 0. □

Example 4.

Let $\mathcal{W}=[0,1]$. Define mappings ${\phi }_{\alpha \beta }:{\mathcal{W}}^3\to [0,\infty )$ and $\alpha ,\beta :{\mathcal{W}}^3\to [1,\infty )$ as follows:

$${\phi }_{\alpha \beta }(\rho ,\varrho ,\eta )=\left\{\begin{array}{c}0,\rho =\varrho =\eta \hfill \\ |max\{\rho ,\varrho \}-\eta |,\mathrm{otherwise}\hfill \end{array}\right.,$$

$$\alpha (\rho ,\varrho ,\eta )=max\{\rho ,\varrho \}+\eta +1,$$

$$\beta (\rho ,\varrho ,\eta )=min\{\rho ,\varrho \}+\eta +2,$$

and

$$\psi \left(v\right)=\frac{1}{2}v.$$

Let T be a self-mapping on $\mathcal{W}$ defined by

$$T\left(\rho \right)=\frac{\rho }{2}.$$

Note that

$$\begin{array}{cc}\hfill {\phi }_{\alpha \beta }(T\rho ,T\varrho ,T\eta )& =\left|max\left\{\frac{\rho }{2},\frac{\varrho }{2}\right\}-\frac{\eta }{2}\right|\hfill \\ \hfill & =\frac{1}{2}{\phi }_{\alpha \beta }(\rho ,\varrho ,\eta )\hfill \\ \hfill & =\psi \left({\phi }_{\alpha \beta }(\rho ,\varrho ,\eta )\right).\hfill \end{array}$$

Furthermore, for any $\rho \in \mathcal{W}$, we have ${\displaystyle {\rho }_n=\frac{\rho }{2^n}}$. So, for any $c\in \mathcal{W}$, we obtain

$$\begin{array}{cc}\hfill \alpha ({\rho }_n,{\rho }_n,c)+\beta ({\rho }_n,{\rho }_n,c)& =max\left\{\frac{\rho }{2^n},\frac{\rho }{2^n}\right\}+c+1+min\left\{\frac{\rho }{2^n},\frac{\rho }{2^n}\right\}+c+1\hfill \\ \hfill & =2c+2+\frac{\rho }{2^{n-1}}\hfill \\ \hfill & \le 4+\frac{1}{2^{n-1}}.\hfill \end{array}$$

Note that we can choose $L=3$ so that $\alpha ({\rho }_n,{\rho }_n,c)+\beta ({\rho }_n,{\rho }_n,c)<\frac{9}{2}$, and in this case, we pick δ to be 9. Therefore, all conditions of the above theorem are satisfied; hence, T has a unique fixed point equal to 0.

4. Applications

We use our results in this part to provide a condensed justification for the existence of a solution to the Fredholm integral equation. For this purpose, let $\mathcal{W}=\mathcal{C}[a,c]$ be the space of all continuous real-valued functions on $[a,c]$. Define three functions ${\phi }_{\alpha \beta }:{\mathcal{W}}^3\to [0,\infty )$ by

$${\phi }_{\alpha \beta }(\rho \left(\zeta \right),\varrho \left(\zeta \right),\eta \left(\zeta \right))={\parallel \rho \left(\zeta \right)-\eta \left(\zeta \right)\parallel }_{\infty }{+\parallel \varrho \left(\zeta \right)-\eta \left(\zeta \right)\parallel }_{\infty }{,\mathrm{where}\parallel a\left(\zeta \right)\parallel }_{\infty }=\underset{\zeta \in [a,c]}{max}\left|a\left(\zeta \right)\right|,$$

and $\alpha ,\beta :{\mathcal{W}}^3\to [1,\infty )$ by

$$\alpha (\rho \left(\zeta \right),\varrho \left(\zeta \right),\eta \left(\zeta \right))=\underset{\zeta \in [a,c]}{max}\left(\underset{\zeta \in [a,c]}{max}\left\{\left|\rho \right(\zeta \left)\right|,\left|\varrho \right(\zeta \left)\right|\right\}+\left|\eta \left(\zeta \right)\right|+1\right),$$

and

$$\beta (\rho \left(\zeta \right),\varrho \left(\zeta \right),\eta \left(\zeta \right)))=\underset{\zeta \in [a,c]}{min}\left(\underset{\zeta \in [a,c]}{min}\left\{\left|\rho \right(\zeta \left)\right|,\left|\varrho \right(\zeta \left)\right|\right\}+\left|\eta \left(\zeta \right)\right|+2\right).$$

It should be noted here that $(\mathcal{W},S)$ is a complete, extended S-metric space of type $(\alpha ,\beta )$. Now, consider the following Fredholm integral equation:

$$\rho \left(\zeta \right)=j\left(\zeta \right)+{\int }_a^{c}M(\zeta ,s,\rho \left(s\right))ds,\zeta ,s\in [a,c],$$

(10)

where $j:[a,c]\to (-\infty ,\infty )$ and $M:[a,c]\times [a,c]\times (-\infty ,\infty )\to (-\infty ,\infty )$ are both continuous functions. Let $T:\mathcal{W}\to \mathcal{W}$ be given by

$$T\left(\rho \left(\zeta \right)\right)=j\left(\zeta \right)+{\int }_a^{c}M(\zeta ,s,\rho \left(s\right))ds,\zeta ,s\in [a,c].$$

In addition, assume that

$$|M(\zeta ,s,{\eta }_1)-M(\zeta ,s,{\eta }_2)|\le \frac{1}{c-a}|{\eta }_1-{\eta }_2|,\mathit{for\; each}\zeta ,s\in [a,c].$$

Then, the above integral equation has a solution. Now, we have to show that the operator T satisfies all conditions of Theorem 2 with $\lambda =\frac{1}{2}$. To do so, we have

$$\begin{array}{cc}\hfill \parallel T\rho -T^2{\rho \parallel }_{\infty }& =\underset{\zeta \in [a,c]}{max}\left|T\rho \left(\zeta \right)-T^2\rho \left(\zeta \right)\right|\hfill \\ \hfill & =\underset{\zeta \in [a,c]}{max}\left|{\int }_a^{c}M(\zeta ,s,\rho \left(s\right))-M(\zeta ,s,T\rho \left(s\right))ds\right|\hfill \\ \hfill & \le \underset{\zeta \in [a,c]}{max}\frac{1}{c-a}\left|\rho \left(\zeta \right)-T\rho \left(\zeta \right)\right|{\int }_a^{c}ds\hfill \\ \hfill & ={\parallel \rho -T\rho \parallel }_{\infty },\hfill \end{array}$$

for any $\rho \in \mathcal{W}$. Consequently, we obtain

$${\phi }_{\alpha \beta }(T\rho ,T\rho ,T^2\rho )\le \frac{1}{2}{\phi }_{\alpha \beta }(\rho ,\rho ,T\rho ),$$

which finishes our justification.

5. Conclusions

In this work, we have investigated a generalization of S-metric spaces that was introduced in [9] and have named it extended S-metric space of type $(\alpha ,\beta )$. This has been achieved by proposing two mappings using the rectangle inequality, which makes it different from other generalizations. As a result, we have introduced some results on the existence and uniqueness of the fixed point. Finally, we have provided an application to justify the existence of the solution to the Fredholm integral equation. In future work, researchers should demonstrate additional fixed-point outcomes in this new space.