Solving a System of Integral Equations in Rectangular Menger Probabilistic Metric Spaces and Rectangular Menger Probabilistic b-Metric Spaces

Abstract

This work introduces the concepts of rectangular Menger probabilistic metric ($RMPM$) space and rectangular Menger probabilistic b-metric ($RMPbM$) space as generalizations of the Menger probabilistic metric space and the Menger probabilistic b-metric space, respectively. Some nonunique fixed-point and coupled-fixed-point results for contractive mappings are provided. The findings extend and improve outcomes presented in the existing literature. The main results are illustrated with examples, and validated by means of an application to a system of integral equations. The importance of spaces with non-Hausdorff topology is high, as is the case of computer science, with the Tarskian approach to programming language semantics.

1. Introduction and Background

Various real-world problems are formulated mathematically in terms of integral and differential equations. On the one hand, symmetry plays an important role in examining different types of such Equations [1,2]. On the other hand, integral and differential equations play a special role in describing many symmetry phenomena, such as symmetry breaking in molecular vibrations [3], nonlinear resonators [4], and others [5].

The idea of b-metric space goes back to the work of Bakhtin [6] and Czerwik [7]. Thereafter, some other authors discussed b-metric spaces and introduced convergent and Cauchy sequences. Furthermore, several fixed-point results were obtained, with applications to nonlinear functional analysis. In [8], stochastic matrix functions were introduced with the objective of stabilizing a nonlinear $\mathsf{\Psi }$-Hilfer fractional Volterra integro-differential equation, while fixed-point theory methods were used to address Ulam–Hyers and Ulam–Hyers–Rassias stability in a matrix MB-space. In [9], the authors considered $C^{*}$-algebra valued fuzzy normed spaces with applications to a random integral equation, namely the problem of Hyers–Ulam stability. In [10], a nonlinear Cauchy problem in the $\mathsf{\Psi }$-Hilfer stochastic fractional derivative was addressed. Using fixed-point theory, the fuzzy Ulam–Hyers–Rassias stability was investigated. In [11], the concept and properties of e-distance on a Menger PGM space were defined and coupled-fixed-point results were demonstrated. In [12], fixed-point theorems were obtained in the scope of set-valued F-contractions in quasi-ordered metric spaces. In [13], the idea of integral type contraction was adopted to show coupled-fixed-point results in ordered G-metric spaces. In [14], fixed-point findings in the framework of F-quasi-metric spaces were presented, a Hausdorff F-distance was introduced, and a coincidence-point theorem was derived. Additional results on fixed-point theory in b-metric spaces, and properties of b-metric spaces are provided in [15,16,17,18,19].

The notion of metric space was developed by Branciari [20] to introduce the generalized Banach contraction theorem and the idea of rectangular metric space. George et al. [21] proposed rectangular b-metric spaces, which generalized the concept of metric space, rectangular metric space, and b-metric space. Under this concept, they proved the analog of the Banach contraction mapping principle. On the other hand, the concept of probabilistic metric ($PM$) space was proposed by Menger [22] in the mid of the last century. Afterwards, Sehgal and Bharucha-Reid [23] proved fixed-point theorems in $PM$ spaces, while Schweizer and Sklar [24] discussed properties of Menger probabilistic metric ($MPM$) spaces. The fixed-point theory in $PM$ spaces for single- and multi-valued mappings was widely studied by several authors. In [25], the authors introduced the concept of monotone generalized contraction in partially ordered probabilistic metric spaces, and derived fixed-point results generalizing some known matters. In [26], the Banach fixed-point principle for general nonlinear contractions in its probabilistic version was provided. In [27], fixed-point theorems were established based on new classes of contractive mappings in MPM spaces. In [28], fixed-point results for probabilistic contractions were extended to multi-valued contractions in generalized Menger spaces. In [29], fixed-point results were introduced for multi- and single-valued mappings, as well as coupled-fixed-point theorems in complete and partially ordered MPM spaces. The results extended and generalized probabilistic versions of the Banach contraction. In [30], the idea of generalized type of rational F-contraction mapping was presented and was adopted to obtain fixed-point results in a complete b-metric space. In [31], fixed-point results with different types of contraction in rectangular b-metric spaces were proved. These results extended some of those already reported in the literature. In [32], fixed-point theorems were extended in rectangular b-metric spaces, establishing the existence and uniqueness of a fixed point for Kannan type mappings. Additional results can be found in [33,34]. Recently, Hasanvand and Khanehgir [35] introduced the Menger probabilistic b-metric ($MPbM$) space, and proved a fixed-point theorem for the single-value operator.

In this paper, we introduce the concepts of rectangular Menger probabilistic metric ($RMPM$) space, which generalizes the $MPM$ space, and rectangular Menger probabilistic b-metric ($RMPbM$) space, which generalizes the $MPbM$ space. These new non-Hausdorff topology spaces play an important role in different areas, namely in the Tarskian view to programming language semantics adopted in computer science [36].

Some useful definitions for the rest of the paper are recalled from the existing literature and are presented in the sequel.

Definition 1

([6,7]). Let us consider that V is a nonempty set and $\mu :V\times V\to [0,+\infty )$ is a functional, satisfying the following:

(i)

$\mu (u,v)=0$ if $u=v$,

(ii)

$\mu (u,v)=\mu (v,u)$ for all $u,v\in V$,

(iii)

there exists a real number $s\ge 1$ such that $\mu (u,n)\le s\left[\mu \right(u,v)+\mu (v,n\left)\right]$ for all $u,v,n\in V$.

Then, μ is a b-metric on V, and $(V,\mu )$ is a b-metric space with coefficient s.

Definition 2

([20]). Assume that $V\ne \varnothing$. Then, $(V,\mu )$ is a rectangular metric space when for every $u,v\in V$ and distinct $x,y\in V\setminus \{u,v\}$,

(i)

$\mu (u,v)=0$ if $u=v$,

(ii)

$\mu (u,v)=\mu (v,u)$,

(iii)

$\mu (u,v)\le \mu (u,x)+\mu (x,y)+\mu (y,v)$.

Definition 3

([21]). Consider $V\ne \varnothing$ with the coefficient $s\ge 1$. Then, $(V,{\mu }_b)$ is a rectangular b-metric space when for each $u,v\in V$ and distinct $u,v\in V\setminus \{u,v\}$,

(i)

${\mu }_b(u,v)=0$ if $u=v$,

(ii)

${\mu }_b(u,v)={\mu }_b(v,u)$,

(iii)

${\mu }_b(u,v)\le s[{\mu }_b(u,u)+{\mu }_b(u,v)+{\mu }_b(v,v)]$.

