Novel Fixed Point Results in Rectangular G_b-Metric Spaces and Some Applications on Fractional Differential Equations

Abstract

In this work, we prove some fixed point theorems in rectangular $G_b$-metric space, which is the generalization of rectangular metric space and $G_b$-metric space. Moreover, we give some examples to support our theoretical findings. Finally, using our main results, we present some applications to obtain solutions of Riemann–Liouville and Atangana–Baleanu fractional integral equations.

1. Introduction

In recent decades, the rapid growth of fixed point theory in metric spaces and the prominent interest of researchers in this field evidence that it is a very important field in modern mathematics, particularly in the branch of nonlinear functional analysis thanks to its wide applications. When we say fixed point theory, we should not forget that the Banach contraction principle [1] is the cornerstone of this field. After Banach laid out this contraction type, many researchers have generalized and extended it, such as the Kannan type [2], Chatterjea type [3], Hardy–Rogers type [4], etc.

On the other hand, researchers are also interested in enriching the literature with new metric space generalizations. For instance, b-metric [5,6] and G-metric space were described by Mustafa and Sims [7]; in 2009, the same authors, in [8], demonstrated some fixed point theorems in G-metric spaces. Afterwards, researchers have obtained many important results; for example, [9,10,11]. In 2014, inspired by the b-metric and G-metric, Aghajani et al. [12] presented $G_b$-metric space. Researchers have since proven many fixed point results in $G_b$-metric spaces. For example, Ege [13] proves a common fixed point theorem in complex valued $G_b$-metric spaces using the concept of $\alpha$-series. For some other works on this topic, see [14,15,16]. In 2022, Li and Cui [17] introduced the concept of rectangular $G_b$-metric space and obtained some important fixed point theorems. Also, the authors have presented the concept of convex rectangular $G_b$-metric space and have proved some fixed points of enriched type contractions in this space.

The outline of this work is given as follows: Section 2, we provide some required information on the paper. In Section 3, we prove Hardy–Rogers-, Kannan-, and Chatterjea-type fixed point results using a weakly G-contraction function. Finally, we give two applications.

2. Preliminaries

First, we give the required background on rectangular $G_b$-metric spaces.

Definition 1

([6]). Let E be a nonempty set. A function B:$E^2\to [0,\infty )$ is named b-metric and $(E,B)$ is named a b-metric space $\left(BMS\right)$ if, for each $\nu ,\iota ,\mu \in E$, it satisfies

$\left(B1\right)$$B(\nu ,\iota )=0$ iff $\nu =\iota$,

$\left(B2\right)$$B(\nu ,\iota )=B(\iota ,\nu )$,

$\left(B3\right)$$B(\nu ,\iota )\le s\left[B\right(\nu ,\mu )+B(\mu ,\iota \left)\right]$ for some constant $s\ge 1$.

Definition 2

([18]). Let E be a nonempty set. A function $R_B$:$E^2\to [0,\infty )$ is named rectangular b-metric and $(E,R_B)$ is named a rectangular b-metric space $\left(RBMS\right)$ if it satisfies the conditions below:

$\left(RB1\right)$$B(\nu ,\iota )=0$ iff $\nu =\iota$,

$\left(RB2\right)$$B(\nu ,\iota )=B(\iota ,\nu )$, for all $\nu ,\iota \in E$,

$\left(RB3\right)$ there exists a real number $s\ge 1$ such that $B(\nu ,\iota )\le s\left[B\right(\nu ,\mu )+B(\mu ,\alpha )+B(\alpha ,\iota \left)\right]$ for all $\nu ,\iota \in E$ and for all distinct points $\mu ,\alpha \in E\setminus \{\nu ,\iota \}$.

Definition 3

([7]). Let E be a nonempty set. A function G:$E^3\to [0,\infty )$ is named G-metric and $(E,G)$ is named a G-metric space $\left(GMS\right)$ if, for each $\nu ,\iota ,\mu ,\alpha \in E$ it satisfies

$\left(G1\right)$$G(\nu ,\iota ,\mu )=0$ if $\nu =\iota =\mu$,

$\left(G2\right)$$0<G(\nu ,\nu ,\iota )$, $\forall \nu ,\iota \in E$ with $\nu \ne \iota$,

$\left(G3\right)$$G(\nu ,\iota ,\iota )\le G(\nu ,\iota ,\mu )$, $\forall \nu ,\iota ,\mu \in E$ with $\iota \ne \mu$,

$\left(G4\right)$$G(\nu ,\iota ,\mu )=G(\nu ,\mu ,\iota )=G(\mu ,\nu ,\iota )=G(\mu ,\iota ,\nu )=G(\iota ,\mu ,\nu )=G(\iota ,\nu ,\mu )$,

$\left(G5\right)$$G(\nu ,\iota ,\mu )\le G(\nu ,\alpha ,\alpha )+G(\alpha ,\iota ,\mu ).$

Definition 4

([17]). Let E be a nonempty set. A function $R_G$: $E^3\to [0,\infty )$ is named rectangular G-metric and $(E,R_G)$ is named a rectangular G-metric space $\left(RGMS\right)$ if, for each $\nu ,\iota ,\mu ,\in E$, it satisfies

$\left(RG1\right)$$R_G(\nu ,\iota ,\mu )=0$ if $\nu =\iota =\mu$,

$\left(RG2\right)$$0<R_G(\nu ,\nu ,\iota )$, $\forall \nu ,\iota \in E$ with $\nu \ne \iota$,

$\left(RG3\right)$$R_G(\nu ,\iota ,\iota )\le R_G(\nu ,\iota ,\mu )$, $\forall \nu ,\iota ,\mu \in E$ with $\iota \ne \mu$,

$\left(RG4\right)$$R_G(\nu ,\iota ,\mu )=R_G(\nu ,\mu ,\iota )=R_G(\mu ,\nu ,\iota )=R_G(\mu ,\iota ,\nu )=R_G(\iota ,\mu ,\nu )=R_G(\iota ,\nu ,\mu )$,

$\left(RG5\right)$$R_G(\nu ,\iota ,\mu )\le R_G(\nu ,\alpha ,\alpha )+R_G(\alpha ,\rho ,\rho )+R_G(\rho ,\iota ,\mu )$ for all $\alpha \ne \rho$, $\alpha ,\rho \in E\setminus \{\nu ,\iota \}.$

Definition 5

([12]). Let E be a nonempty set. A function $G_b:E^3\to [0,\infty )$ is named $G_b$-metric and $(E,G_b)$ is named a $G_b$-metric space $\left(GBMS\right)$ if, for each $\nu ,\iota ,\mu ,\alpha \in E$ it satisfies

$\left(Gb1\right)$$G_b(\nu ,\iota ,\mu )=0$ if $\nu =\iota =\mu$,

$\left(Gb2\right)$$0<G_b(\nu ,\iota ,\iota )$, $\forall \nu ,\iota \in E$ with $\nu \ne \iota$,

$\left(Gb3\right)$$G_b(\nu ,\nu ,\iota )\le G_b(\nu ,\iota ,\mu )$, $\forall \nu ,\iota ,\mu \in E$ with $\iota \ne \mu$,

$\left(Gb4\right)$$G_b(\nu ,\iota ,\mu )=G_b(\nu ,\mu ,\iota )=G_b(\mu ,\nu ,\iota )=G_b(\mu ,\iota ,\nu )=G_b(\iota ,\mu ,\nu )=G_b(\iota ,\nu ,\mu )$,

$\left(Gb5\right)$$G_b(\nu ,\iota ,\mu )\le e[G_b(\nu ,\alpha ,\alpha )+G_b(\alpha ,\iota ,\mu )]$ for a given real number $e\ge 1$.

Remark 1.

It is easy to see that when $e=1$, every $\left(GBMS\right)$ is just a $\left(GMS\right)$. But every $\left(GBMS\right)$ does not need to be a $\left(GMS\right)$.

Definition 6

([17]). Let E be a nonempty set. A function $R_{G_b}:E^3\to [0,\infty )$ is named rectangular $G_b$-metric and $(E,R_G)$ is named a $R_{G_b}$-metric space $\left(RGBMS\right)$ if, for each $\nu ,\iota ,\mu \in E$, it satisfies

$\left(RGb1\right)$$R_{G_b}(\nu ,\iota ,\mu )=0$ if $\nu =\iota =\mu$,

$\left(RGb2\right)$$0<R_{G_b}(\nu ,\nu ,\iota )$, $\forall \nu ,\iota \in E$ with $\nu \ne \iota$,

$\left(RGb3\right)$$R_{G_b}(\nu ,\nu ,\iota )\le R_{G_b}(\nu ,\iota ,\mu )$, $\forall \nu ,\iota ,\mu \in E$ with $\iota \ne \mu$,

$\left(RGb4\right)$$R_{G_b}(\nu ,\iota ,\mu )=R_{G_b}(\nu ,\mu ,\iota )=R_{G_b}(\mu ,\nu ,\iota )=R_{G_b}(\mu ,\iota ,\nu )=R_{G_b}(\iota ,\mu ,\nu )=R_{G_b}(\iota ,\nu ,\mu )$,

$\left(RGb5\right)$ there exists a real number $e\ge 1$ such that $R_{G_b}(\nu ,\iota ,\mu )\le e[R_{G_b}(\nu ,\alpha ,\alpha )+R_{G_b}(\alpha ,\rho ,\rho )+R_{G_b}(\rho ,\iota ,\mu )]$ for all $\alpha \ne \rho$, $\alpha ,\rho \in E\setminus \{\nu ,\iota \}.$

Remark 2

([17]). (1) It is easy to see that when $e=1$, every $\left(RGBMS\right)$ is a $\left(RGMS\right)$.

• (2) Every $\left(GBMS\right)$ with coefficient e is an $\left(RGBMS\right)$ with coefficient $e^2$.

Example 1

([17]). Let $E=\mathbb{R}$ and the function $R_{G_b}:E^3\to [0,\infty )$ defined by

$$R_{G_b}(\nu ,\iota ,\mu )={\left(\right|\nu -\iota |+|\iota -\mu |+|\mu -\nu \left|\right)}^u,(u\ge 1).$$

Then $(E,R_{G_b})$ is a $\left(RGBMS\right)$ with $e=3^{u-1}$.

Example 2.

