Fixed-Point Theorems in Sequential G-Metric Spaces and Their Application to the Solvability of a System of Integral Equations

Abstract

The goal of this work is to demonstrate some fixed-point theorems for $JS$-contractions over sequential G-metric spaces that are not necessarily continuous in any variable and do not enjoy the triangle inequality. This new space is a generalization of standard G-metric spaces and $G_b$-metric spaces. Additionally, a related application and an illustrated example are included to verify the accuracy of the findings.

1. Introduction

Fixed-point theory results play a crucial role across various branches of mathematics and applied sciences, such as numerical analysis, economics, and algebraic geometry. For instance, in algebraic geometry, fixed-point concepts are applied to the study of automorphisms of vector bundle moduli spaces over compact Riemann surfaces and the study of algebraic curves [1,2]. Also, in order to demonstrate the existence and uniqueness of solutions to a wide range of mathematical problems, such as integral and differential equations and their fractional forms, fixed-point theorems are crucial tools.

If we take the metric as the absolute value metric in $\mathbb{R}$, the contraction mapping theorem states that if for both arbitrary points x and y in the domain of the function f, and the length of the line segment connecting the points $fx$ and $fy$ is less than or equal to a multiple $(\in [0,1\left)\right)$ of the distance between the points x and y, then this function has a unique fixed point. The length of a line segment is one of the geometric concepts that can be changed by the perimeter, area, and volume. The contraction mapping theorem in $\mathbb{R}$ with a G-metric (which is a generalization of the ordinary metric) expresses the same geometric fact in which, for example, instead of the length of the line segment, the perimeter of the triangle made by the points $fx$, $fy$, and $fz$ is considered.

Mustafa and Sims introduced the idea of a generalized metric space, known as a G-metric space [3]. Several fixed-point theorems for mappings that meet distinct contractive requirements on G-metric spaces have been obtained by numerous researchers in recent years. We refer the reader to [3,4,5,6,7,8,9,10,11,12], and the references cited therein for an overview of fixed-point theory, its applications, various contractive conditions, and related subjects in G-metric spaces.

Definition 1

(G-Metric Space [3]). Consider a nonempty set $\mathfrak{L}$ together with a mapping $G:{\mathfrak{L}}^3\to {\mathbb{R}}^{+}$ that fulfills the following properties:

(G1)

$G(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″})=0\iff \mathfrak{b}={\mathfrak{b}}^{\prime }={\mathfrak{b}}^{″}$;

(G2)

For every $\mathfrak{b},{\mathfrak{b}}^{\prime }\in \mathfrak{L}$ with $\mathfrak{b}\ne {\mathfrak{b}}^{\prime }$, one has $0<G(\mathfrak{b},\mathfrak{b},{\mathfrak{b}}^{\prime })$;

(G3)

For all $\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″}\in \mathfrak{L}$ with ${\mathfrak{b}}^{\prime }\ne {\mathfrak{b}}^{″}$,

$$G(\mathfrak{b},\mathfrak{b},{\mathfrak{b}}^{\prime })\le G(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″});$$

(G4)

The function G is symmetric in all three arguments, i.e.,

$$G(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″})=G(\mathfrak{b},{\mathfrak{b}}^{″},{\mathfrak{b}}^{\prime })=G({\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″},\mathfrak{b})=\cdots ;$$

(G5)

For any $\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″},a\in \mathfrak{L}$,

$$G(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″})\le G(\mathfrak{b},a,a)+G(a,{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″}),$$

which is referred to as the rectangle inequality.

The structure $(\mathfrak{L},G)$ is called a G-metric space, and G is said to be a G-metric on $\mathfrak{L}$.

In recent years, many authors have generalized metric spaces and articulated them in different ways. In fact, authors working in fixed-point theory have changed the principles of defining a metric to arrive at generalized metric spaces. Some authors remove the symmetry condition and study quasi-metric spaces, others remove the condition that the distance of a point to itself is zero, and others modify the triangular inequality. Some authors also extend the domain of the generalized metric mapping from the double Cartesian product to the triple Cartesian product. Some also consider two or more changes in the principles of defining ordinary metrics at the same time. In sequential metric spaces, the triangle inequality is removed and replaced by a sequential inequality.