Definition 4

([25]). A triangular norm, or t-norm, is a binary operation τ on the interval $[0,1]$, satisfying:

(i)

Associative and commutative properties;

(ii)

Continuity;

(iii)

$\tau (a,1)=a$ for all $a\in [0,1]$;

(iv)

$\tau (a,b)\le \tau (c,d)$ when $a\le c$, and $b\le a$, for each $a,b,c,d\in [0,1].$

Definition 5

([33]). An $MPM$ space is a triplet $(V,F,\tau )$, where V denotes a nonempty set, τ stands for a continuous t-norm, and F corresponds to a mapping from $V\times V$ into $D^{+}$, with $D^{+}$ being the set of Menger distance distribution functions, such that if $F_{u,v}$ is the value of F at $(u,v)$:

(i)

$F_{u,v}\left(t\right)=1$ for all $t>0\iff u=v$;

(ii)

$F_{u,v}\left(t\right)=F_{v,u}\left(t\right)$ for all $u,v\in V$ and $t>0$;

(iii)

$F_{u,n}(t+s)\ge \tau (F_{u,v}\left(t\right),F_{v,n}\left(s\right))$ for all $u,v,n\in V$ and $t,s\ge 0$.

Definition 6

([35]). Assume that $V\ne \varnothing$, $\mathcal{T}$ is a continuous t-norm, $\mathcal{F}:V\times V\to D^{+}$ is a mapping, and $\alpha \in (0,1]$. Then, $(V,\mathcal{F},\mathcal{T})$ is a $MPbM$ space when for every $u,v,z\in V$ and $t,s>0$, as follows:

(i)

${\mathcal{F}}_{u,v}\left(t\right)=1$ if $u=v$;

(ii)

${\mathcal{F}}_{u,v}\left(t\right)={\mathcal{F}}_{v,u}\left(t\right)$;

(iii)

${\mathcal{F}}_{u,z}(t+s)\ge \mathcal{T}({\mathcal{F}}_{u,v}\left(\alpha t\right),{\mathcal{F}}_{v,z}\left(\alpha s\right))$.

The $MPbM$ space is a $MPM$ space with $\alpha =1$. However, the converse is not true. Therefore, the class of $MPbM$ spaces is wider than the one of $MPM$ spaces. For further notions on $MPM$ spaces, please see reference [35].

Definition 7

([37]). A function $\phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is called a φ-function, whenever the following terms are met:

(i)

$\phi \left(q\right)=0$ if $q=0$;

(ii)

$\phi \left(q\right)$ is strictly monotone non-decreasing and ${lim}_{q\to \infty }\phi \left(q\right)=\infty$;

(iii)

φ is left-continuous in $(0,\infty )$;

(iv)

φ is continuous at 0.

In the follow-up, we introduce the notion of $RMPM$ and $RMPbM$ spaces, and obtain new fixed-point and coupled-fixed-point results in these spaces.

Section 2 introduces the $RMPM$ space and establishes a fixed-point theorem for single-valued operators. Section 3 presents the $RMPbM$ space and establishes a coupled-fixed-point theorem. Section 4 gives an application to the existence of solution for a system of the integral Equations ($SIE$). Finally, Section 5 summarizes the main conclusions.

2. New Fixed-Point Theorem for Single-Valued Operators in $\mathbf{RMPM}$ Space

We start by introducing some useful definitions under the framework of $RMPM$ space.

Definition 8.

Assume that $V\ne \varnothing$, $\mathcal{T}$ is a continuous t-norm and $\mathbb{E}:V\times V\to D^{+}$. Then, $(V,\mathbb{E},\mathcal{T})$ is a $RMPM$ space when for every $u,v\in V$ and distinct $x,y\in V\setminus \{u,v\}$:

$\left(\mathbb{E}1\right)$

${\mathbb{E}}_{u,v}\left(t\right)=1$ if $u=v$;

$\left(\mathbb{E}2\right)$

${\mathbb{E}}_{u,v}\left(t\right)={\mathbb{E}}_{v,u}\left(t\right)$;

$\left(\mathbb{E}3\right)$

${\mathbb{E}}_{u,v}(t+s+z)\ge \mathcal{T}(\mathcal{T}({\mathbb{E}}_{u,x}\left(t\right),{\mathbb{E}}_{x,y}\left(s\right)),{\mathbb{E}}_{y,v}\left(z\right))$.

It should be noted that each $PM$ space is a $RPM$ space. However, the opposite is not necessarily true.

Example 1.

Consider $V=\{1,2,3,4\}$ and $\mathcal{T}(u,v)=min\{u,v\}$. Introduce $\mathbb{E}:V\times V\to D^{+}$ by

$${\mathbb{E}}_{u,v}\left(t\right)=1ifu=v,$$

$${\mathbb{E}}_{u,v}\left(t\right)={\mathbb{E}}_{v,u}\left(t\right)forallu,v\in V,$$

$${\mathbb{E}}_{1,2}\left(t\right)=\frac{1}{\alpha },$$

$${\mathbb{E}}_{1,4}\left(t\right)={\mathbb{E}}_{2,4}\left(t\right)={\mathbb{E}}_{3,4}\left(t\right)=\frac{1}{4\alpha },$$

$${\mathbb{E}}_{1,3}\left(t\right)={\mathbb{E}}_{2,3}\left(t\right)=\frac{1}{\beta },$$

where $\alpha \ge \beta >1$. We show that $\mathbb{E}$ is a $RMPM$ space. As points $\left(\mathbb{E}1\right)$ and $\left(\mathbb{E}2\right)$ are trivial, focusing in $\left(\mathbb{E}3\right)$, we obtain

$${\mathbb{E}}_{1,2}(t+s+z)=\frac{1}{\alpha }\ge min\{{\mathbb{E}}_{1,3}\left(t\right),{\mathbb{E}}_{2,4}\left(s\right),{\mathbb{E}}_{3,4}\left(z\right)\}=\frac{1}{4\alpha }.$$

So, for every $u,v\in V$ with distinct $x,y\in V\setminus \{u,v\}$, we obtain

$${\mathbb{E}}_{u,v}(t+s+z)\ge \mathcal{T}\left(\mathcal{T}({\mathbb{E}}_{u,x}\left(t\right),{\mathbb{E}}_{x,y}\left(s\right)){\mathbb{E}}_{y,v}\left(z\right)\right).$$

Then, $(V,\mathbb{E},\mathcal{T})$ is a $RMPM$ space. However, $(V,\mathbb{E},\mathcal{T})$ is not a $PM$ space because

$${\mathbb{E}}_{1,2}(t+s)=\frac{1}{\alpha }\ngeqq min\{{\mathbb{E}}_{1,3}\left(t\right),{\mathbb{E}}_{3,2}\left(s\right)\}=\frac{1}{\beta }.$$

Lemma 1.