Let $E=\{{\displaystyle \frac{1}{n}}:n\ge 2,n\in \mathbb{N}\}\bigcup {\mathbb{Z}}^{+}$ and define $R_{G_b}:E^3\to [0,\infty )$ such that

$$R_{G_b}(\nu ,\iota ,\mu )=R_{G_b}(\nu ,\mu ,\iota )=R_{G_b}(\mu ,\nu ,\iota )=R_{G_b}(\mu ,\iota ,\nu )=R_{G_b}(\iota ,\mu ,\nu )=R_{G_b}(\iota ,\nu ,\mu )$$

for all $\nu ,\iota ,\mu \in E$ and

$$R_{G_b}(\nu ,\iota ,\mu )=\left\{\begin{array}{cc}0,\hfill & if \nu =\iota =\mu \hfill \\ 3t,\hfill & if \nu \ne \iota \ne \mu , for all \nu ,\iota ,\mu \in {\mathbb{Z}}^{+}\hfill \\ {\displaystyle \frac{t}{n+1}},\hfill & if \nu \in {\mathbb{Z}}^{+} and \iota ,\mu \in \{{\displaystyle \frac{1}{\mathrm{n}}}:\mathrm{n}=2,3,\dots \}\hfill \\ t,\hfill & otherwise,\hfill \end{array}\right.$$

where $t>0$ is a constant. Then $(E,R_{G_b})$ is a $\left(RGBMS\right)$ with coefficient $e=2$. However $(E,R_{G_b})$ is not a $\left(RGMS\right)$ since

$$\begin{array}{ccc}\hfill R_{G_b}(1,2,3)& \le & R_{G_b}(1,{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}})+R_{G_b}({\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{3}},{\displaystyle \frac{1}{3}})+R_{G_b}({\displaystyle \frac{1}{3}},2,3)\hfill \\ \hfill 3t& \le & {\displaystyle \frac{t}{3}}+t+t\hfill \\ \hfill 3t& >& {\displaystyle \frac{7t}{3}}.\hfill \end{array}$$

Definition 7

([17]). If $R_{G_b}(\nu ,\iota ,\iota )=R_{G_b}(\iota ,\iota ,\nu )$ for all $\nu ,\iota \in E$, then we say the $R_{G_b}$-metric is a symmetric rectangular $G_b$-metric.

Definition 8

([17]). Let $(E,R_{G_b})$ be an $\left(RGBMS\right)$. Consider a sequence $\left\{{\lambda }_n\right\}$ in E and a point $\lambda \in E$. Then

(1) if for any $\delta >0$, given a positive integer $l_0\in \mathbb{N}$ such that $R_{G_b}({\lambda }_n,{\lambda }_q,\lambda )<\delta$ for all $n,q\ge l_0,$ we say $\left\{{\lambda }_n\right\}$ is a $R_{G_b}$-convergent in E to λ,

(2) if for any $\delta >0$, given a positive integer $l_0\in \mathbb{N}$ such that $R_{G_b}({\lambda }_n,{\lambda }_q,{\lambda }_l)<\delta$ for all $q,n,l\ge l_0$, we say $\left\{{\lambda }_n\right\}$ is a $R_{G_b}$-Cauchy in E,

(3) if every $R_{G_b}$-Cauchy sequence in E converges to some λ in E, we say $(E,R_{G_b})$ is a complete $\left(RGBMS\right)$.

Definition 9

Let $(E,R_{G_b})$ and $(E^{\prime },R_{G_b}^{\prime })$ be two rectangular $G_b$-metric spaces. A mapping $g:E\to E^{\prime }$ is $R_{G_b}$-continuous at a point $\lambda \in E$ if and only if it is $R_{G_b}$-sequentially continuous at λ, meaning that, when a sequence ${\lambda }_n$ is convergent to λ in E, then the sequence $g\left({\lambda }_n\right)$ is convergent to $g\left(\lambda \right)$ in $E^{\prime }$.

Definition 10

([9]). Let ψ be a function defined by $\psi :[0,\infty )\to [0,\infty )$ and let the properties below be satisfied. Then, ψ is an altering distance function.

(1) ψ is continuous and increasing;

(2) $\psi \left(v\right)=0$ if and only if $v=0$.

Definition 11

([9]). Let $(E,G)$ be a $\left(GMS\right)$ and $H:E\to E$ be a function satisfying

• if, for all $\nu ,\iota ,\mu \in E$,

$$\begin{array}{cc}\hfill \psi \left(G\right(H\nu ,H\iota ,H\mu \left)\right)\le & \psi \left({\displaystyle \frac{1}{3}}[G(\nu ,H\iota ,H\iota )+G(\iota ,H\mu ,H\mu )+G(\mu ,H\nu ,H\nu )]\right)\hfill \\ \hfill & -\varphi \left(G(\nu ,H\iota ,H\iota ),G(\iota ,H\mu ,H\mu ),G(\mu ,H\nu ,H\nu )\right),\hfill \end{array}$$

(1)

where $\varphi :{[0,\infty )}^3\to [0,\infty )$ is a continuous function with $\varphi (\nu ,\iota ,\mu )=0$ iff $\nu =\iota =\mu$. We say H is a weakly G-contraction type function.

Proposition 1

([17]). Let $(E,R_{G_b})$ be an $\left(RGBMS\right)$ and $\delta >0$, then for each $\nu ,\iota ,\mu \in E$,

(1) if $R_{G_b}(\nu ,\iota ,\mu )=0$, then $\nu =\iota =\mu$,

(2) if $R_{G_b}({\lambda }_n,{\lambda }_q,{\lambda }_l)<\delta$ for any $n,q,l\in \mathbb{N}$, then $R_{G_b}({\lambda }_n,{\lambda }_q,{\lambda }_q)<\delta$ and $R_{G_b}({\lambda }_n,{\lambda }_n,{\lambda }_q)<\delta$.

Lemma 1

([17]). Let $(E,R_{G_b})$ be an $\left(RGBMS\right)$ with the coefficient $e\ge 1$, $H:E\to E$ a self-mapping and ${\lambda }_n$ a sequence in E defined by ${\lambda }_{n+1}=H{\lambda }_n$ such that

$$R_{G_b}({\lambda }_n,{\lambda }_{n+1},{\lambda }_{n+2})\le \kappa R_{G_b}({\lambda }_{n-1},{\lambda }_n,{\lambda }_{n+1}),$$

where $\kappa \in [0,1)$ and $n\in \mathbb{N}$. Then ${\lambda }_n$ is a $R_{G_b}$-Cauchy sequence in E.

Theorem 1

([17]). Let $(E,R_{G_b})$ be a complete $\left(RGBMS\right)$ with a coefficient $e\ge 1$, for all $\nu ,\iota ,\mu \in E$ and the function $H:E\to E$ satisfy the condition below:

$$R_{G_b}(H\nu ,H\iota ,H\mu )\le \theta R_{G_b}(\nu ,\iota ,\mu ),$$

(2)

where $\theta \in (0,1]$ is a constant. Then H has a unique fixed point in E.

3. Main Results

In this section, we prove various fixed point theorems in $\left(RGBMS\right)$. Firstly, we begin with the Hardy–Rogers type fixed point theorem.

Theorem 2.

Let $(E,R_{G_b})$ be a complete $\left(RGBMS\right)$ with a coefficient $e\ge 1$, for all $\nu ,\iota ,\mu \in E$ and the function $H:E\to E$ satisfy the condition below:

$$\begin{array}{cc}\hfill R_{G_b}(H\nu ,H\iota ,H\mu )\le & a_1{R}_{G_b}(\nu ,\iota ,\mu )+a_2{R}_{G_b}(\nu ,H\nu ,H\nu )+a_3{R}_{G_b}(\iota ,H\iota ,H\iota )\hfill \\ \hfill & +a_4{R}_{G_b}(\mu ,H\mu ,H\mu )+a_5{R}_{G_b}(\nu ,H\iota ,H\iota )+a_6{R}_{G_b}(\iota ,H\mu ,H\mu )\hfill \\ \hfill & +a_{7}G(\mu ,H\nu ,H\nu ),\hfill \end{array}$$

(3)

where ${\sum }_{i=1}^7{a}_i<1$ and $e(a_3+a_4+a_6)<1$, then H has a unique fixed point in E.

Proof. 

Let ${\lambda }_0\in E$ and define a sequence ${\lambda }_n$ in E by $H{\lambda }_{n-1}={\lambda }_n$, $\forall n\in {\mathbb{N}}^{*}$. If ${\lambda }_{n-1}={\lambda }_n$, then ${\lambda }_n$ is the fixed point of H. So, we assume that ${\lambda }_{n-1}\ne {\lambda }_n$, $\forall n\in {\mathbb{N}}^{*}$. It follows from the inequality (3) and $\left(RGb4\right)$ and $\left(RGb3\right)$ in (6) that