Based on the foregoing discussion, several variants of metric-type structures have been studied in the literature. Examples include extended b-metric spaces [13], $JS$-metric spaces [14], strong $JS$-metric spaces [15], rectangular metric spaces [16], and b-metric-like spaces [17], among others. By modifying the settings of D-metric and G-metric spaces, Sedghi et al. [18] proposed the concept of S-metric spaces. Later, in 2016, the class of $S_b$-metric spaces was introduced as a further extension of S-metric spaces [19]. More recently, Beg et al. [20] put forward the idea of $S^{JS}$-metric spaces. However, this latter structure does not necessarily satisfy both the symmetry condition and the rectangle inequality.

Sequential S-metric spaces and their strong form can be found in [21,22]. Generalized metric spaces allow for the study of problems in more flexible and inclusive settings. These structures have been extensively studied in the literature due to their applicability in areas where standard metric principles are too restrictive, including functional analysis, topology, and applied fields such as optimization and control theory. The presentation of new fixed-point results in generalized metric spaces enriches the theoretical framework and broadens the range of applications in both pure and applied fields.

The class of $JS$-metric spaces, introduced in 2015, represents a broad extension of abstract metric spaces. This framework encompasses several well-known structures, as every metric space, b-metric space, dislocated metric space, and modular metric space with the Fatou property can be viewed as particular instances (see [14]).

In the present work, we put forward the notion of sequential G-metric (s.g.m.) spaces as a new generalization within the metric-space setting. Furthermore, we derive several fixed-point results for self-mappings that are $JS$-contractive in the framework of s.g.m. spaces.

2. Preliminaries

First, we should remember some definitions.

Definition 2

([23]). Let $s\ge 1$ and let Λ denote a nonempty set. A mapping $\mho :{\Lambda }^2\to [0,+\infty )$ is referred to as a b-metric on Λ provided that the following hold:

1.

$\mho (\mathfrak{b},{\mathfrak{b}}^{\prime })=0$ if and only if $\mathfrak{b}={\mathfrak{b}}^{\prime }$ for all $b,b^{\prime } in \Lambda$;

2.

$\mho (\mathfrak{b},{\mathfrak{b}}^{\prime })=\mho ({\mathfrak{b}}^{\prime },\mathfrak{b})$ for all $b,b^{\prime } in \Lambda$ (symmetry);

3.

For all $\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″}\in \Lambda$,

$$\mho (\mathfrak{b},{\mathfrak{b}}^{\prime })\le s\left(\mho (\mathfrak{b},{\mathfrak{b}}^{″})+\mho ({\mathfrak{b}}^{″},{\mathfrak{b}}^{\prime })\right).$$

In this case, the pair $(\Lambda ,\mho )$ is termed a b-metric space.

Let $\Lambda$ be a nonempty set, and consider a mapping $\rho :{\Lambda }^2\to [0,\infty ]$. For each $\varsigma \in \Lambda$, define

$$C_{\varsigma }(\rho ,\Lambda )=\left\{\left\{{\varsigma }_n\right\}\subset \Lambda :\underset{n\to \infty }{lim}\rho ({\varsigma }_n,\varsigma )=0\right\}.$$

(1)

The notion of a generalized metric space was introduced by Jleli and Samet as a natural broadening of the classical metric space framework.

Definition 3

([14]). Let $\rho :{\Lambda }^2\to [0,\infty ]$ be a mapping satisfying the following conditions:

1.

For all $b,b^{\prime } in \Lambda$, $\rho (\mathfrak{b},{\mathfrak{b}}^{\prime })=0$ implies $\mathfrak{b}={\mathfrak{b}}^{\prime }$;

2.

For every $\mathfrak{b},{\mathfrak{b}}^{\prime }\in \Lambda$, one has $\rho (\mathfrak{b},{\mathfrak{b}}^{\prime })=\rho ({\mathfrak{b}}^{\prime },\mathfrak{b})$;

3.

there exists a constant $\omega >0$ such that, whenever $(\mathfrak{b},{\mathfrak{b}}^{\prime })\in {\Lambda }^2$ and $\left\{{\mathfrak{b}}_n\right\}\in C_{\mathfrak{b}}(\rho ,\Lambda )$, it holds that

$$\rho (\mathfrak{b},{\mathfrak{b}}^{\prime })\le \omega \underset{n\to \infty }{lim\; sup}\rho ({\mathfrak{b}}_n,{\mathfrak{b}}^{\prime }).$$

In this setting, the pair $(\Lambda ,\rho )$ is called a $JS$-metric space.