Consider that $(V,\mathbb{E},\mathcal{T})$ is a $RMPM$ space and φ is a φ-function. If we obtain ${\mathbb{E}}_{u,v}\left(\phi \left(t\right)\right)\ge {\mathbb{E}}_{u,v}\left(\phi \left(\frac{t}{c}\right)\right)$ for every $u,v\in V$, $t>0$ and $c\in (0,1)$, then ${\mathbb{E}}_{u,v}\left(t\right)=1$.

Proof. 

Since ${\mathbb{E}}_{u,v}\left(\phi \left(t\right)\right)\ge {\mathbb{E}}_{u,v}\left(\phi \left(\frac{t}{c}\right)\right)$, we have

$${\mathbb{E}}_{u,v}\left(\phi \left(t\right)\right)\ge {\mathbb{E}}_{u,v}\left(\phi \left(\frac{t}{c}\right)\right)\ge {\mathbb{E}}_{u,v}\left(\phi \left(\frac{t}{c^2}\right)\right)\ge \dots \ge {\mathbb{E}}_{u,v}\left(\phi \left(\frac{t}{c^n}\right)\right)$$

for every $n\in \mathbb{N}$ and $t>0$. By letting $n\to \infty$, we obtain ${\mathbb{E}}_{u,v}\left(\phi \left(t\right)\right)\ge 1$, so $u=v$. □

The convergence, Cauchy sequence and completeness are introduced in the follow-up. We assume that there exists a $N\in \mathbb{N}$ such that $ϵ,\delta >0$. Thus, for $n\ge N$, we have ${\mathbb{E}}_{u_n,u}\left(ϵ\right)>1-\delta$. Then, the sequence $\left\{u_n\right\}$ in V converges to u in V. If, for every $ϵ,\delta >0$, there exists a $N\in \mathbb{N}$ such that ${\mathbb{E}}_{u_n,u_m}\left(ϵ\right)>1-\delta$ for every $m,n\ge N$, then $\left\{u_n\right\}$ in V is a Cauchy sequence. If every Cauchy sequence is convergent in a $RMPM$ space, then it is a complete space.

For every $p\in V$ and $\lambda >0$, the $ϵ$- neighborhood of p is

$$N_p\left(\lambda \right)=\{q\in V:{\mathbb{E}}_{p,q}\left(ϵ\right)>1-ϵ\}.$$

Additionally, $U\subseteq V$ is called bounded if there is a $\delta >0$ and $ϵ\in (0,1)$ such that ${\mathbb{E}}_{u,v}\left(\delta \right)>1-ϵ$ for every $u,v\in U$.

Lemma 2.

Suppose that $(V,\mathbb{E},\mathcal{T})$ is a $RMPM$ space and $\left\{u_n\right\}$ is a Cauchy sequence so that $u_n\ne u_m$, with $m\ne n$. Then, $\left\{u_n\right\}$ converges at most to one point.

Proof. 

Suppose that a sequence $\left\{u_n\right\}$ in V has two limit points $u,v\in V$, that is, ${lim}_{n\to \infty }{\mathbb{E}}_{u_n,u}\left(t\right)=1$ and ${lim}_{n\to \infty }{\mathbb{E}}_{u_n,v}\left(t\right)=1$. So, from $\left(\mathbb{E}3\right)$,

$${\mathbb{E}}_{u,v}(t+s+z)\ge \mathcal{T}(\mathcal{T}({\mathbb{E}}_{u,u_n}\left(t\right),{\mathbb{E}}_{u_n,u_m}\left(s\right)),{\mathbb{E}}_{u_m,v}\left(z\right)),$$

and, as $n,m\to \infty$, we obtain ${\mathbb{E}}_{u,v}(t+s+z)\ge 1$, so $u=v$. □

Theorem 1.

Consider that $(V,\mathbb{E},\mathcal{T})$ is a complete $RMPM$ space, $\mathcal{T}(a,a)\ge a$, with $a\in [0,1]$, and $g:V\to V$ is a continuous operator, satisfying the following:

$$\begin{array}{cc}\hfill {\mathbb{E}}_{gu,gv}\left(\phi \left(t\right)\right)& \ge \lambda min\{{\mathbb{E}}_{u,v}\left(\phi \left(\frac{t}{c}\right)\right),{\mathbb{E}}_{u,gu}\left(\phi \left(\frac{t}{c}\right)\right),{\mathbb{E}}_{gv,v}\left(\phi \left(\frac{t}{c}\right)\right),\hfill \\ \hfill & {\mathbb{E}}_{u,gv}\left(\phi \left(\frac{t}{c}\right)\right),{\mathbb{E}}_{gu,v}\left(\phi \left(\frac{t}{c}\right)\right)\},\hfill \end{array}$$

where $\lambda \ge 1$. Then, g has a fixed point.

Proof. 

With the initial point $u_0\in V$, construct an iterative sequence $\left\{u_n\right\}$ by:

$$u_1=f{u}_0,u_2=f^2{u}_0,u_3=f^3{u}_0,\cdots ,u_n=f^n{u}_0,\dots .$$

Since $\phi$ is continuous at 0, we can find $r>0$ such that $t>\phi \left(r\right)$. So, we obtain

$$\begin{array}{cc}\hfill {\mathbb{E}}_{u_n,u_{n+1}}\left(\phi \left(r\right)\right)& ={\mathbb{E}}_{f{u}_{n-1},f{u}_n}\left(\phi \left(r\right)\right)\hfill \\ \hfill & \ge \lambda min\{{\mathbb{E}}_{u_{n-1},f{u}_{n-1}}\left(\phi \left(\frac{r}{c}\right)\right),{\mathbb{E}}_{u_{n-1},u_n}\left(\phi \left(\frac{r}{c}\right)\right),\hfill \\ \hfill & {\mathbb{E}}_{f{u}_n,u_n}\left(\phi \left(\frac{r}{c}\right)\right),{\mathbb{E}}_{u_n,f{u}_{n-1}}\left(\phi \left(\frac{r}{c}\right)\right),{\mathbb{E}}_{f{u}_n,u_{n-1}}\left(\phi \left(\frac{r}{c}\right)\right)\}\hfill \\ \hfill & \ge min\{{\mathbb{E}}_{u_{n-1},u_n}\left(\phi \left(\frac{r}{c}\right)\right),{\mathbb{E}}_{u_n,u_{n+1}}\left(\phi \left(\frac{r}{c}\right)\right)\}.\hfill \end{array}$$