$$\begin{array}{ccc}\hfill R_{G_b}({\lambda }_n,{\lambda }_{n+1},{\lambda }_{n+2})& =& R_{G_b}(H{\lambda }_{n-1},H{\lambda }_n,H{\lambda }_{n+1})\hfill \\ & \le & a_1{R}_{G_b}({\lambda }_{n-1},{\lambda }_n,{\lambda }_{n+1})+a_2{R}_{G_b}({\lambda }_{n-1},H{\lambda }_{n-1},H{\lambda }_{n-1})\hfill \\ & & +a_3{R}_{G_b}({\lambda }_n,H{\lambda }_n,H{\lambda }_n)+a_4{R}_{G_b}({\lambda }_{n+1},H{\lambda }_{n+1},H{\lambda }_{n+1})\hfill \\ & & +a_5{R}_{G_b}({\lambda }_{n-1},H{\lambda }_n,H{\lambda }_n)+a_6{R}_{G_b}({\lambda }_n,H{\lambda }_{n+1},H{\lambda }_{n+1})\hfill \\ & & +a_7{R}_{G_b}({\lambda }_{n+1},H{\lambda }_{n-1},H{\lambda }_{n-1})\hfill \\ & =& a_1{R}_{G_b}({\lambda }_{n-1},{\lambda }_n,{\lambda }_{n+1})+a_2{R}_{G_b}({\lambda }_{n-1},{\lambda }_n,{\lambda }_n)\hfill \\ & & +a_3{R}_{G_b}({\lambda }_n,{\lambda }_{n+1},{\lambda }_{n+1})+a_4{R}_{G_b}({\lambda }_{n+1},{\lambda }_{n+2},{\lambda }_{n+2})\hfill \\ & & +a_5{R}_{G_b}({\lambda }_{n-1},{\lambda }_{n+1},{\lambda }_{n+1})+a_6{R}_{G_b}({\lambda }_n,{\lambda }_{n+2},{\lambda }_{n+2})\hfill \\ & & +a_7{R}_{G_b}({\lambda }_{n+1},{\lambda }_n,{\lambda }_n)\hfill \\ & \le & a_1{R}_{G_b}({\lambda }_{n-1},{\lambda }_n,{\lambda }_{n+1})+a_2{R}_{G_b}({\lambda }_n,{\lambda }_n,{\lambda }_{n-1})\hfill \\ & & +a_3{R}_{G_b}({\lambda }_{n+1},{\lambda }_{n+1},{\lambda }_n)+a_4{R}_{G_b}({\lambda }_{n+2},{\lambda }_{n+2},{\lambda }_{n+1})\hfill \\ & & +a_5{R}_{G_b}({\lambda }_{n+1},{\lambda }_{n+1},{\lambda }_{n-1})+a_6{R}_{G_b}({\lambda }_{n+2},{\lambda }_{n+2},{\lambda }_n)\hfill \\ & & +a_7{R}_{G_b}({\lambda }_n,{\lambda }_n,{\lambda }_{n+1})\hfill \\ & \le & a_1{R}_{G_b}({\lambda }_{n-1},{\lambda }_n,{\lambda }_{n+1})+a_2{R}_{G_b}({\lambda }_{n-1},{\lambda }_n,{\lambda }_{n+1})\hfill \\ & & +a_3{R}_{G_b}({\lambda }_n,{\lambda }_{n+1},{\lambda }_{n+2})+a_4{R}_{G_b}({\lambda }_n,{\lambda }_{n+1},{\lambda }_{n+2})\hfill \\ & & +a_5{R}_{G_b}({\lambda }_{n-1},{\lambda }_n,{\lambda }_{n+1})+a_6{R}_{G_b}({\lambda }_n,{\lambda }_{n+1},{\lambda }_{n+2})\hfill \\ & & +a_7{R}_{G_b}({\lambda }_n,{\lambda }_{n+1},{\lambda }_{n+2}).\hfill \end{array}$$

Hence, we have

$$R_{G_b}({\lambda }_n,{\lambda }_{n+1},{\lambda }_{n+2})\le {\displaystyle \frac{(a_1+a_2+a_5)}{(1-a_3-a_4-a_6-a_7)}}{R}_{G_b}({\lambda }_{n-1},{\lambda }_n,{\lambda }_{n+1}).$$

Let $k={\displaystyle \frac{(a_1+a_2+a_5)}{(1-a_3-a_4-a_6-a_7)}}$, where $k\in [0,1)$. Now, let we see that ${\lambda }_n\ne {\lambda }_{n+t}$ for all $t\ge 2$. Indeed, if ${\lambda }_n={\lambda }_{n+t}$ by (3), we get

$$\begin{array}{ccc}\hfill R_{G_b}({\lambda }_n,{\lambda }_{n+1},{\lambda }_{n+1})& =& R_{G_b}({\lambda }_n,H{\lambda }_n,H{\lambda }_{n+1})\hfill \\ & =& R_{G_b}({\lambda }_{n+t},H{\lambda }_{n+t},H{\lambda }_{n+t})\hfill \\ & \le & k^{t-1}{R}_{G_b}({\lambda }_n,{\lambda }_{n+1},{\lambda }_{n+1})\hfill \end{array}$$

a contradiction. So let ${\lambda }_n\ne {\lambda }_q$ for all $n\ne q$. By Lemma 1, ${\lambda }_n$ is a Cauchy sequence in E. Since $(E,R_{G_b})$ is a complete $\left(RGBMS\right)$, for any $\delta >0$ let a given $n_0\in \mathbb{N}$ and $j\in E$ such that

$$R_{G_b}({\lambda }_n,{\lambda }_q,j)<{\displaystyle \frac{\delta }{5e(1+a_1+a_2+a_5+a_7)}},\forall n,q\ge n_0$$

and from Proposition 1, we can write the following:

$$R_{G_b}(j,{\lambda }_n,{\lambda }_n)<{\displaystyle \frac{\delta }{5e}},R_{G_b}({\lambda }_n,j,j)<{\displaystyle \frac{\delta }{5a_{1}e}},R_{G_b}({\lambda }_n,{\lambda }_{n+1},{\lambda }_{n+1})<{\displaystyle \frac{\delta }{5(1+a_2)e}},$$

$$R_{G_b}(j,{\lambda }_{n+1},{\lambda }_{n+1})<{\displaystyle \frac{\delta }{5a_{7}e}},\forall n\ge n_0.$$

In addition, from Theorem 1, we obtain $R_{G_b}({\lambda }_n,Hj,Hj)=R_{G_b}(H{\lambda }_{n-1},Hj,Hj)\le \theta R_{G_b}({\lambda }_{n-1},j,j)<{\displaystyle \frac{\theta \delta }{5a_{5}e}}$, $\forall n\ge n_0$.

Now, for any $n\in \mathbb{N}$, by $\left(RGb5\right)$ in (6) and (3), we have

$$\begin{array}{ccc}\hfill R_{G_b}(j,Hj,Hj)& \le & e[R_{G_b}(j,{\lambda }_n,{\lambda }_n)+R_{G_b}({\lambda }_n,{\lambda }_{n+1},{\lambda }_{n+1})\hfill \\ & & +R_{G_b}({\lambda }_{n+1},Hj,Hj)]\hfill \\ & \le & e{R}_{G_b}(j,{\lambda }_n,{\lambda }_n)+e{R}_{G_b}({\lambda }_n,{\lambda }_{n+1},{\lambda }_{n+1})\hfill \\ & & +e{a}_1{R}_{G_b}({\lambda }_n,j,j)+e{a}_2{R}_{G_b}({\lambda }_n,H{\lambda }_n,H{\lambda }_n)\hfill \\ & & +e{a}_3{R}_{G_b}(j,Hj,Hj)+e{a}_4{R}_{G_b}(j,Hj,Hj)\hfill \\ & & +e{a}_5{R}_{G_b}({\lambda }_n,Hj,Hj)+e{a}_6{R}_{G_b}(j,Hj,Hj)\hfill \\ & & +e{a}_7{R}_{G_b}(j,H{\lambda }_n,H{\lambda }_n)\hfill \\ \hfill (1-e(a_3+a_4+a_6))R_{G_b}(j,Hj,Hj)& <& e[R_{G_b}(j,{\lambda }_n,{\lambda }_n)+(1+a_2)R_{G_b}({\lambda }_n,{\lambda }_{n+1},{\lambda }_{n+1})\hfill \\ & & +a_1{R}_{G_b}({\lambda }_n,j,j)+a_5{R}_{G_b}({\lambda }_n,Hj,Hj)\hfill \\ & & +a_7{R}_{G_b}(j,{\lambda }_{n+1},{\lambda }_{n+1})]\hfill \\ \hfill (1-e(a_3+a_4+a_6))R_{G_b}(j,Hj,Hj)& \le & e[{\displaystyle \frac{\delta }{5e}}+(1+a_2){\displaystyle \frac{\delta }{5(1+a_2)e}}+a_1{\displaystyle \frac{\delta }{5a_{1}e}}+a_5{\displaystyle \frac{\theta \delta }{5a_{5}e}}\hfill \\ & & +a_7{\displaystyle \frac{\delta }{5a_{7}e}}]\hfill \\ & \le & {\displaystyle \frac{\delta (4+\theta )}{5}}\hfill \\ & \le & \delta .\hfill \end{array}$$

Since $e(a_3+a_4+a_6)<1$,

$$\begin{array}{c}\hfill R_{G_b}(j,Hj,Hj)<{\displaystyle \frac{\delta }{1-e(a_3+a_4+a_6)}},\end{array}$$

we deduce that $R_{G_b}(j,Hj,Hj)=0$, that is, $Hj=j$. So j is a fixed point of H.

Now, let us see that j is a unique fixed point of H. If there exists another fixed point such that $j\ne m$, by (3) and $\left(RGb4\right)$ in (6), we have

$$\begin{array}{ccc}\hfill R_{G_b}(j,j,m)& \le & a_1{R}_{G_b}(j,j,m)+a_2{R}_{G_b}(j,Hj,Hj)\hfill \\ & & +a_3{R}_{G_b}(j,Hj,Hj)+a_4{R}_{G_b}(m,Hm,Hm)\hfill \\ & & +a_5{R}_{G_b}(j,Hj,Hj)+a_6{R}_{G_b}(j,Hm,Hm)\hfill \\ & & +a_7{R}_{G_b}(m,Hj,Hj)\hfill \\ & \le & (a_1+a_7)R_{G_b}(j,j,m)+a_6{R}_{G_b}(j,m,m)\hfill \\ & \le & (a_1+a_6+a_7)max\{R_{G_b}(j,j,m),R_{G_b}(j,m,m)\}\hfill \\ \hfill max\{R_{G_b}(j,j,m),R_{G_b}(j,m,m)\}& \le & (a_1+a_6+a_7)max\{R_{G_b}(j,j,m),R_{G_b}(j,m,m)\}.\hfill \end{array}$$

Similarly, we obtain

$$\begin{array}{ccc}\hfill R_{G_b}(j,m,m)& \le & a_1{R}_{G_b}(j,m,m)+a_2{R}_{G_b}(j,Hj,Hj)\hfill \\ & & +a_3{R}_{G_b}(m,Hm,Hm)+a_4{R}_{G_b}(m,Hm,Hm)\hfill \\ & & +a_5{R}_{G_b}(j,Hm,Hm)+a_6{R}_{G_b}(m,Hm,Hm)\hfill \\ & & +a_7{R}_{G_b}(m,Hj,Hj)\hfill \\ & \le & (a_1+a_5)R_{G_b}(j,m,m)+a_7{R}_{G_b}(m,j,j)\hfill \\ & \le & (a_1+a_5+a_7)max\{R_{G_b}(j,j,m),R_{G_b}(j,m,m)\}\hfill \\ & \le & (a_1+a_5+a_6+a_7)max\{R_{G_b}(j,j,m),R_{G_b}(j,m,m)\}.\hfill \end{array}$$

Then, we find

$$max\{R_{G_b}(j,j,m),R_{G_b}(j,m,m)\}\le (a_1+a_5+a_6+a_7)max\{R_{G_b}(j,j,m),R_{G_b}(j,m,m)\}.$$

Since $(a_1+a_6+a_7)<1$, we find $max\{R_{G_b}(j,j,m),R_{G_b}(j,m,m)\}=0$. Therefore

$$R_{G_b}(j,j,m)=R_{G_b}(j,m,m)=0,$$

which means that $j=m$. □

We remark that the Kannan- and Chatterjea-type fixed point results in $\left(RGBMS\right)$ are just corollaries of Theorem 2.