Remark 1

([20]). It is worth mentioning that rectangular metric spaces are not necessarily $JS$-metric spaces; see Example 2.2 in [20] for clarification. On the other hand, every metric space, b-metric space, and Hitzler–Seda (dislocated) metric space can be regarded as particular cases of $JS$-metric spaces.

Within the setting of Branciari (rectangular) metric spaces, Jleli and Samet proposed a broader version of the Banach fixed-point theorem.

Definition 4

([24]). Let Φ denote the family of all functions $\nu :(0,\infty )\to (1,\infty )$ that satisfy the following conditions:

($\nu 1$)

ν is monotone and increasing;

($\nu 2$)

For every sequence $\left\{{\mathfrak{b}}_n\right\}\subset (0,\infty )$, one has

$$\underset{n\to \infty }{lim}\nu \left({\mathfrak{b}}_n\right)=1\iff \underset{n\to \infty }{lim}{\mathfrak{b}}_n=0;$$

($\nu 3$)

There exist $\chi \in (0,1)$ and $\tau \in (0,\infty ]$ such that

$$\underset{u\to 0^{+}}{lim}{\displaystyle \frac{\nu \left(u\right)-1}{u^{\chi }}}=\tau .$$

As an extension of the Banach fixed-point theorem, Jleli and Samet [24] established the following result:

Theorem 1

([24]). Let $(\Lambda ,d)$ be a metric space and let $\Re :\Lambda \to \Lambda$ be a mapping.

Suppose there exist $\nu \in \Phi$ and $\mu \in (0,1)$ such that, for all $\mathfrak{b},{\mathfrak{b}}^{\prime }\in \Lambda$,

$$\begin{array}{c}\hfill d(\Re \mathfrak{b},\Re {\mathfrak{b}}^{\prime })\ne 0\Rightarrow \nu \left(d(\Re \mathfrak{b},\Re {\mathfrak{b}}^{\prime })\right)\le {\left(\nu \left(d(\mathfrak{b},{\mathfrak{b}}^{\prime })\right)\right)}^{\mu }.\end{array}$$

(2)

Then, the mapping ℜ has a unique fixed point in Λ.

Definition 5

(Generalized Contraction Mappings). Let $(\Lambda ,d)$ be a metric space and let $\Re :\Lambda \to \Lambda$ be a self-mapping. Then ℜ is said to be

1.

A Kannan contraction [25,26] if there exists $\lambda \in [0,{\displaystyle \frac{1}{2}})$ such that

$$d(\Re \mathfrak{b},\Re {\mathfrak{b}}^{\prime })\le \lambda \left[d(\mathfrak{b},\Re \mathfrak{b})+d({\mathfrak{b}}^{\prime },\Re {\mathfrak{b}}^{\prime })\right],\forall \mathfrak{b},{\mathfrak{b}}^{\prime }\in \Lambda .$$

2.

A Chatterjea contraction [27] if there exists $\lambda \in [0,{\displaystyle \frac{1}{2}})$ such that

$$d(\Re \mathfrak{b},\Re {\mathfrak{b}}^{\prime })\le \lambda \left[d(\mathfrak{b},\Re {\mathfrak{b}}^{\prime })+d({\mathfrak{b}}^{\prime },\Re \mathfrak{b})\right],\forall \mathfrak{b},{\mathfrak{b}}^{\prime }\in \Lambda .$$

3.

A Reich contraction [28] if there exist constants $\alpha ,\beta ,\gamma \ge 0$ with $\alpha +\beta +\gamma <1$ such that

$$d(\Re \mathfrak{b},\Re {\mathfrak{b}}^{\prime })\le \alpha d(\mathfrak{b},{\mathfrak{b}}^{\prime })+\beta d(\mathfrak{b},\Re \mathfrak{b})+\gamma d({\mathfrak{b}}^{\prime },\Re {\mathfrak{b}}^{\prime }),\forall \mathfrak{b},{\mathfrak{b}}^{\prime }\in \Lambda .$$

4.