Now we show that

$${\mathbb{E}}_{u_n,u_{n+1}}\left(\phi \left(r\right)\right)\ge {\mathbb{E}}_{u_{n-1},u_n}\left(\phi \left(\frac{r}{c}\right)\right).$$

(1)

Suppose that ${\mathbb{E}}_{u_n,u_{n+1}}\left(\phi \left(\frac{r}{c}\right)\right)$ is the minimum. By Lemma 1, since we obtain $u_n=u_{n+1}$, we reach a contradiction with $u_n\ne u_{n+1}$. Therefore, ${\mathbb{E}}_{u_{n-1},u_n}\left(\phi \left(\frac{r}{c}\right)\right)$ is the minimum and (1) is true. From expression (1), we obtain

$$\begin{array}{c}\hfill {\mathbb{E}}_{u_n,u_{n+1}}\left(t\right)\ge {\mathbb{E}}_{u_n,u_{n+1}}\left(\phi \left(r\right)\right)\ge {\mathbb{E}}_{u_{n-1},u_n}\left(\phi \left(\frac{r}{c}\right)\right)\ge \cdots \ge {\mathbb{E}}_{u_0,u_1}\left(\phi \left(\frac{r}{c^n}\right)\right),\end{array}$$

that is, ${\mathbb{E}}_{u_n,u_{n+1}}\left(t\right)\ge {\mathbb{E}}_{u_0,u_1}\left(\phi \left(\frac{r}{c^n}\right)\right)$ for desired $n\in \mathbb{N}$. Next, suppose that $m,n\in \mathbb{N}$ where $m>n$. Then, by $\left(\mathbb{E}3\right)$ and the strictly non-decreasing property of $\phi$, we obtain

$$\begin{array}{cc}\hfill {\mathbb{E}}_{u_n,u_m}\left((m-n)t\right)& \ge min\{{\mathbb{E}}_{u_n,u_{n+1}}\left(t\right),\cdots ,{\mathbb{E}}_{u_{m-1},u_{m-2}}\left(t\right),{\mathbb{E}}_{u_{m-1},u_m}\left(t\right)\}\hfill \\ \hfill & \ge min\{{\mathbb{E}}_{u_0,u_1}\left(\phi \left(\frac{r}{c^n}\right)\right),\cdots ,{\mathbb{E}}_{u_0,u_1}\left(\phi \left(\frac{r}{c^{m-2}}\right)\right),{\mathbb{E}}_{u_0,u_1}\left(\phi \left(\frac{r}{c^{m-1}}\right)\right)\}\hfill \\ \hfill & ={\mathbb{E}}_{u_0,u_1}\left(\phi \left(\frac{r}{c^n}\right)\right).\hfill \end{array}$$

Since $\phi \left(\frac{r}{c^n}\right)\to \infty$ as $n\to \infty$, there exists a number $n_0\in \mathbb{N}$ such that ${\mathbb{E}}_{u_0,u_1}\left(\phi \left(\frac{r}{c^n}\right)\right)>1-ϵ$ for fixed $ϵ\in (0,1)$, whenever $n\ge n_0$. This implies that ${\mathbb{E}}_{u_n,u_m}\left((m-n)t\right)>1-ϵ$ for each $m>n\ge n_0$. Since $t>0$ and $ϵ\in (0,1)$ are desired, we conclude that $\left\{u_n\right\}$ is a Cauchy sequence in the complete $RMPM$ space $(V,\mathbb{E},\mathcal{T})$. Hence, there exists $u\in V$ such that $u_n\to u$ as $n\to \infty$.

We will demonstrate that f has as a fixed point, u. By the continuous property of f, we obtain

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{u}_{n+1}=\underset{n\to \infty }{lim}f{u}_n=f\left(\underset{n\to \infty }{lim}{u}_n\right)=fu,\end{array}$$

thus, $fu=u$. □

Example 2.

Suppose that $V={\mathbb{R}}^{+}$ and $\mathcal{T}(u,v)=min\{u,v\}$. Introduce $\mathbb{E}:V\times V\to D^{+}$ by

$${\mathbb{E}}_{u,v}\left(t\right)=\left\{\begin{array}{c}\frac{t}{t+|u-v|},ift>0,\hfill \\ 0otherwise,\hfill \end{array}\right.$$

for every $u,v\in V$[35]. Then, $(V,\mathbb{E},\mathcal{T})$ is a complete $RMPM$ space. Introduce the mapping $f:V\to V$ by $fu=\frac{u}{4}$ and $\phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ by $\phi \left(t\right)=t$, also $c=\frac{1}{2}$ and $\lambda =1$. From

$$\begin{array}{cc}\hfill {\mathbb{E}}_{fu,fv}\left(\phi \left(t\right)\right)& \ge \lambda min\{{\mathbb{E}}_{u,v}(\phi \left(\frac{t}{c}\right),{\mathbb{E}}_{u,fu}(\phi \left(\frac{t}{c}\right),{\mathbb{E}}_{fv,v}(\phi \left(\frac{t}{c}\right),\hfill \\ \hfill & {\mathbb{E}}_{u,fv}(\phi \left(\frac{t}{c}\right),{\mathbb{E}}_{fu,v}\left(\phi \left(\frac{t}{c}\right)\right\},\hfill \end{array}$$

Thus, we obtain

$$\begin{array}{cc}\hfill {\mathbb{E}}_{fu,fv}\left(\phi \left(t\right)\right)=\frac{t}{t+|\frac{u}{4}-\frac{v}{4}{|}^2}& =\frac{t}{t+2^{-2}{|u-v|}^2}\hfill \\ \hfill & ={\mathbb{E}}_{u,v}\left(\phi \left(\frac{t}{c}\right)\right)\hfill \\ \hfill & \ge min\{{\mathbb{E}}_{u,v}(\phi \left(\frac{t}{c}\right),{\mathbb{E}}_{u,fu}(\phi \left(\frac{t}{c}\right),{\mathbb{E}}_{fv,v}(\phi \left(\frac{t}{c}\right),\hfill \\ & {\mathbb{E}}_{u,fv}(\phi \left(\frac{t}{c}\right),{\mathbb{E}}_{fu,v}\left(\phi \left(\frac{t}{c}\right)\right\}\hfill \\ \hfill & =\lambda min\{{\mathbb{E}}_{u,v}(\phi \left(\frac{t}{c}\right),{\mathbb{E}}_{u,fu}(\phi \left(\frac{t}{c}\right),\hfill \\ & {\mathbb{E}}_{fv,v}(\phi \left(\frac{t}{c}\right),{\mathbb{E}}_{u,fv}(\phi \left(\frac{t}{c}\right),{\mathbb{E}}_{fu,v}\left(\phi \left(\frac{t}{c}\right)\right\}.\hfill \end{array}$$

Theorem 1 is satisfied, so f has a fixed point.