Corollary 1.

Let $(E,R_{G_b})$ be a complete $\left(RGBMS\right)$ with coefficient $e\ge 1$ and the function $H:E\to E$. Suppose there exists $\gamma \in [0,{\displaystyle \frac{1}{max\{2e,3\}}})$ such that

$$R_{G_b}(H\nu ,H\iota ,H\mu )\le \gamma \{R_{G_b}(\nu ,H\nu ,H\nu )+R_{G_b}(\iota ,H\iota ,H\iota )+R_{G_b}(\mu ,H\mu ,H\mu )\}.$$

Then H has a unique fixed point in E.

Proof. 

The assertion follows when we take $a_1=a_5=a_6=a_7=0$ and $a_2=a_3=a_4=\gamma$ in Theorem 2. □

Corollary 2.

Let $(E,R_{G_b})$ be a complete $\left(RGBMS\right)$ with the coefficient $e\ge 1$ and $H:E\to E$ a self-mapping. Suppose there exists $\gamma \in [0,{\displaystyle \frac{1}{2e+1}})$ such that

$$R_{G_b}(H\nu ,H\iota ,H\mu )\le \gamma \{R_{G_b}(\nu ,H\iota ,H\iota )+R_{G_b}(\iota ,H\mu ,H\mu )+R_{G_b}(\mu ,H\nu ,H\nu )\},$$

then H has a unique fixed point in E.

Proof. 

The assertion follows when we take $a_1=a_2=a_3=a_4=0$ and $a_5=a_6=a_7=\gamma$ in Theorem 2. □

The next example is given to support Theorem 2.

Example 3.

We will use the same (RGBMS) as in Example 1. Let $(E,R_{G_b})$ be a $\left(RGBMS\right)$ where $E=\mathbb{R}$ and $R_{G_b}:E^3\to [0,\infty )$ defined by

$$R_{G_b}(\nu ,\iota ,\mu )={\left(\right|\nu -\iota |+|\iota -\mu |+|\mu -\nu \left|\right)}^2.$$

Consider a function $H:E\to E$ defined by

$$H\left(t\right)={\displaystyle \frac{t}{2}}+3$$

for all $t\in E$. If we take $a_1=0.3, a_2=0.2, a_3=0.01, a_4=0.02, a_5=0.1, a_6=0.05,$ and $a_7=0.15$, we find that the conditions below are satisfied:

$$\sum _{i=1}^7{a}_i=0.83<1ande(a_3+a_4+a_6)=0.24<1,$$

and for all $\nu ,\iota ,\mu \in E$,

$$\begin{array}{ccc}\hfill R_{G_b}(H\nu ,H\iota ,H\mu )& =& R_{G_b}({\displaystyle \frac{\nu }{2}}+3,{\displaystyle \frac{\iota }{2}}+3,{\displaystyle \frac{\mu }{2}}+3)\hfill \\ & =& {\displaystyle \frac{1}{4}}{\left(\right|\nu -\iota |+|\iota -\mu |+|\nu -\mu \left|\right)}^2\hfill \\ & \le & 0.3{\left(\right|\nu -\iota |+|\iota -\mu |+|\nu -\mu \left|\right)}^2\hfill \\ & =& 0.3R_{G_b}(\nu ,\iota ,\mu )\hfill \\ & \le & 0.3R_{G_b}(\nu ,\iota ,\mu )+0.2R_{G_b}(\nu ,H\nu ,H\nu )+0.01R_{G_b}(\iota ,H\iota ,H\iota )\hfill \\ & & +0.02R_{G_b}(\mu ,H\mu ,H\mu )+0.1R_{G_b}(\nu ,H\iota ,H\iota )+0,05R_{G_b}(\iota ,H\mu ,H\mu )\hfill \\ & & +0.15G(\mu ,H\nu ,H\nu ).\hfill \end{array}$$

Hence, it is easy to see that Theorem 2 is verified and so H has a unique fixed point as 6.

Example 4.

Let $E=\{{\displaystyle \frac{1}{2}}\}\bigcup [1,2,3,4]$ and $R_{G_b}:E^3\to [0,\infty )$ be a function such that

$$R_{G_b}(\nu ,\iota ,\mu )=R_{G_b}(\nu ,\mu ,\iota )=R_{G_b}(\mu ,\nu ,\iota )=R_{G_b}(\mu ,\iota ,\nu )=R_{G_b}(\iota ,\mu ,\nu )=R_{G_b}(\iota ,\nu ,\mu )$$

for all $\nu ,\iota ,\mu \in E$ and

$$R_{G_b}(\nu ,\iota ,\mu )=\left\{\begin{array}{cc}0,\hfill & \mathit{if}\nu =\iota =\mu \hfill \\ 3,\hfill & \mathit{if}\nu ={\displaystyle \frac{1}{2}},\iota ,\mu \in [1,2,3,4]\hfill \\ 1,\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

Then $(E,R_{G_b})$ is a $\left(RGBMS\right)$ with coefficient $e=1$. Consider a function $H:E\to E$ defined as follows:

$$H\left(t\right)=\left\{\begin{array}{cc}1,\hfill & \mathit{if}t\in [1,2,3,4]\hfill \\ 2,\hfill & \mathit{if}t={\displaystyle \frac{1}{2}}.\hfill \end{array}\right.$$

For $\gamma \in (0,{\displaystyle \frac{1}{3}})$, we have the following cases:

• Case-1: $\nu ={\displaystyle \frac{1}{2}}$ and $\iota ,\mu \in [2,3,4]$,

$$\begin{array}{ccc}\hfill R_{G_b}(H\nu ,H\iota ,H\mu )& \le & \gamma \{R_{G_b}(\nu ,H\nu ,H\nu )+R_{G_b}(\iota ,H\iota ,H\iota )+R_{G_b}(\mu ,H\mu ,H\mu )\}\hfill \\ \hfill R_{G_b}(2,1,1)& \le & \gamma \{R_{G_b}({\displaystyle \frac{1}{2}},2,2)+R_{G_b}(\iota ,1,1)+R_{G_b}(\mu ,1,1)\}\hfill \\ \hfill 1& \le & \gamma [3+1+1].\hfill \end{array}$$

Case-2: $\nu ={\displaystyle \frac{1}{2}}$ and $\iota =\mu =1$,

$$\begin{array}{ccc}\hfill R_{G_b}(H\nu ,H\iota ,H\mu )& \le & \gamma \{R_{G_b}(\nu ,H\nu ,H\nu )+R_{G_b}(\iota ,H\iota ,H\iota )+R_{G_b}(\mu ,H\mu ,H\mu )\}\hfill \\ \hfill R_{G_b}(2,1,1)& \le & \gamma \{R_{G_b}({\displaystyle \frac{1}{2}},2,2)+R_{G_b}(1,1,1)+R_{G_b}(1,1,1)\}\hfill \\ \hfill 1& \le & \gamma [3+0+0].\hfill \end{array}$$

Case-3: $\nu ,\iota ,\mu \in [1,2,3,4]$,

$$\begin{array}{ccc}\hfill R_{G_b}(H\nu ,H\iota ,H\mu )& \le & \gamma \{R_{G_b}(\nu ,H\nu ,H\nu )+R_{G_b}(\iota ,H\iota ,H\iota )+R_{G_b}(\mu ,H\mu ,H\mu )\}\hfill \\ \hfill R_{G_b}(1,1,1)& \le & \gamma \{R_{G_b}(\nu ,1,1)+R_{G_b}(\iota ,1,1)+R_{G_b}(\mu ,1,1)\}\hfill \\ \hfill 0& \le & \gamma \{R_{G_b}(\nu ,1,1)+R_{G_b}(\iota ,1,1)+R_{G_b}(\mu ,1,1)\}.\hfill \end{array}$$

So we can say that the condition in Corollary 1 is verified in each case and hence H has a unique fixed point as 1.

Theorem 3.

Let $(E,R_{G_b})$ be a complete $\left(RGBMS\right)$ with the coefficient $e\ge 1$, $R_{G_b}$ be a continuous function, and $H:E\to E$ be a mapping such that

$$\begin{array}{cc}\hfill \psi \left(R_{G_b}(H\nu ,H\iota ,H\mu )\right)\le & \psi \left({\displaystyle \frac{1}{3}}[G(\nu ,H\iota ,H\iota )+R_{G_b}(\iota ,H\mu ,H\mu )+R_{G_b}(\mu ,H\nu ,H\nu )]\right)\hfill \\ \hfill & -\varphi \left(R_{G_b}(\nu ,H\iota ,H\iota ),R_{G_b}(\iota ,H\mu ,H\mu ),R_{G_b}(\mu ,H\nu ,H\nu )\right),\hfill \end{array}$$

(4)

where ψ is the altering distance function and $\varphi :{[0,\infty )}^3\to [0,\infty )$ is a continuous function with $\varphi (u,v,z)=0$ iff $u=v=z$. Then, H has a unique fixed point in E.

Proof. 