A Ćirić contraction [29] if there exist nonnegative constants $\alpha ,\beta ,\gamma ,\delta$ with $\alpha +\beta +\gamma +2\delta <1$ such that

$$d(\Re \mathfrak{b},\Re {\mathfrak{b}}^{\prime })\le \alpha d(\mathfrak{b},{\mathfrak{b}}^{\prime })+\beta d(\mathfrak{b},\Re \mathfrak{b})+\gamma d({\mathfrak{b}}^{\prime },\Re {\mathfrak{b}}^{\prime })+\delta [d(\mathfrak{b},\Re {\mathfrak{b}}^{\prime })+d({\mathfrak{b}}^{\prime },\Re \mathfrak{b})],\forall \mathfrak{b},{\mathfrak{b}}^{\prime }\in \Lambda .$$

5.

A Hardy–Rogers contraction [30] if there exist non-negative constants $\alpha ,\beta ,\gamma ,\delta ,\eta$ with $\alpha +\beta +\gamma +\delta +\eta <1$ such that

$$d(\Re \mathfrak{b},\Re {\mathfrak{b}}^{\prime })\le \alpha d(\mathfrak{b},{\mathfrak{b}}^{\prime })+\beta d(\mathfrak{b},\Re \mathfrak{b})+\gamma d({\mathfrak{b}}^{\prime },\Re {\mathfrak{b}}^{\prime })+\delta d(\mathfrak{b},\Re {\mathfrak{b}}^{\prime })+\eta d({\mathfrak{b}}^{\prime },\Re \mathfrak{b}),\forall \mathfrak{b},{\mathfrak{b}}^{\prime }\in \Lambda .$$

Now, we will go through our new structure and we will try to present some fixed-point results.

3. Sequential $\mathbf{G}$-Metric Spaces

Let $\Lambda$ be a non-empty set and let

$$\underline{\mathcal{G}}:{\Lambda }^3\to [0,\infty ]$$

be a mapping. For any $\varsigma \in \Lambda$, define the set

$$C_{\varsigma }(\underline{\mathcal{G}},\Lambda )=\left\{\left\{{\varsigma }_n\right\}\subset \Lambda :\underset{n\to \infty }{lim}\underline{\mathcal{G}}(\varsigma ,{\varsigma }_n,{\varsigma }_n)=0\right\}.$$

(3)

Definition 6

(Sequential G-Metric Space). Let Λ be a non-empty set and let

$$\underline{\mathcal{G}}:{\Lambda }^3\to {\mathbb{R}}^{+}$$

be a mapping satisfying the following conditions:

(G1)

$\mathfrak{b}={\mathfrak{b}}^{\prime }={\mathfrak{b}}^{″}$ if $\underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″})=0$ for all $b,b^{\prime },b^{″} in \Lambda$;

(G2)

$\underline{\mathcal{G}}(\mathfrak{b},\mathfrak{b},{\mathfrak{b}}^{\prime })>0$ for all $\mathfrak{b},{\mathfrak{b}}^{\prime }\in \Lambda$ with $\mathfrak{b}\ne {\mathfrak{b}}^{\prime }$;

(G3)

$\underline{\mathcal{G}}(\mathfrak{b},\mathfrak{b},{\mathfrak{b}}^{\prime })\le \underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″})$ for all $\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″}\in \Lambda$ with ${\mathfrak{b}}^{\prime }\ne {\mathfrak{b}}^{″}$;

(G4)

$\underline{\mathcal{G}}$ is symmetric in all three variables, i.e.,

$$\underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″})=\underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}^{″},{\mathfrak{b}}^{\prime })=\underline{\mathcal{G}}({\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″},\mathfrak{b})=\cdots ;$$

(G5)

There exists $\omega >0$ such that, for every $\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″}\in \Lambda$ and every sequence $\left\{{\mathfrak{b}}_n\right\}\in C_{\mathfrak{b}}(\underline{\mathcal{G}},\Lambda )$,

$$\underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″})\le \omega \underset{n\to \infty }{lim\; inf}\underline{\mathcal{G}}({\mathfrak{b}}_n,{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″}).$$

The pair $(\Lambda ,\underline{\mathcal{G}})$ is called a sequential $\underline{\mathcal{G}}$-metric space (s.g.m. space).