3. New Coupled-Fixed-Point Theorem in $\mathbf{RMPbM}$ Space

This section presents useful definitions in the scope of $RMPbM$ spaces.

Definition 9.

Assume that $V\ne \varnothing$, $\mathcal{T}$ is a continuous t-norm and ${\mathbb{E}}^b:V\times V\to D^{+}$. Then, $(V,{\mathbb{E}}^b,\mathcal{T})$ is a rectangular probabilistic b-metric ($RMPbM$, for short) space when for every $u,v\in V$, $\alpha \in (0,1]$ and distinct $x,y\in V\setminus \{u,v\}$,

$\left(Eb1\right)$

${\mathbb{E}}_{u,v}^b\left(t\right)=1$ if $u=v$,

$(Eb2$)

${\mathbb{E}}_{u,v}^b\left(t\right)={\mathbb{E}}_{v,u}\left(t\right)$,

$\left(Eb3\right)$

${\mathbb{E}}_{u,v}^b(t+s+z)\ge \mathcal{T}(\mathcal{T}({\mathbb{E}}_{u,x}\left(\alpha t\right),{\mathbb{E}}_{x,y}\left(\alpha s\right)),{\mathbb{E}}_{y,v}\left(\alpha z\right))$.

Note that each probabilistic b-metric space with coefficient $\alpha$ is a $RMPbM$ space with coefficient ${\alpha }^2$. However, the opposite may not be true.

Lemma 3.

Consider that $(V,{\mathbb{E}}^b,\mathcal{T})$ is a $RMPbM$ space having coefficient α, and that φ is a φ-function. If we obtain ${\mathbb{E}}_{u,v}^b\left({\alpha }^k\phi \left(t\right)\right)\ge {\mathbb{E}}_{u,v}^b\left({\alpha }^{k-1}\phi \left(\frac{t}{c}\right)\right)$ for every $u,v\in V$, $t>0$, $c\in (0,1)$ and $k\in \mathbb{N}$, then ${\mathbb{E}}_{u,v}^b\left(t\right)=1$.

Proof. 

Since ${\mathbb{E}}_{u,v}^b\left({\alpha }^k\phi \left(t\right)\right)\ge {\mathbb{E}}^b\_u,v\left({\alpha }^{k-1}\phi \left(\frac{t}{c}\right)\right)$, we have

$${\mathbb{E}}_{u,v}^b\left({\alpha }^k\phi \left(t\right)\right)\ge {\mathbb{E}}_{u,v}^b\left({\alpha }^{k-1}\phi \left(\frac{t}{c}\right)\right)\ge {\mathbb{E}}_{u,v}^b\left({\alpha }^{k-2}\phi \left(\frac{t}{c^2}\right)\right)\ge \dots \ge {\mathbb{E}}_{u,v}\left({\alpha }^{k-n}\phi \left(\frac{t}{c^n}\right)\right)$$

for every $t>0$ and $n\in \mathbb{N}$. By taking $n\to \infty$, we obtain ${\mathbb{E}}_{u,v}\left({\alpha }^k\phi \left(t\right)\right)\ge 1$, so $u=v$. □

Consider that $(V,{\mathbb{E}}^b,\mathcal{T})$ is a $RMPbM$ space. We assume that there exists a number $N\in \mathbb{N}$ such that $ϵ,\delta >0$. Thus, for $n\ge N$, we have ${\mathbb{E}}_{u_n,u}^b\left(ϵ\right)>1-\delta$. Then, the sequence $\left\{u_n\right\}$ in V converges to u in V. If, for every $ϵ,\delta >0$, there exists a $N\in \mathbb{N}$${\mathbb{E}}_{u_n,u_m}^b\left(ϵ\right)>1-\delta$ for every $m,n\ge N$, then we say that $\left\{u_n\right\}$ in V is a Cauchy sequence. A $RMPbM$ space is a complete space if every Cauchy sequence is convergent in it.

For every $p\in V$ and $\lambda >0$, the $ϵ$- neighborhood of p is defined as

$$N_p\left(\lambda \right)=\{q\in V:{\mathbb{E}}_{p,q}^b\left(ϵ\right)>1-ϵ\}.$$

Additionally, $U\subseteq V$ is called bounded if there is $\delta >0$ and $ϵ\in (0,1)$ such that ${\mathbb{E}}_{u,v}^b\left(\delta \right)>1-ϵ$ for every $u,v\in U$.

Lemma 4.

Consider that $(V,{\mathbb{E}}^b,\mathcal{T})$ is a $RMPbM$ space with coefficient α. So, ${\mathbb{E}}^b$ is a lower semi-continuous function of points, that is, for each fixed $t>0$ and each two convergent sequences $\left\{u_n\right\},\left\{v_n\right\}$ in V, such that $u_n\to u$ and $v_n\to v$, it holds that $li{m}_{n\to \infty }inf{\mathbb{E}}_{u_n,v_n}^b\left(t\right)={\mathbb{E}}_{u,v}^b\left(t\right)$.

Proof. 