Let ${\lambda }_0\in E$ and define a sequence ${\lambda }_n$ in E by $H{\lambda }_{n-1}={\lambda }_n$, $\forall n\in {\mathbb{N}}^{*}$. If ${\lambda }_{n-1}={\lambda }_n$, then H has a fixed point. So suppose that ${\lambda }_{n-1}\ne {\lambda }_n$ for all $n\in {\mathbb{N}}^{*}$. By (4), we get

$$\begin{array}{ccc}\hfill \psi \left(R_{G_b}({\lambda }_n,{\lambda }_{n+1},{\lambda }_{n+2})\right)& =& \psi (R_{G_b}(H{\lambda }_{n-1},H{\lambda }_n,H{\lambda }_{n+1})\hfill \\ & \le & \psi ({\displaystyle \frac{1}{3}}[R_{G_b}({\lambda }_{n-1},H{\lambda }_n,H{\lambda }_n)+R_{G_b}({\lambda }_n,H{\lambda }_{n+1},H{\lambda }_{n+1})\hfill \\ & & +R_{G_b}({\lambda }_{n+1},H{\lambda }_{n-1},H{\lambda }_{n-1})\left]\right)-\varphi (R_{G_b}({\lambda }_{n-1},H{\lambda }_n,H{\lambda }_n),\hfill \\ & & R_{G_b}({\lambda }_n,H{\lambda }_{n+1},H{\lambda }_{n+1}),R_{G_b}({\lambda }_{n+1},H{\lambda }_{n-1},H{\lambda }_{n-1}))\hfill \\ & =& \psi ({\displaystyle \frac{1}{3}}[R_{G_b}({\lambda }_{n-1},{\lambda }_{n+1},{\lambda }_{n+1})+R_{G_b}({\lambda }_n,{\lambda }_{n+2},{\lambda }_{n+2})\hfill \\ & & +R_{G_b}({\lambda }_{n+1},{\lambda }_n,{\lambda }_n)\left]\right)-\varphi (R_{G_b}({\lambda }_{n-1},{\lambda }_{n+1},{\lambda }_{n+1}),\hfill \\ & & R_{G_b}({\lambda }_n,{\lambda }_{n+2},{\lambda }_{n+2}),R_{G_b}({\lambda }_{n+1},{\lambda }_n,{\lambda }_n)).\hfill \end{array}$$

From $\left(RGb3\right)$ in (6), we have

$$\begin{array}{cc}\hfill \psi \left(R_{G_b}({\lambda }_n,{\lambda }_{n+1},{\lambda }_{n+2})\right)\le & \psi ({\displaystyle \frac{1}{3}}[R_{G_b}({\lambda }_{n-1},{\lambda }_n,{\lambda }_{n+1})+R_{G_b}({\lambda }_n,{\lambda }_{n+1},{\lambda }_{n+2})\hfill \\ \hfill & +R_{G_b}({\lambda }_n,{\lambda }_{n+1},{\lambda }_{n+2})\left]\right)-\varphi (R_{G_b}({\lambda }_{n-1},{\lambda }_n,{\lambda }_{n+1}),R_{G_b}({\lambda }_n,{\lambda }_{n+1},{\lambda }_{n+2}),\hfill \\ \hfill & R_{G_b}({\lambda }_n,{\lambda }_{n+1},{\lambda }_{n+2})),\hfill \end{array}$$

(5)

and

$$\psi \left(R_{G_b}({\lambda }_n,{\lambda }_{n+1},{\lambda }_{n+2})\right)\le \psi \left({\displaystyle \frac{1}{3}}[R_{G_b}({\lambda }_{n-1},{\lambda }_n,{\lambda }_{n+1})+R_{G_b}({\lambda }_n,{\lambda }_{n+1},{\lambda }_{n+2})+R_{G_b}({\lambda }_n,{\lambda }_{n+1},{\lambda }_{n+2})]\right).$$

Since $\psi$ is an increasing function, we find

$$R_{G_b}({\lambda }_n,{\lambda }_{n+1},{\lambda }_{n+2})\le \left({\displaystyle \frac{1}{3}}[R_{G_b}({\lambda }_{n-1},{\lambda }_n,{\lambda }_{n+1})+R_{G_b}({\lambda }_n,{\lambda }_{n+1},{\lambda }_{n+2})+R_{G_b}({\lambda }_n,{\lambda }_{n+1},{\lambda }_{n+2})]\right)$$

(6)

and so

$$R_{G_b}({\lambda }_n,{\lambda }_{n+1},{\lambda }_{n+2})\le R_{G_b}({\lambda }_{n-1},{\lambda }_n,{\lambda }_{n+1}).$$

Therefore $\{R_{G_b}({\lambda }_n,{\lambda }_{n+1},{\lambda }_{n+2}):n\in \mathbb{N}\}$ is a nonincreasing sequence. Thus there exists $\vartheta \ge 0$ such that ${lim}_{n\to \infty }{R}_{G_b}({\lambda }_n,{\lambda }_{n+1},{\lambda }_{n+2})=\vartheta$. Taking $(n\to \infty$) in (6), we have

$$\vartheta \le {\displaystyle \frac{1}{3}}li{m}_{n\to \infty }{R}_{G_b}({\lambda }_{n-1},{\lambda }_n,{\lambda }_{n+1})+{\displaystyle \frac{2}{3}}\vartheta .$$

Hence

$$\underset{n\to \infty }{lim}{R}_{G_b}({\lambda }_n,{\lambda }_{n+1},{\lambda }_{n+2})\le \vartheta .$$

If we take $(n\to \infty )$ in (5), we get

$$\begin{array}{ccc}\hfill \psi \left(\vartheta \right)& \le & \psi ({\displaystyle \frac{1}{3}}\vartheta +{\displaystyle \frac{1}{3}}\vartheta +{\displaystyle \frac{1}{3}}\vartheta )-\varphi (\vartheta ,\vartheta ,\vartheta )\hfill \\ \hfill \psi \left(\vartheta \right)& \le & \psi \left(\vartheta \right)-\varphi (\vartheta ,\vartheta ,\vartheta )\hfill \\ \hfill \varphi (\vartheta ,\vartheta ,\vartheta )& \le & 0.\hfill \end{array}$$

So $\varphi (\vartheta ,\vartheta ,\vartheta )=0$, which means that $\vartheta =0$. Therefore

$$\underset{n\to \infty }{lim}{R}_{G_b}({\lambda }_n,{\lambda }_{n+1},{\lambda }_{n+2})=0.$$

(7)

Next, we show that ${\lambda }_n$ is an $R_{G_b}$-Cauchy sequence in E. Let ${\lambda }_n\ne {\lambda }_q$ such that $q<n$. By (4), $\left(RGb3\right)$, $\left(RGb4\right)$, and $\left(RGb5\right)$ in (6), we obtain

$$\begin{array}{ccc}\hfill \psi \left(R_{G_b}({\lambda }_q,{\lambda }_n,{\lambda }_n)\right)& =& \psi \left(R_{G_b}(H{\lambda }_{q-1},H{\lambda }_{n-1},H{\lambda }_{n-1})\right)\hfill \\ & \le & \psi ({\displaystyle \frac{1}{3}}[R_{G_b}({\lambda }_{q-1},H{\lambda }_{n-1},H{\lambda }_{n-1})+R_{G_b}({\lambda }_{n-1},H{\lambda }_{n-1},H{\lambda }_{n-1})\hfill \\ & & +R_{G_b}({\lambda }_{n-1},H{\lambda }_{q-1},H{\lambda }_{q-1})\left]\right)-\varphi (R_{G_b}({\lambda }_{q-1},H{\lambda }_{n-1},H{\lambda }_{n-1}),\hfill \\ & & R_{G_b}({\lambda }_{n-1},H{\lambda }_{n-1},H{\lambda }_{n-1}),R_{G_b}({\lambda }_{n-1},H{\lambda }_{q-1},H{\lambda }_{q-1}))\hfill \\ & =& \psi \left({\displaystyle \frac{1}{3}}[R_{G_b}({\lambda }_{q-1},{\lambda }_n,{\lambda }_n)+R_{G_b}({\lambda }_{n-1},{\lambda }_n,{\lambda }_n)+R_{G_b}({\lambda }_{n-1},{\lambda }_q,{\lambda }_q)]\right)\hfill \\ & & -\varphi (R_{G_b}({\lambda }_{q-1},{\lambda }_n,{\lambda }_n),R_{G_b}({\lambda }_{n-1},{\lambda }_n,{\lambda }_n),R_{G_b}({\lambda }_{n-1},{\lambda }_q,{\lambda }_q))\hfill \\ & \le & \psi \left({\displaystyle \frac{1}{3}}[R_{G_b}({\lambda }_{q-1},{\lambda }_n,{\lambda }_n)+R_{G_b}({\lambda }_{n-1},{\lambda }_n,{\lambda }_n)+R_{G_b}({\lambda }_{n-1},{\lambda }_q,{\lambda }_q)]\right)\hfill \\ & \le & \psi ({\displaystyle \frac{1}{3}}[e{R}_{G_b}({\lambda }_{q-1},{\lambda }_q,{\lambda }_q)+e{R}_{G_b}({\lambda }_q,{\lambda }_{n-1},{\lambda }_{n-1})+e{R}_{G_b}({\lambda }_{n-1},{\lambda }_n,{\lambda }_n)\hfill \\ & & +R_{G_b}({\lambda }_{n-1},{\lambda }_n,{\lambda }_n)+R_{G_b}({\lambda }_{n-1},{\lambda }_q,{\lambda }_q)\left]\right)\hfill \\ & =& \psi ({\displaystyle \frac{e}{3}}{R}_{G_b}({\lambda }_{q-1},{\lambda }_q,{\lambda }_q)+{\displaystyle \frac{e+1}{3}}{R}_{G_b}({\lambda }_{n-1},{\lambda }_n,{\lambda }_n)\hfill \\ & & +{\displaystyle \frac{e}{3}}{R}_{G_b}({\lambda }_q,{\lambda }_{n-1},{\lambda }_{n-1})+{\displaystyle \frac{1}{3}}{R}_{G_b}({\lambda }_{n-1},{\lambda }_q,{\lambda }_q))\hfill \\ & \le & \psi ({\displaystyle \frac{e}{3}}{R}_{G_b}({\lambda }_q,{\lambda }_q,{\lambda }_{q-1})+{\displaystyle \frac{e+1}{3}}{R}_{G_b}({\lambda }_n,{\lambda }_n,{\lambda }_{n-1})\hfill \\ & & +{\displaystyle \frac{e}{3}}{R}_{G_b}({\lambda }_{n-1},{\lambda }_{n-1},{\lambda }_q)+{\displaystyle \frac{1}{3}}{R}_{G_b}({\lambda }_q,{\lambda }_q,{\lambda }_{n-1}))\hfill \\ & \le & \psi ({\displaystyle \frac{e}{3}}{R}_{G_b}({\lambda }_{q-1},{\lambda }_q,{\lambda }_{q+1})+{\displaystyle \frac{e+1}{3}}{R}_{G_b}({\lambda }_{n-1},{\lambda }_n,{\lambda }_{n+1})\hfill \\ & & +{\displaystyle \frac{e^2}{3}}[R_{G_b}({\lambda }_q,{\lambda }_{q+1},{\lambda }_{q+1})+R_{G_b}({\lambda }_{q+1},{\lambda }_n,{\lambda }_n)\hfill \\ & & +R_{G_b}({\lambda }_n,{\lambda }_{n-1},{\lambda }_{n-1})]+{\displaystyle \frac{1}{3}}{R}_{G_b}({\lambda }_{n-1},{\lambda }_q,{\lambda }_q)).\hfill \end{array}$$