Furthermore, the s.g.m. space $(\Lambda ,\underline{\mathcal{G}})$ is said to be symmetric if

$$\underline{\mathcal{G}}(\mathfrak{b},\mathfrak{b},{\mathfrak{b}}^{\prime })=\underline{\mathcal{G}}({\mathfrak{b}}^{\prime },{\mathfrak{b}}^{\prime },\mathfrak{b}),\forall \mathfrak{b},{\mathfrak{b}}^{\prime }\in \Lambda .$$

It is noteworthy that there exist many subclasses of s.g.m. spaces. In particular, every G-metric space, every $G_b$-metric space, and every $G_b$-metric-like space can be regarded as an s.g.m. space.

Definition 7.

Let $(\Lambda ,\underline{\mathcal{G}})$ be an s.g.m. space, and let $\left\{{\mathfrak{b}}_n\right\}$ be a sequence in Λ with $\mathfrak{b}\in \Lambda$. Then,

(i)

If $\left\{{\mathfrak{b}}_n\right\}\in C_{\mathfrak{b}}(\underline{\mathcal{G}},\Lambda )$, the sequence $\left\{{\mathfrak{b}}_n\right\}$ converges to $\mathfrak{b}$.

(ii)

If

$$\underset{n,m\to \infty }{lim}\underline{\mathcal{G}}({\mathfrak{b}}_n,{\mathfrak{b}}_m,{\mathfrak{b}}_m)=0,$$

then $\left\{{\mathfrak{b}}_n\right\}$ is a Cauchy sequence in $(\Lambda ,\underline{\mathcal{G}})$.

(iii)

The space $(\Lambda ,\underline{\mathcal{G}})$ is called complete if every Cauchy sequence in it converges to a point in Λ.

Example 1.

Let $\Lambda =\mathbb{R}$ and $\underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″})=max\left\{\right|\mathfrak{b}|,|{\mathfrak{b}}^{\prime }|,|{\mathfrak{b}}^{″}\left|\right\}$, $\forall \mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″}\in \Lambda$.

Evidently, $\underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″})=0$ implies that $\mathfrak{b}={\mathfrak{b}}^{\prime }={\mathfrak{b}}^{″}=0$. Subsequently, Definition 6’s first requirement is met.

Let $\mathfrak{b}\in \Lambda$, and $\left\{{\mathfrak{b}}_n\right\}$ is a convergent sequence in Λ such that

$$\underset{n\to \infty }{lim}\underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}_n,{\mathfrak{b}}_n)=\underset{n\to \infty }{lim}max\left\{\right|\mathfrak{b}|,|{\mathfrak{b}}_n|,|{\mathfrak{b}}_n\left|\right\}=max\left\{\right|\mathfrak{b}|,\underset{n\to \infty }{lim}|{\mathfrak{b}}_n|,\underset{n\to \infty }{lim}|{\mathfrak{b}}_n\left|\right\}=0.$$

So,

$$\underset{n\to \infty }{lim}{\mathfrak{b}}_n=\mathfrak{b}=0.$$

Then we have

$$\begin{array}{cc}\hfill \underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″})& =max\left\{\right|\mathfrak{b}|,|{\mathfrak{b}}^{\prime }|,|{\mathfrak{b}}^{″}\left|\right\}=max\{\underset{n\to \infty }{lim}|{\mathfrak{b}}_n|,|{\mathfrak{b}}^{\prime }|,|{\mathfrak{b}}^{″}\left|\right\}\hfill \\ & =\underset{n\to \infty }{lim}max\left\{\right|{\mathfrak{b}}_n|,|{\mathfrak{b}}^{\prime }|,|{\mathfrak{b}}^{″}\left|\right\}.\hfill \end{array}$$

Then

$$\begin{array}{c}\hfill \underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″})\le \underset{n\to \infty }{lim\; inf}\underline{\mathcal{G}}({\mathfrak{b}}_n,{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″}).\end{array}$$

After that, the presumptions of Definition 6 are all met. As a result, $\underline{\mathcal{G}}$ is an s.g.m. space.