Suppose that $t>0$ and $ϵ>0$ are given. Since ${\mathbb{E}}_{u,v}^b$ is left-continuous at t, there exists $0<{\delta }_1<t$ for which ${\mathbb{E}}_{u,v}^b\left(t\right)-{\mathbb{E}}_{u,v}^b(t-{\delta }_1)<ϵ$. Let $h\in \mathbb{R}$ be fixed with $0<2h<{\delta }_1$, so ${\mathbb{E}}_{u,v}^b\left(t\right)-{\mathbb{E}}_{u,v}^b(t-2h)<ϵ$. Adopting again the left-continuity of ${\mathbb{E}}_{u,v}^b$ at $t-{\delta }_1$, there exists ${\delta }_2>0$ for which ${\mathbb{E}}_{u,v}^b(t-{\delta }_1)-{\mathbb{E}}_{u,v}^b(t-{\delta }_2)<ϵ$. By repeating this method, $k\in \mathbb{N}$, ${\delta }_u,{\delta }_{u+1}>0$ $(u=1,\cdots ,k)$, where ${\mathbb{E}}_{u,v}^b(t-{\delta }_u)-{\mathbb{E}}_{u,v}^b(t-{\delta }_{u+1})<ϵ$ and ${\alpha }^{2}t-2{\alpha }^{2}h\in (t-{\delta }_k,t-{\delta }_{k+1})$. Thus, we conclude that

$$\begin{array}{cc}\hfill & {\mathbb{E}}_{u,v}^b\left(t\right)-{\mathbb{E}}_{u,v}^b({\alpha }^{2}t-2{\alpha }^{2}h)\hfill \\ \hfill & =({\mathbb{E}}_{u,v}^b\left(t\right)-{\mathbb{E}}_{u,v}^b(t-{\delta }_1))+({\mathbb{E}}_{u,v}^b(t-{\delta }_1)-{\mathbb{E}}_{u,v}^b(t-{\delta }_2))+\hfill \\ \hfill & \dots +({\mathbb{E}}_{u,v}^b(t-{\delta }_k)-{\mathbb{E}}_{u,v}^b({\alpha }^{2}t-2{\alpha }^{2}h))\hfill \\ \hfill & <(k+1)ϵ.\hfill \end{array}$$

(2)

Set ${\mathbb{E}}_{u,v}^b({\alpha }^{2}t-2{\alpha }^{2}h)=a$. From the properties of $\mathcal{T}$ and $\mathcal{T}(a,1)=a$, there exists $0<l<1$, such that $\mathcal{T}(a,l)>a-\frac{ϵ}{3}$ and $\mathcal{T}(a-\frac{ϵ}{3},l)>a-\frac{2ϵ}{3}$. Meanwhile, since $u_n\to u$ and $v_n\to v$, there is $M_{h,l}\in \mathbb{Z}$ for which ${\mathbb{E}}_{u_n,u}^b(t-{\alpha }^{2}t)>l$ and ${\mathbb{E}}_{v_n,v}^b\left(2{\alpha }^{2}h\right)>l$, whenever $n>M_{h,l}$. Now, by $\left(Eb3\right)$

$${\mathbb{E}}_{u_n,v_n}^b\left(t\right)\ge \mathcal{T}(\mathcal{T}({\mathbb{E}}_{u_n,u}^b(t-{\alpha }^{2}t),{\mathbb{E}}_{u,v}^b({\alpha }^{2}t-2{\alpha }^{2}h)),{\mathbb{E}}_{v_n,v}^b\left(2{\alpha }^{2}h\right)),$$

thus, we obtain

$${\mathbb{E}}_{u_n,v_n}^b\left(t\right)\ge \mathcal{T}(\mathcal{T}(l,a),l)>\mathcal{T}(a-\frac{ϵ}{3},l).$$

From (2), we obtain

$${\mathbb{E}}_{u_n,v_n}^b\left(t\right)\ge \mathcal{T}(a-\frac{ϵ}{3},l)>a-\frac{2ϵ}{3}>{\mathbb{E}}_{u,v}^b\left(t\right)-\frac{(3k+5)ϵ}{3}.$$

□

Theorem 2.

Suppose that $(V,{\mathbb{E}}^b,\mathcal{T})$ represents a complete $RMPbM$ space with coefficient α, which satisfies $\mathcal{T}(a,a)\ge a$ with $a\in [0,1]$. Consider that $f:V\times V\to V$ is a continuous operator such that for every $(u,v),(x,y)\in V^2$, $t>0$, $\lambda \ge 1$, $\phi \in \phi$ and $0<c<1$, we have

$$\begin{array}{cc}\hfill {\mathbb{E}}_{f(u,v),f(x,y)}^b\left({\alpha }^k\phi \left(t\right)\right)& \ge \lambda min\{{\mathbb{E}}_{u,x}^b({\alpha }^{k-1}\phi \left(\frac{t}{c}\right),{\mathbb{E}}_{v,y}^b\left({\alpha }^{k-1}\phi \left(\frac{t}{c}\right)\right\}.\hfill \end{array}$$

Then, f possesses a coupled fixed point.

Proof. 

There is $(u_0,v_0)\in V^2$ for which $u_1=f(u_0,v_0)$ and $v_1=f(v_0,u_0)$. Suppose that $u_2=f(u_1,v_1)$ and $v_2=f(v_1,u_1)$, we write

$$\begin{array}{c}\hfill f^2(u_0,v_0)=f(f(u_0,v_0),f(v_0,u_0))=f(u_1,v_1)=u_2,\\ \hfill f^2(v_0,u_0)=f(f(v_0,u_0),f(u_0,v_0))=f(v_1,u_1)=v_2.\end{array}$$

By repeating this process, for every $n\ge 0$,

$$(u_{n+1},v_{n+1})=(f^{n+1}(u_0,v_0),f^{n+1}(v_0,u_0))=(f(f^n(u_0,v_0),f^n(v_0,u_0)),f(f^n(v_0,u_0),f^n(u_0,v_0)).$$

If $(u_{n+1},v_{n+1})=(u_n,v_n)$ for every $n\in \mathbb{N}$, then we obtain nothing to show. Consider $(u_{n+1},v_{n+1})\ne (u_n,v_n)$ for every $n=1,2,\cdots$. Based on the continuity property of $\phi$ at 0, there is $r>0$ for which $t>\phi \left(r\right)$. Consequently, we obtain

$$\begin{array}{cc}\hfill {\mathbb{E}}_{u_n,u_{n+1}}^b\left(t\right)& \ge {\mathbb{E}}_{f(u_{n-1},v_{n-1}),f(u_n,v_n)}^b\left({\alpha }^k\phi \left(r\right)\right)\hfill \\ \hfill & \ge \lambda min\{{\mathbb{E}}_{u_{n-1},u_n}^b({\alpha }^{k-1}\phi \left(\frac{t}{c}\right),{\mathbb{E}}_{v_{n-1},v_n}^b\left({\alpha }^{k-1}\phi \left(\frac{t}{c}\right)\right\}.\hfill \end{array}$$

By simplifying, we obtain

$${\mathbb{E}}_{u_n,u_{n+1}}^b\left({\alpha }^{k}t\right)\ge min\left\{{\mathbb{E}}_{u_0,u_1}^b\right({\alpha }^{k-n}\phi \left(\frac{t}{c^n}\right),{\mathbb{E}}_{v_0,v_1}^b\left({\alpha }^{k-n}\phi \left(\frac{t}{c^n}\right)\right\}$$

for each $n\in \mathbb{N}$. Suppose that $m,n\in \mathbb{N}$ with $m>n$, by (Eb3) and the strictly non-decreasing property of $\phi$, we obtain