If we continue in the same way, we have

$$\begin{array}{ccc}& \le & \psi ({\displaystyle \frac{e}{3}}{R}_{G_b}({\lambda }_{q-1},{\lambda }_q,{\lambda }_{q+1})+{\displaystyle \frac{e+1}{3}}{R}_{G_b}({\lambda }_{n-1},{\lambda }_n,{\lambda }_{n+1})+{\displaystyle \frac{e^2}{3}}{R}_{G_b}({\lambda }_q,{\lambda }_{q+1},{\lambda }_{q+2})\hfill \\ & & +{\displaystyle \frac{e^2}{3}}{R}_{G_b}({\lambda }_{n-1},{\lambda }_n,{\lambda }_{n+1})+{\displaystyle \frac{e^3}{3}}{R}_{G_b}({\lambda }_{q+1},{\lambda }_{q+2},{\lambda }_{q+3})+\dots +\dots \hfill \\ & & +{\displaystyle \frac{e^{n-q}}{3}}{R}_{G_b}({\lambda }_{n-2},{\lambda }_{n-1},{\lambda }_n)+{\displaystyle \frac{1}{3}}{R}_{G_b}({\lambda }_{n-1},{\lambda }_q,{\lambda }_q)).\hfill \end{array}$$

Similarly, we can continue to apply $\left(RGb3\right)$ and $\left(RGb5\right)$ on $R_{G_b}({\lambda }_{n-1},{\lambda }_q,{\lambda }_q)$. Since $\psi$ is an increasing function,

$$\begin{array}{ccc}\hfill R_{G_b}({\lambda }_q,{\lambda }_n,{\lambda }_n)& \le & {\displaystyle \frac{e}{3}}{R}_{G_b}({\lambda }_{q-1},{\lambda }_q,{\lambda }_{q+1})+{\displaystyle \frac{e+1}{3}}{R}_{G_b}({\lambda }_{n-1},{\lambda }_n,{\lambda }_{n+1})\hfill \\ & & +{\displaystyle \frac{e^2}{3}}[R_{G_b}({\lambda }_q,{\lambda }_{q+1},z{\lambda }_{q+2})+{\displaystyle \frac{e^2}{3}}{R}_{G_b}({\lambda }_{n-1},{\lambda }_n,{\lambda }_{n+1})\hfill \\ & & +{\displaystyle \frac{e^3}{3}}{R}_{G_b}({\lambda }_{q+1},{\lambda }_{q+2},{\lambda }_{q+3})+\dots +\dots \hfill \\ & & +{\displaystyle \frac{e^n}{3}}{R}_{G_b}({\lambda }_{n-2},{\lambda }_{n-1},{\lambda }_n)+{\displaystyle \frac{e}{3}}{R}_{G_b}({\lambda }_q,{\lambda }_{q+1},{\lambda }_{q+2})+\dots \hfill \end{array}$$

Taking the limit in (7), we have

$$\underset{q,n\to \infty }{lim}{R}_{G_b}({\lambda }_q,{\lambda }_n,{\lambda }_n)=0.$$

Then, $\left\{{\lambda }_n\right\}$ is an $R_{G_b}$-Cauchy sequence in E. From the fact that $R_{G_b}$ is a complete metric space, there exists $j\in E$ such that

$$\underset{n\to \infty }{lim}{R}_{G_b}({\lambda }_n,{\lambda }_n,j)=0.$$

Below, we see that j is a fixed point of H. From (4), we get

$$\begin{array}{ccc}\hfill \psi \left(R_{G_b}({\lambda }_{n+1},{\lambda }_{n+1},Hj)\right)& =& \psi \left(R_{G_b}(H{\lambda }_n,H{\lambda }_n,Hj)\right)\hfill \\ & \le & \psi ({\displaystyle \frac{1}{3}}[R_{G_b}({\lambda }_n,H{\lambda }_n,H{\lambda }_n)+R_{G_b}({\lambda }_n,Hj,Hj)\hfill \\ & & +R_{G_b}(j,H{\lambda }_n,H{\lambda }_n)\left]\right)-\varphi (R_{G_b}({\lambda }_n,H{\lambda }_n,H{\lambda }_n),\hfill \\ & & R_{G_b}({\lambda }_n,Hj,Hj),R_{G_b}(j,H{\lambda }_n,H{\lambda }_n))\hfill \\ & \le & \psi ({\displaystyle \frac{1}{3}}[R_{G_b}({\lambda }_n,{\lambda }_{n+1},{\lambda }_{n+1})+R_{G_b}({\lambda }_n,Hj,Hj)\hfill \\ & & +R_{G_b}(j,{\lambda }_{n+1},{\lambda }_{n+1})\left]\right)-\varphi (R_{G_b}({\lambda }_n,{\lambda }_{n+1},{\lambda }_{n+1}),\hfill \\ & & R_{G_b}({\lambda }_n,Hj,Hj),R_{G_b}(j,{\lambda }_{n+1},{\lambda }_{n+1})\hfill \\ & \le & \psi ({\displaystyle \frac{1}{3}}[R_{G_b}({\lambda }_n,{\lambda }_{n+1},{\lambda }_{n+1})+R_{G_b}({\lambda }_n,Hj,Hj)\hfill \\ & & +R_{G_b}(j,{\lambda }_{n+1},{\lambda }_{n+1})\left]\right).\hfill \end{array}$$

By the fact that $\psi$ is an increasing function, we have

$$\left(R_{G_b}({\lambda }_{n+1},{\lambda }_{n+1},Hj)\right)\le \left({\displaystyle \frac{1}{3}}[R_{G_b}({\lambda }_n,{\lambda }_{n+1},{\lambda }_{n+1})+R_{G_b}({\lambda }_n,Hj,Hj)+R_{G_b}(j,{\lambda }_{n+1},{\lambda }_{n+1})]\right).$$

Taking $(n\to \infty )$, we get

$$\begin{array}{ccc}\hfill R_{G_b}(j,j,Hj)& \le & {\displaystyle \frac{1}{3}}{R}_{G_b}(j,Hj,Hj)\hfill \\ \hfill max\{R_{G_b}(j,j,Hj),R_{G_b}(j,Hj,Hj)\}& \le & {\displaystyle \frac{1}{3}}max\{R_{G_b}(j,j,Hj),R_{G_b}(j,Hj,Hj)\}.\hfill \end{array}$$

Therefore

$$max\{R_{G_b}(j,j,Hj),R_{G_b}(j,Hj,Hj)\}=0.$$

Then, $R_{G_b}(j,j,Hj)=R_{G_b}(j,Hj,Hj)=0$ so $j=Hj$, which means that j is a fixed point of H. Finally, we show that j is a unique fixed point of H. So, let m be another fixed point of H, then

$$\begin{array}{ccc}\hfill \psi \left(R_{G_b}(j,m,m)\right)& =& \psi \left(R_{G_b}(Hj,Hm,Hm)\right)\hfill \\ & \le & \psi \left({\displaystyle \frac{1}{3}}[R_{G_b}(j,Hm,Hm)+R_{G_b}(m,Hm,Hm)+R_{G_b}(m,Hj,Hj)]\right)\hfill \\ & & -\varphi (R_{G_b}(j,Hm,Hm),R_{G_b}(m,Hm,Hm),R_{G_b}(m,Hj,Hj))\hfill \\ & =& \psi \left({\displaystyle \frac{1}{3}}[R_{G_b}(j,j,m)+R_{G_b}(m,j,j)]\right)\hfill \\ & & -\varphi (R_{G_b}(j,m,m),0,R_{G_b}(m,j,j))\hfill \\ & \le & \psi \left({\displaystyle \frac{1}{3}}[R_{G_b}(j,m,m)+R_{G_b}(m,j,j)]\right).\hfill \end{array}$$

From the fact that $\psi$ is an increasing function, we have

$$\begin{array}{ccc}\hfill R_{G_b}(j,m,m)& \le & {\displaystyle \frac{1}{3}}[R_{G_b}(j,m,m)+R_{G_b}(m,j,j)]\hfill \\ & \le & {\displaystyle \frac{2}{3}}max[R_{G_b}(j,m,m),R_{G_b}(m,j,j)].\hfill \end{array}$$

Similarly, we obtain

$$\begin{array}{ccc}\hfill \psi \left(R_{G_b}(m,j,j)\right)& =& \psi \left(R_{G_b}(Hm,Hj,Hj)\right)\hfill \\ & \le & \psi \left({\displaystyle \frac{1}{3}}[R_{G_b}(m,Hj,Hj)+R_{G_b}(j,Hj,Hj)+R_{G_b}(j,Hm,Hm)]\right)\hfill \\ & & -\varphi (R_{G_b}(m,Hj,Hj),R_{G_b}(j,Hj,Hj),R_{G_b}(j,Hm,Hm))\hfill \\ & =& \psi \left({\displaystyle \frac{1}{3}}[R_{G_b}(m,j,j)+R_{G_b}(j,m,m)]\right)\hfill \\ & & -\varphi (R_{G_b}(m,j,j),0,R_{G_b}(j,m,m))\hfill \\ & \le & \psi \left({\displaystyle \frac{1}{3}}[R_{G_b}(m,j,j)+R_{G_b}(j,m,m)]\right).\hfill \end{array}$$

By the fact that $\psi$ is an increasing function, we get

$$\begin{array}{ccc}\hfill R_{G_b}(m,j,j)& \le & {\displaystyle \frac{1}{3}}[R_{G_b}(m,j,j)+R_{G_b}(j,m,m)]\hfill \\ & \le & {\displaystyle \frac{2}{3}}max[R_{G_b}(m,j,j),R_{G_b}(j,m,m)]\hfill \end{array}$$

and so

$$max\{R_{G_b}(m,j,j),R_{G_b}(j,m,m)\}\le {\displaystyle \frac{2}{3}}max\{R_{G_b}(m,j,j),R_{G_b}(j,m,m)\}.$$

Hence

$$max\{R_{G_b}(m,j,j),R_{G_b}(j,m,m)\}=0,$$

and consequently, $R_{G_b}(j,m,m)=R_{G_b}(m,j,j)=0$, that is, $j=m$, which means that j is a unique fixed point of H. □

Example 5.