Note that $\underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″})=\left|\mathfrak{b}\right|+|{\mathfrak{b}}^{\prime }|+|{\mathfrak{b}}^{″}|$, $\forall \mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″}\in \Lambda$ is not an s.g.m..

Example 2.

Let $\Lambda =\mathbb{R}$ and $\underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″})=e^{|\mathfrak{b}-{\mathfrak{b}}^{\prime }|+|{\mathfrak{b}}^{\prime }-{\mathfrak{b}}^{″}|+|{\mathfrak{b}}^{″}-\mathfrak{b}|}-1$, $\forall \mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″}\in \Lambda$. Evidently, $\underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″})=0$ implies that $|\mathfrak{b}-{\mathfrak{b}}^{\prime }|+|{\mathfrak{b}}^{\prime }-{\mathfrak{b}}^{″}|+|{\mathfrak{b}}^{″}-\mathfrak{b}|=0$, which gives us $\mathfrak{b}={\mathfrak{b}}^{\prime }={\mathfrak{b}}^{″}$. Subsequently, Definition 6’s first requirement is met.

Let $\mathfrak{b}\in \Lambda$, and $\left\{{\mathfrak{b}}_n\right\}$ is a convergent sequence in Λ such that

$$\underset{n\to \infty }{lim}\underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}_n,{\mathfrak{b}}_n)=0.$$

So,

$$\underset{n\to \infty }{lim}{\mathfrak{b}}_n=\mathfrak{b}.$$

Then we have

$$\begin{array}{cc}\hfill \underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″})& =e^{|\mathfrak{b}-{\mathfrak{b}}^{\prime }|+|{\mathfrak{b}}^{\prime }-{\mathfrak{b}}^{″}|+|{\mathfrak{b}}^{″}-\mathfrak{b}|}-1\le e^{|\mathfrak{b}-{\mathfrak{b}}_n+{\mathfrak{b}}_n-{\mathfrak{b}}^{\prime }|+|{\mathfrak{b}}^{\prime }-{\mathfrak{b}}^{″}|+|{\mathfrak{b}}^{″}-{\mathfrak{b}}_n+{\mathfrak{b}}_n-\mathfrak{b}|}-1\hfill \\ & \le e^{|\mathfrak{b}-{\mathfrak{b}}_n|+|{\mathfrak{b}}_n-{\mathfrak{b}}^{\prime }|+|{\mathfrak{b}}^{\prime }-{\mathfrak{b}}^{″}|+|{\mathfrak{b}}^{″}-{\mathfrak{b}}_n|+|{\mathfrak{b}}_n-\mathfrak{b}|}-1.\hfill \end{array}$$

Then

$$\begin{array}{c}\hfill \underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″})\le \underset{n\to \infty }{lim\; inf}\underline{\mathcal{G}}({\mathfrak{b}}_n,{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″}).\end{array}$$

After that, the presumptions of Definition 6 are all met. As a result, $\underline{\mathcal{G}}$ is a symmetric s.g.m. space.

Taking $\mathfrak{b}=5$, ${\mathfrak{b}}^{\prime }=0$ and ${\mathfrak{b}}^{″}=0$ and $a=1$, we see that

$$G(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″})-G(\mathfrak{b},a,a)-G(a,{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″})=e^{10}-1-(e^8-1)-(e^2-1)=19039.1187517$$

which shows that this space is not a usual G-metric space.

The following is an example of a non-symmetric s.g.m. space which is not a standard G-metric space.

Example 3.

Let $\Lambda =\{1,2,3\}$. We define a function $\underline{\mathcal{G}}:{\Lambda }^3\to [0,\infty )$ by the values given in

Table 1. The values of s.g.m. $\underline{\mathcal{G}}$

$\mathfrak{b}$	${\mathfrak{b}}^{\prime }$	${\mathfrak{b}}^{″}$	$\underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″})$
1	1	1	0
2	2	2	0
3	3	3	0
1	1	2	1
1	1	3	1.5
1	2	2	1
1	3	3	0.5
2	2	3	1.5
2	3	3	0.8
1	2	3	3

below.