Case 1: If $m-n$ is an even number

$$\begin{array}{cc}\hfill {\mathbb{E}}_{u_n,u_m}^b\left((m-n)t\right)& \ge min\{{\mathbb{E}}_{u_n,u_{n+1}}^b\left(\alpha t\right),{\mathbb{E}}_{u_{n+1},u_{n+2}}^b\left(\alpha t\right)\cdots ,{\mathbb{E}}_{u_{m-1},u_{m-2}}^b\left({\alpha }^{m-n-1}t\right),{\mathbb{E}}_{u_{m-1},u_m}^b\left({\alpha }^{m-n-1}t\right)\}\hfill \\ \hfill & \ge min\{min\{{\mathbb{E}}_{u_0,u_1}^b\left({\alpha }^{1-n}\phi \left(\frac{r}{c^n}\right)\right),{\mathbb{E}}_{v_0,v_1}^b\left({\alpha }^{1-n}\phi \left(\frac{r}{c^n}\right)\right)\},\cdots ,\hfill \\ \hfill & min\{{\mathbb{E}}_{u_0,u_1}^b\left({\alpha }^{-n}\phi \left(\frac{r}{c^{m-1}}\right)\right),{\mathbb{E}}_{v_0,v_1}^b\left({\alpha }^{-n}\phi \left(\frac{r}{c^{m-1}}\right)\right)\}\}\hfill \\ \hfill & =min\{{\mathbb{E}}_{u_0,u_1}^b\left({\alpha }^{1-n}\phi \left(\frac{r}{c^n}\right)\right),{\mathbb{E}}_{v_0,v_1}^b\left({\alpha }^{1-n}\phi \left(\frac{r}{c^n}\right)\right)\}.\hfill \end{array}$$

Since ${\alpha }^{1-n}\phi \left(\frac{r}{c^n}\right)\to \infty$ as $n\to \infty$, there exists $n_0\in \mathbb{N}$ so that ${\mathbb{E}}_{u_0,u_1}^b\left({\alpha }^{1-n}\phi \left(\frac{r}{c^n}\right)\right)>1-ϵ$ and ${\mathbb{E}}_{v_0,v_1}^b\left({\alpha }^{1-n}\phi \left(\frac{r}{c^n}\right)\right)>1-ϵ$ for fixed $ϵ\in (0,1)$ whenever $n\ge n_0$. This results in ${\mathbb{E}}_{u_n,u_m}^b\left((m-n)t\right)>1-ϵ$ for each $m>n\ge n_0$. Since $t>0$ and $ϵ\in (0,1)$ are desired, we conclude that $\left\{u_n\right\}$ denotes a Cauchy sequence. Closely, $\left\{v_n\right\}$ is a Cauchy sequence.

Case 2: If $m-n$ is an odd number, one can show that $\left\{u_n\right\}$ and $\left\{v_n\right\}$ represent Cauchy sequences. Now, by the completeness of V, there exist $u,v\in V$ such that $\underset{n\to \infty }{lim}{u}_n=u$ and $\underset{n\to \infty }{lim}{v}_n=v$. In the sequel, we will show that f has a coupled fixed point in V. Since $u_{n+1}=f(u_n,v_n)$, f is continuous. Using the limit as $n\to \infty$, we obtain $f(u,v)=u$. Closely, we obtain $f(v,u)=v$. Thus, f has a coupled fixed point $(u,v)$. □

Example 3.

Consider V, ${\mathbb{E}}^b$, $\mathcal{T}$, φ and c as in Example 2. Introduce $f:V\times V\to V$ by $f(u,v)=\frac{i}{4}$. We have

$$\begin{array}{cc}\hfill {\mathbb{E}}_{f(u,v),f(x,y)}^b\left({\alpha }^k\phi \left(t\right)\right)& \ge \lambda min\{{\mathbb{E}}_{u,x}^b({\alpha }^{k-1}\phi \left(\frac{t}{c}\right),{\mathbb{E}}_{v,y}^b\left({\alpha }^{k-1}\phi \left(\frac{t}{c}\right)\right\}.\hfill \end{array}$$

By means of the definition of ${\mathbb{E}}^b$ in Example 2, we obtain

$$\begin{array}{cc}\hfill {\mathbb{E}}_{f(u,v),f(x,y)}^b\left({\alpha }^k\phi \left(t\right)\right)& =\frac{\frac{1}{2^k}t}{\frac{1}{2^k}t+{|\frac{u}{4}-\frac{x}{4}|}^2}\hfill \\ \hfill & =\frac{\frac{1}{2^k}t}{\frac{1}{2^k}t+\frac{1}{2^4}{|u-x|}^2}\hfill \\ \hfill & =\frac{t}{t+\frac{2^k}{2^4}{|u-x|}^2}\hfill \\ \hfill & =\frac{t}{t+2^{k-4}{|u-x|}^2}\hfill \\ \hfill & \ge \frac{t}{t+2^{k-2}{|u-x|}^2}={\mathbb{E}}_{u,x}^b({\alpha }^{k-1}\phi \left(\frac{t}{c}\right)\hfill \\ \hfill & \ge min\{{\mathbb{E}}_{u,x}^b({\alpha }^{k-1}\phi \left(\frac{t}{c}\right),{\mathbb{E}}_{v,y}^b\left({\alpha }^{k-1}\phi \left(\frac{t}{c}\right)\right\}\hfill \\ \hfill & =\lambda min\{{\mathbb{E}}_{u,x}^b({\alpha }^{k-1}\phi \left(\frac{t}{c}\right),{\mathbb{E}}_{v,y}^b\left({\alpha }^{k-1}\phi \left(\frac{t}{c}\right)\right\}.\hfill \end{array}$$

By $\lambda =1$, Theorem 2 is satisfied and f has a coupled fixed point.

4. An Application to a System of Integral Equations

Let us consider the $SIE$:

$$\left\{\begin{array}{c}u\left(t\right)={\int }_a^{b}G(t,s)K(s,u\left(s\right),v\left(s\right))ds,\hfill \\ v\left(t\right)={\int }_a^{b}G(t,s)K(s,v\left(s\right),u\left(s\right))ds,\hfill \end{array}\right.$$

(3)

for every $t\in I=[a,b]$, for which $a<b$, $K\in C(I\times \mathbb{R}\times \mathbb{R},\mathbb{R})$ and $G\in C(I\times I,[0,\infty \left)\right)$.