Let $E=[0,1]$ and $R_{G_b}:E^3\to [0,\infty )$ be a $\left(RGBMS\right)$ such that

$$R_{G_b}(\nu ,\iota ,\mu )={\left(\right|\nu -\iota |+|\iota -\mu |+|\mu -\nu \left|\right)}^2$$

with the coefficient $e=3$ where $\nu ,\iota ,\mu \in E$. Let $\psi :[0,\infty )\to [0,\infty )$ such that $\psi \left(\nu \right)=3\nu$ is an altering distance function and $\varphi :{[0,\infty )}^3\to [0,\infty )$ is a continuous function with $\varphi (\nu ,\iota ,\mu )=0$ if $\nu =\iota =\mu$ such that

$$\varphi (\nu ,\iota ,\mu )={\displaystyle \frac{\nu +\iota +\mu }{2}}.$$

If we define $H\left(\nu \right)=1$, we get $\psi \left(R_{G_b}(H\nu ,H\iota ,H\mu )\right)=0$. It can be easily seen that H is a weakly G-contraction mapping. So, from Theorem 3, H has one fixed point as 1.

4. Some Applications on Fractional Integrals

4.1. Application to Riemann–Liouville Fractional Integrals

As an application of Theorem 2, we demonstrate that the Riemann–Liouville fractional integral has just one solution. The form of the Riemann–Liouville fractional integral is

$${}_a{}^{RL}I_{\varrho }^{\varsigma }={\displaystyle \frac{1}{\Gamma \left(\varsigma \right)}}{\int }_a^{\varrho }h\left(\mu \right){(\varrho -\mu )}^{\varsigma -1}d\mu ;\Gamma \left(\varsigma \right)>0,$$

(8)

where $\varsigma \in \mathbb{R}$, $\Omega =${ h : h is a continuous function, $h:[0,1]\to \mathbb{R}$)} and $\varrho ,\mu \in [0,1]$. Using Example 1, we define $R_{G_b}:{\Omega }^3\to [0,\infty )$ by

$$R_{G_b}(h,l,d)={\left(\right|h-l|+|l-h|+|h-d\left|\right)}^2$$

with $s=3$ for all $h\left(\varrho \right),l\left(\varrho \right),d\left(\varrho \right)\in \Omega$ and $\varrho \in [0,1]$, where $R_{G_b}$ is a rectangular $G_b$-metric. Now, if we accept the condition below, we can show that (8) has a unique solution.

$${\displaystyle \frac{1}{{\Gamma }^2(\varrho +1)}}.{\displaystyle \frac{{(\varrho -\mu )}^{\varsigma -1}{(\varrho -a)}^{2\varsigma }}{{|(\varrho -\mu )}^{\varsigma -1}|}}=k,\mathrm{where}k<{\displaystyle \frac{1}{45}}\mathrm{and}\varrho \ne \mu .$$

Firstly, let $H:\Omega \to \Omega$ be an operator defined by

$$Hh\left(\varrho \right)={\displaystyle \frac{1}{\Gamma \left(\varsigma \right)}}{\int }_a^{\varrho }h\left(\mu \right){(\varrho -\mu )}^{\varsigma -1}d\mu .$$

(9)

Then, we obtain the following:

$$\begin{array}{ccc}\hfill R_{G_b}(Hh\left(\varrho \right),Hh\left(\varrho \right),Hl\left(\varrho \right))& =& {\left(\right|Hh\left(\varrho \right)-Hh\left(\varrho \right)|+|Hh\left(\varrho \right)-Hl\left(\varrho \right)|+|Hl\left(\varrho \right)-Hh\left(\varrho \right)\left|\right)}^2\hfill \\ & =& 4\left(\right|Hh\left(\varrho \right)-{Hl\left(\varrho \right)\left|\right)}^2\hfill \\ & =& 4{\left|{\displaystyle \frac{1}{\Gamma \left(\varsigma \right)}}{\int }_a^{\varrho }h\left(\mu \right){(\varrho -\mu )}^{\varsigma -1}d\mu -{\displaystyle \frac{1}{\Gamma \left(\varsigma \right)}}{\int }_a^{\varrho }l\left(\mu \right){(\varrho -\mu )}^{\varsigma -1}d\mu \right|}^2\hfill \\ & \le & 4{\left({\displaystyle \frac{1}{\Gamma \left(\varsigma \right)}}{\int }_a^{\varrho }{(\varrho -\mu )}^{\varsigma -1}d\mu \right)}^2{|h\left(\mu \right)-l\left(\mu \right)|}^2\hfill \\ & \le & {\displaystyle \frac{4}{{\Gamma }^2\left(\varsigma \right)}}({\int }_a^{\varrho }{|(\varrho -\mu )}^{\varsigma -1}|d\mu {)}^2{|h\left(\mu \right)-l\left(\mu \right)|}^2\hfill \\ & \le & {\displaystyle \frac{4}{{\Gamma }^2\left(\varsigma \right)}}({\int }_a^{\varrho }{|(\varrho -\mu )}^{\varsigma -1}|d\mu {)}^2\left(\right|h\left(\mu \right)-Hh\left(\varrho \right)|+|Hh\left(\varrho \right)-l\left(\mu \right)|{)}^2\hfill \\ & \le & {\displaystyle \frac{4}{{\Gamma }^2\left(\varsigma \right)}}\left({\int }_a^{\varrho }{|(\varrho -\mu )}^{\varsigma -1}|d\mu {)}^2\right(|h\left(\mu \right)-Hh\left(\varrho \right)|\hfill \\ & & +|Hh\left(\varrho \right)-Hl\left(\varrho \right)|+|Hl\left(\varrho \right)-l\left(\mu \right)|{)}^2\hfill \\ & \le & {\displaystyle \frac{4}{{\Gamma }^2\left(\varsigma \right)}}\left({\int }_a^{\varrho }{|(\varrho -\mu )}^{\varsigma -1}|d\mu {)}^{2}3\right({|h\left(\mu \right)-Hh\left(\varrho \right)|}^2\hfill \\ & & {+|Hh\left(\varrho \right)-Hl\left(\varrho \right)|}^2+{|Hh\left(\varrho \right)-l\left(\mu \right)|}^2)\hfill \\ & \le & {\displaystyle \frac{4}{{\Gamma }^2\left(\varsigma \right)}}\left({\int }_a^{\varrho }{|(\varrho -\mu )}^{\varsigma -1}|d\mu {)}^2\right(3{|h\left(\varrho \right)-Hh\left(\varrho \right)|}^2\hfill \\ & & +3{\left(\left|Hh\right(\varrho )-l(\mu \left)\right|+\left|l\right(\mu )-Hl(\varrho \left)\right|\right)}^2\hfill \\ & & +3{\left(\left|Hl\right(\varrho )-h(\mu \left)\right|+\left|h\right(\varrho )-l(\mu \left)\right|\right)}^2)\hfill \\ & \le & {\displaystyle \frac{4}{{\Gamma }^2\left(\varsigma \right)}}\left({\int }_a^{\varrho }{|(\varrho -\mu )}^{\varsigma -1}|d\mu {)}^2\right(3{|h\left(\mu \right)-Hh\left(\varrho \right)|}^2\hfill \end{array}$$

$$\begin{array}{ccc}& & {+6|Hh\left(\varrho \right)-l\left(\mu \right)|}^2+6{|l\left(\mu \right)-Hl\left(\varrho \right)|}^2\hfill \\ & & {+6|Hl\left(\varrho \right)-h\left(\mu \right)|}^2+6{|h\left(\mu \right)-l\left(\mu \right)|}^2)\hfill \\ & =& {\displaystyle \frac{1}{{\Gamma }^2\left(\varsigma \right)}}{\displaystyle \frac{{(\varrho -\mu )}^{\varsigma -1}}{{|(\varrho -\mu )}^{\varsigma -1}|}}({\int }_a^{\varrho }{|(\varrho -\mu )}^{\varsigma -1}|d\mu {)}^2.4(3{|h\left(\mu \right)-Hh\left(\varrho \right)|}^2\hfill \\ & & {+6|Hh\left(\varrho \right)-l\left(\mu \right)|}^2{+6|l\left(\mu \right)-Hl\left(\varrho \right)|}^2+6{|Hl\left(\varrho \right)-h\left(\mu \right)|}^2\hfill \\ & & {+6|h\left(\mu \right)-l\left(\mu \right)|}^2)\hfill \\ & =& {\displaystyle \frac{1}{{\Gamma }^2\left(\varsigma \right)}}{\displaystyle \frac{{(\varrho -\mu )}^{\varsigma -1}}{{|(\varrho -\mu )}^{\varsigma -1}|}}({\displaystyle \frac{{(\varrho -\mu )}^{\varsigma }}{\varsigma }}{|}_a^{\varrho }{)}^2.4(3{|h\left(\mu \right)-Hh\left(\varrho \right)|}^2\hfill \\ & & {+6|Hh\left(\varrho \right)-l\left(\mu \right)|}^2+6{|l\left(\mu \right)-Hl\left(\varrho \right)|}^2\hfill \\ & & {+6|Hl\left(\varrho \right)-h\left(\mu \right)|}^2+6{|h\left(\mu \right)-l\left(\mu \right)|}^2)\hfill \\ & =& {\displaystyle \frac{1}{\Gamma (\varsigma +1)}}{\displaystyle \frac{{(\varrho -\mu )}^{\varsigma -1}{(\varrho -a)}^{2\varsigma }}{{|(\varrho -\mu )}^{\varsigma -1}|}}.4(3{|h\left(\varrho \right)-Hh\left(\varrho \right)|}^2\hfill \\ & & {+6|Hh\left(\varrho \right)-l\left(\mu \right)|}^2+6{|l\left(\mu \right)-Hl\left(\varrho \right)|}^2\hfill \\ & & {+6|Hl\left(\varrho \right)-h\left(\mu \right)|}^2+6{|h\left(\mu \right)-l\left(\mu \right)|}^2)\hfill \\ & =& {6k.4|h\left(\mu \right)-l\left(\mu \right)|}^2+3k.4{|h\left(\mu \right)-Hh\left(\varrho \right)|}^2\hfill \\ & & {+6k.4|l\left(\mu \right)-Hl\left(\varrho \right)|}^2+6k.4{|Hl\left(\varrho \right)-h\left(\mu \right)|}^2\hfill \\ & & {+6k.4|Hh\left(\varrho \right)-l\left(\mu \right)|}^2\hfill \\ & =& 6k{R}_{G_b}(h\left(\mu \right),h\left(\mu \right),l\left(\mu \right)+3k{R}_{G_b}(h\left(\mu \right),Hh\left(\varrho \right),Hh\left(\varrho \right)\hfill \\ & & +6k{R}_{G_b}(l\left(\mu \right),Hl\left(\varrho \right),Hl\left(\varrho \right))+6k{R}_{G_b}(h\left(\mu \right),Hl\left(\varrho \right),Hl\left(\varrho \right))\hfill \\ & & +6k{R}_{G_b}(l\left(\mu \right),Hh\left(\varrho \right),Hh\left(\varrho \right)).\hfill \end{array}$$