The function $\underline{\mathcal{G}}$ satisfies the properties $G1$-$G4$ of Definition 1 However, this space is not symmetric, that is, $\underline{\mathcal{G}}(\mathfrak{b},\mathfrak{b},{\mathfrak{b}}^{\prime })=\underline{\mathcal{G}}({\mathfrak{b}}^{\prime },{\mathfrak{b}}^{\prime },\mathfrak{b})$ does not hold for all $\mathfrak{b},{\mathfrak{b}}^{\prime }\in \Lambda$. For instance, $\underline{\mathcal{G}}(1,1,3)=1.5\ne 0.5=\underline{\mathcal{G}}(3,3,1).$ Moreover, the classical triangle inequality for G-metrics (condition $G5$) does not hold here, so $\underline{\mathcal{G}}$ is not a standard G-metric. Instead, $(\Lambda ,\underline{\mathcal{G}})$ forms an s.g.m. space.

Example 4.

Let $\Lambda =\mathbb{R}$ and $\underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″})={sinh}^{-1}\left(max\left\{\right|\mathfrak{b}-{\mathfrak{b}}^{\prime }|,|{\mathfrak{b}}^{\prime }-{\mathfrak{b}}^{″}|,|{\mathfrak{b}}^{″}-\mathfrak{b}\left|\right\}\right)$, $\forall \mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″}\in \Lambda$.

Obviously, $\underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″})=0$ yields that $max\left\{\right|\mathfrak{b}-{\mathfrak{b}}^{\prime }|,|{\mathfrak{b}}^{\prime }-{\mathfrak{b}}^{″}|,|{\mathfrak{b}}^{″}-\mathfrak{b}\left|\right\}=0$, which gives us $\mathfrak{b}={\mathfrak{b}}^{\prime }={\mathfrak{b}}^{″}$.

Let $\mathfrak{b}\in \Lambda$ and $\left\{{\mathfrak{b}}_n\right\}$ be a convergent sequence in Λ such that

$$\underset{n\to \infty }{lim}\underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}_n,{\mathfrak{b}}_n)=0.$$

So,

$$\underset{n\to \infty }{lim}|{\mathfrak{b}}_n-\mathfrak{b}|=0.$$

Then we have

$$\begin{array}{cc}\hfill \underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″})& ={sinh}^{-1}\left(max\left\{\right|\mathfrak{b}-{\mathfrak{b}}^{\prime }|,|{\mathfrak{b}}^{\prime }-{\mathfrak{b}}^{″}|,|{\mathfrak{b}}^{″}-\mathfrak{b}\left|\right\}\right)\hfill \\ & \le {sinh}^{-1}\left(max\left\{\right|\mathfrak{b}-{\mathfrak{b}}_n+{\mathfrak{b}}_n-{\mathfrak{b}}^{\prime }|,|{\mathfrak{b}}^{\prime }-{\mathfrak{b}}^{″}|,|{\mathfrak{b}}^{″}-{\mathfrak{b}}_n+{\mathfrak{b}}_n-\mathfrak{b}\left|\right\}\right)\hfill \\ & \le {sinh}^{-1}\left(max\left\{\right|\mathfrak{b}-{\mathfrak{b}}_n|+|{\mathfrak{b}}_n-{\mathfrak{b}}^{\prime }|,|{\mathfrak{b}}^{\prime }-{\mathfrak{b}}^{″}|,|{\mathfrak{b}}^{″}-{\mathfrak{b}}_n|+|{\mathfrak{b}}_n-\mathfrak{b}\left|\right\}\right).\hfill \end{array}$$

Then

$$\begin{array}{c}\hfill \underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″})\le \underset{n\to \infty }{lim\; inf}\underline{\mathcal{G}}({\mathfrak{b}}_n,{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″}).\end{array}$$

As a result, $\underline{\mathcal{G}}$ is a symmetric s.g.m. space.

We now present some other examples of s.g.m. spaces.

Proposition 1.

Let $(\Lambda ,G)$ be a G-metric space. Then, G also defines an s.g.m. on Λ.

Proof. 

Assume that $(\Lambda ,G)$ is a G-metric space. We will show that G satisfies condition (G5) in Definition 6.