Suppose that $\mathsf{\Xi }=C(I,\mathbb{R})$ is the Banach space of all real continuous functions introduced on I with ${\left|\right|u\left|\right|}_{\infty }=\underset{t\in I}{max}\left|u\left(t\right)\right|$ for $u\in \mathsf{\Xi }$ and induced b-metric $\eta (u,v)=\left|\right|u-{v\left|\right|}^2$ for every $u,v\in \mathsf{\Xi }$. It is worth mentioning that $\eta$ is complete b-metric with $s=2$. Introduce the mapping ${\mathbb{E}}^b:\mathsf{\Xi }\times \mathsf{\Xi }\to D^{+}$ by ${\mathbb{E}}_{u,v}^b\left(t\right)=\chi (t-\eta (u,v))$ for $u,v\in \mathsf{\Xi }$ and $t>0$, where

$$\chi \left(t\right)=\left\{\begin{array}{c}1ift>0\hfill \\ 0ift\le 0.\hfill \end{array}\right.$$

Then, the space $(\mathsf{\Xi },{\mathbb{E}}^b,\mathcal{T})$ with $\mathcal{T}(a,b)=min\{a,b\}$ is a complete $RMPbM$ space with coefficient $\alpha =\frac{1}{4}$ (see [35]).

Theorem 3.

Suppose that $(\mathsf{\Xi },{\mathbb{E}}^b,\mathcal{T})$ is a $RMPbM$ space. Assume that $f:\mathsf{\Xi }\times \mathsf{\Xi }\to \mathsf{\Xi }$ is an operator such that $f(u,v)t={\int }_a^{b}G(t,s)K(s,u\left(s\right),v\left(s\right))ds$, where $G\in C(I\times I,[0,\infty \left)\right)$, and $K\in C(I\times \mathbb{R}\times \mathbb{R},\mathbb{R})$ are two operators fulfilling:

(i)

${\left|\right|K\left|\right|}_{\infty }=\underset{s\in I,u,v\in \mathsf{\Xi }}{sup}\left|K(s,u\left(s\right),v\left(s\right))\right|<\infty$;

(ii)

For every $u,v\in \mathsf{\Xi }$ and all $s\in I$, we obtain

$$\left|\right|K(s,u\left(s\right),v\left(s\right))-K(s,x\left(s\right),y\left(s\right))\left|\right|\le max\{\parallel u\left(s\right)-x\left(s\right){\parallel }^2,\parallel v\left(s\right)-y\left(s\right){\parallel }^2\};$$

(iii)

$\underset{t\in I}{max}{\int }_a^{b}G(t,s)ds<1$.

Then, the $SIE$ (3) has a solution in $\mathsf{\Xi }\times \mathsf{\Xi }$.

Proof. 

Assume that $\eta (u,v)={max}_{t\in I}{\left(\right|u\left(t\right)-v\left(t\right)|}^2)$, for every $u,v\in \mathsf{\Xi }$. As mentioned before, $(\mathsf{\Xi },{\mathbb{E}}^b,\mathcal{T})$ is a complete $RMPbM$ space. Thus, for every $u,v\in \mathsf{\Xi }$, we obtain

$$\begin{array}{cc}\hfill \eta \left(f\right(u,v),f(x,y\left)\right)& \le \underset{t\in I}{max}{\int }_a^{b}G(t,s)|K(s,u\left(s\right)v\left(s\right))-K(s,x\left(s\right),y\left(s\right))|ds\hfill \\ \hfill & \le \underset{t\in I}{max}{\int }_a^b{G(t,s)max\{\parallel u\left(s\right)-x\left(s\right)\parallel }^2{,\parallel v\left(s\right)-y\left(s\right)\parallel }^2\}ds\hfill \\ \hfill & =max\{\eta (u,x),\eta (v,y)\}\underset{t\in I}{max}{\int }_a^{b}G(t,s)ds.\hfill \end{array}$$

Set $c=\underset{t\in I}{max}{\int }_a^{b}G(t,s)ds$. Then, for any $r>0$ and $k\in \mathbb{N}$ and $u,v\in \mathsf{\Xi }$,

$$\begin{array}{cc}\hfill {\mathbb{E}}_{f(u,v),f(x,y)}^b\left(\frac{r}{2^k}\right)& =\chi (\frac{r}{2^k}-\eta (f(u,v),f(x,y))\hfill \\ \hfill & \ge \chi (\frac{r}{2^k}-\frac{c}{2}{max\{\parallel u\left(s\right)-x\left(s\right)\parallel }^2{,\parallel v\left(s\right)-y\left(s\right)\parallel }^2\left\}\right)\hfill \\ \hfill & =\chi (\frac{r}{2^{k-1}c}{-max\{\parallel u\left(s\right)-x\left(s\right)\parallel }^2{,\parallel v\left(s\right)-y\left(s\right)\parallel }^2\left\}\right)\hfill \\ \hfill & =min\{{\mathbb{E}}_{u,x}^b\left(\frac{r}{2^{k-1}c}\right),{\mathbb{E}}_{v,y}^b\left(\frac{r}{2^{k-1}c}\right)\}.\hfill \end{array}$$

So, by making use of the Theorem 2 with $\phi \left(r\right)=r$ for every $r>0$ and $\lambda =1$ for every $u,v\in \mathsf{\Xi }$ and $t>0$, we conclude that there is a coupled fixed point for f which is the solution for $SIE$ (3). □

5. Conclusions

The rectangular—and rectangular b-metric—spaces are relatively new concepts reported in the existing literature. In this paper, we sought to further enrich these important notions by introducing the $RMPM$ and $RMPbM$ spaces. This was accomplished by combining the ideas of rectangular metric, and b-metric, spaces with the concepts of $MPM$ and $MPbM$, spaces. As such, the $RMPM$ and $RMPbM$ spaces are generalizations of the $MPM$ and $MPbM$ spaces, respectively. Then, by using the new $RMPM$ and $RMPbM$ spaces, fixed-point and coupled-fixed-point theorems were proven. Finally, adopting the proposed coupled-fixed-point results, the existence of solution for a $SIE$ in the $RMPbM$ space was demonstrated. Spaces with non-Hausdorff topology have real-world relevance, as in the Tarskian approach to programming language semantics adopted in computer science. Further work will seek new generalizations and applications to different SIEs, of either integer or fractional order.