If we take $6k=a_1=a_4=a_6=a_7$ and $a_2+a_3+a_5=3k$, where ${\sum }_{i=1}^7{a}_i<1$, we get

$$\begin{array}{ccc}\hfill R_{G_b}(Hh\left(\varrho \right),Hh\left(\varrho \right),Hl\left(\varrho \right))& \le & a_1{R}_{G_b}(h\left(\mu \right),h\left(\mu \right),l\left(\mu \right)+(a_2+a_3+a_5)R_{G_b}(h\left(\mu \right),Hh\left(\varrho \right),Hh\left(\varrho \right))\hfill \\ & & +a_4{R}_{G_b}(l\left(\mu \right),Hl\left(\varrho \right),Hl\left(\varrho \right))+a_6{R}_{G_b}(h\left(\mu \right),Hl\left(\varrho \right),Hl\left(\varrho \right))\hfill \\ & & +a_7{R}_{G_b}(l\left(\mu \right),Hh\left(\varrho \right),Hh\left(\varrho \right)).\hfill \end{array}$$

Hence, it is easy to see that Theorem 2 is verified and so H has a unique fixed point which means that the Riemann–Liouville fractional integral Equation (9) has just one solution.

4.2. Application to Atangana–Baleanu Fractional Integrals

As an application to Corollary 2, we demonstrate that the Atangana and Baleanu fractional integral ([19,20,21,22]) has just one solution. The general form of the Atangana and Baleanu fractional integral is

$${}_a{}^{AB}I_{\varrho }^{\varsigma }h\left(\varrho \right)={\displaystyle \frac{1-\varsigma }{\beta \left(\varsigma \right)}}h\left(\varrho \right)+{\displaystyle \frac{\varsigma }{\beta \left(\varsigma \right)\Gamma \left(\varsigma \right)}}{\int }_a^{\varrho }h\left(\mu \right){(\varrho -\mu )}^{\varsigma -1}d\mu ,$$

(10)

where $\varsigma \in \mathbb{R}$, $\Omega =${ h : h is a continuous function, $h:[0,1]\to \mathbb{R}$)} and $\varrho ,\mu \in [0,1]$. Using Example 1, we define $R_{G_b}:{\Omega }^3\to [0,\infty )$ by

$$R_{G_b}(h,l,d)={\left(\right|h-l|+|l-d|+|d-l\left|\right)}^2$$

with $s=3$ for all $h\left(\mu \right),l\left(\mu \right),d\left(\mu \right)\in \Omega$ and $\mu \in [0,1]$, where $R_{G_b}$ is a rectangular $G_b$-metric.

If we accept the condition below, we can show that (10) has a unique solution.

$$\gamma =3\left({\displaystyle \frac{1-\varsigma }{\beta \left(\varsigma \right)}}+{\displaystyle \frac{a^{\varsigma }}{\beta \left(\varsigma \right)\Gamma \left(\varsigma \right)}}\right)\in [0,{\displaystyle \frac{1}{7}}).$$

(11)

Define an operator $H:\Omega \to \Omega$ by

$$Hh\left(\varrho \right)={\displaystyle \frac{1-\varsigma }{\beta \left(\varsigma \right)}}h\left(\varrho \right)+{\displaystyle \frac{\varsigma }{\beta \left(\varsigma \right)\Gamma \left(\varsigma \right)}}{\int }_a^{\varrho }h\left(\mu \right){(\varrho -\mu )}^{\varsigma -1}d\mu .$$

(12)

Then, we have the following statements:

$$\begin{array}{ccc}\hfill R_{G_b}(Hh\left(\varrho \right),Hh\left(\varrho \right),Hl\left(\varrho \right))& =& {\left(\left|Hh\right(\varrho )-Hh(\varrho \left)\right|+\left|Hh\right(\varrho )-Hl(\varrho \left)\right|+\left|Hl\right(\varrho )-Hh(\varrho \left)\right|\right)}^2\hfill \\ & =& 4{\left(\left|Hh\right(\varrho )-Hl(\varrho \left)\right|\right)}^2\hfill \\ & =& 4\left|\right({\displaystyle \frac{1-\varsigma }{\beta \left(\varsigma \right)}}h\left(\varrho \right)+{\displaystyle \frac{\varrho }{\beta \left(\varsigma \right)\Gamma \left(\varsigma \right)}}{\int }_a^{\varrho }h\left(\mu \right){(\varrho -\mu )}^{\varsigma -1}d\mu )\hfill \\ & & -({\displaystyle \frac{1-\varsigma }{\beta \left(\varsigma \right)}}l\left(\varrho \right)+{\displaystyle \frac{\varsigma }{\beta \left(\varsigma \right)\Gamma \left(\varsigma \right)}}{\int }_a^{\varrho }l\left(\mu \right){(\varrho -\mu )}^{\varsigma -1}d\mu ){|}^2\hfill \\ & =& 4\left|\right({\displaystyle \frac{1-\varsigma }{\beta \left(\varsigma \right)}}[h\left(\varrho \right)-l\left(\varrho \right)]+{\displaystyle \frac{\varsigma }{\beta \left(\varsigma \right)\Gamma \left(\varsigma \right)}}{\int }_a^{\varrho }{(\varrho -\mu )}^{\varsigma -1}d\mu [h\left(\mu \right)\hfill \\ & & -l\left(\mu \right)\left]\right){|}^2\hfill \\ & \le & 4\left|\right({\displaystyle \frac{1-\varsigma }{\beta \left(\varsigma \right)}}|h\left(\varrho \right)-l\left(\varrho \right)|+{\displaystyle \frac{\varsigma }{\beta \left(\varsigma \right)\Gamma \left(\varsigma \right)}}{\int }_a^{\varrho }{(\varrho -\mu )}^{\varsigma -1}d\mu |h\left(\mu \right)\hfill \\ & & -l\left(\mu \right)\left|\right){|}^2\hfill \\ & =& 4\left|\right({\displaystyle \frac{1-\varsigma }{\beta \left(\varsigma \right)}}|h\left(\varrho \right)-l\left(\varrho \right)|-{\displaystyle \frac{\varsigma }{\beta \left(\varsigma \right)\Gamma \left(\varsigma \right)}}[{\displaystyle \frac{{(\varrho -\mu )}^{\varsigma }}{\varsigma }}{|}_a^{\varrho }\left]\right|h\left(\mu \right)\hfill \\ & & -l\left(\mu \right)\left|\right){|}^2\hfill \\ & =& 4\left|\right({\displaystyle \frac{1-\varsigma }{\beta \left(\varsigma \right)}}|h\left(\varrho \right)-l\left(\varrho \right)|-{\displaystyle \frac{\varsigma }{\beta \left(\varsigma \right)\Gamma \left(\varsigma \right)}}.{\displaystyle \frac{{(\varrho -a)}^{\varsigma }}{\varsigma }}|h\left(\mu \right)-l\left(\mu \right)|){|}^2\hfill \\ & \le & 4|\left({\displaystyle \frac{1-\varsigma }{\beta \left(\varsigma \right)}}+{\displaystyle \frac{\varsigma }{\beta \left(\varsigma \right)\Gamma \left(\varsigma \right)}}.{\displaystyle \frac{{(\varrho -a)}^{\varsigma }}{\varsigma }}|h\left(\varrho \right)-l\left(\varrho \right)|\right){|}^2\hfill \\ & \le & 4({\displaystyle \frac{1-\varsigma }{\beta \left(\varsigma \right)}}+{\displaystyle \frac{\varsigma }{\beta \left(\varsigma \right)\Gamma \left(\varsigma \right)}}.{\displaystyle \frac{{(\varrho -a)}^{\varsigma }}{\varsigma }}[|h\left(\varrho \right)-Hh\left(\varrho \right)|\hfill \\ & & +|Hh\left(\varrho \right)-l\left(\varrho \right)|+|h\left(\varrho \right)-Hl\left(\varrho \right)|]{)}^2\hfill \\ & \le & 4({\displaystyle \frac{1-\varsigma }{\beta \left(\varsigma \right)}}+{\displaystyle \frac{\varsigma }{\beta \left(\varsigma \right)\Gamma \left(\varsigma \right)}}.{\displaystyle \frac{{(\varrho -a)}^{\varsigma }}{\varsigma }}{)}^2.3[{|h\left(\varrho \right)-Hh\left(\varrho \right)|}^2\hfill \end{array}$$

$$\begin{array}{ccc}& & {+|Hh\left(\varrho \right)-l\left(\varrho \right)|}^2+{|h\left(\varrho \right)-Hl\left(\varrho \right)|}^2]\hfill \\ & \le & 3({\displaystyle \frac{1-\varsigma }{\beta \left(\varsigma \right)}}+{\displaystyle \frac{a^{\varsigma }}{\beta \left(\varsigma \right)\Gamma \left(\varsigma \right)}}{)}^2[R_{G_b}(h\left(\varrho \right),Hh\left(\varrho \right),Hh\left(\varrho \right))\hfill \\ & & +R_{G_b}(h\left(\varrho \right),Hl\left(\varrho \right),Hl\left(\varrho \right))+R_{G_b}(l\left(\varrho \right),Hh\left(\varrho \right),Hh\left(\varrho \right))]\hfill \\ & =& \gamma [R_{G_b}(h\left(\varrho \right),Hh\left(\varrho \right),Hh\left(\varrho \right))+R_{G_b}(h\left(\varrho \right),Hl\left(\varrho \right),Hl\left(\varrho \right))\hfill \\ & & +R_{G_b}(l\left(\varrho \right),Hh\left(\varrho \right),Hh\left(\varrho \right))].\hfill \end{array}$$

Since Corollary 2 is verified, we conclude that H has a unique fixed point and we can say the Atangana–Baleanu fractional integral Equation (12) has just one solution.