Take any $\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″}\in \Lambda$ and a sequence $\left\{{\mathfrak{b}}_n\right\}$ converging to $\mathfrak{b}$. By applying the triangle inequality, we obtain

$$\begin{array}{cc}\hfill \underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″})& \le \underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}_n,{\mathfrak{b}}_n)+\underline{\mathcal{G}}({\mathfrak{b}}_n,{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″})\hfill \end{array}$$

for all natural numbers n. Taking the limit inferior as $n\to \infty$, it follows that

$$\underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″})\le \underset{n\to \infty }{lim\; inf}\underline{\mathcal{G}}({\mathfrak{b}}_n,{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″}).$$

Therefore, condition (G5) of Definition 6 holds with $\omega =1$. □

Proposition 2.

Let $(\Lambda ,G)$ be a $G_b$-metric space with the constant $D\ge 1$. Then, the function $G_b$ also defines an s.g.m. on Λ.

Proof. 

Assume that $(\Lambda ,G)$ is a $G_b$-metric space. We need to verify that $G_b$ satisfies condition (G5) in Definition 6.

Take any $\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″}\in \Lambda$ and a sequence $\left\{{\mathfrak{b}}_n\right\}$ such that ${\mathfrak{b}}_n\to \mathfrak{b}$. Using the triangle inequality, we obtain

$$\begin{array}{cc}\hfill \underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″})& \le D\left(\underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}_n,{\mathfrak{b}}_n)+\underline{\mathcal{G}}({\mathfrak{b}}_n,{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″})\right),\hfill \end{array}$$

for all natural numbers n. Taking the limit inferior as $n\to \infty$, it follows that

$$\underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″})\le D\underset{n\to \infty }{lim\; inf}\underline{\mathcal{G}}({\mathfrak{b}}_n,{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″}).$$

Hence, condition (G5) of Definition 6 is satisfied with $\omega =D$. □

Proposition 3.

Let $(\Lambda ,\mathcal{D})$ be a b-metric space with the constant $D\ge 1$. Then, there exists a mapping $\underline{\mathcal{G}}$ on Λ that defines an s.g.m.

Proof. 

Consider the mapping $\underline{\mathcal{G}}:{\Lambda }^3\to \mathbb{R}$ defined by

$$\underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″})=\mathcal{D}(\mathfrak{b},{\mathfrak{b}}^{\prime })+\mathcal{D}({\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″})+\mathcal{D}({\mathfrak{b}}^{″},\mathfrak{b}),$$

for all $\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″}\in \Lambda$. This construction forms a $G_b$-metric on $\Lambda$. Therefore, by Proposition 2, the desired result holds. □

Remark 2.

Observe that if $(\Lambda ,\mathcal{D})$ defines a b-metric-like structure on Λ, then according to Proposition 3, $(\Lambda ,\mathcal{D})$ also leads to the generation of an s.g.m. space.

Proposition 4.

Let $(\Lambda ,\underline{\mathcal{G}})$ be a symmetric s.g.m., and consider a sequence $\left\{{\mathfrak{b}}_n\right\}\subset \Lambda$ such that $\left\{{\mathfrak{b}}_n\right\}\in C_{\mathfrak{b}}(\underline{\mathcal{G}},\Lambda )$. Then, it holds that

$$\underline{\mathcal{G}}(\mathfrak{b},\mathfrak{b},\mathfrak{b})=0.$$

Proof. 

By condition (G5) of Definition 6, we have

$$\underline{\mathcal{G}}(\mathfrak{b},\mathfrak{b},\mathfrak{b})\le \omega \underset{n\to \infty }{lim\; inf}\underline{\mathcal{G}}({\mathfrak{b}}_n,\mathfrak{b},\mathfrak{b}).$$

Since $\left\{{\mathfrak{b}}_n\right\}\in C_{\mathfrak{b}}(\underline{\mathcal{G}},\Lambda )$, the right-hand side tends to zero, which immediately gives

$$\underline{\mathcal{G}}(\mathfrak{b},\mathfrak{b},\mathfrak{b})=0.$$

□

Now, we introduce a topology on s.g.m. spaces.

Let $(\Lambda ,\underline{\mathcal{G}})$ be an s.g.m. space. For any $\mathfrak{b}\in \Lambda$ and $\eta >0$, define the set

$$\mathcal{B}(\mathfrak{b},\eta ):=\left\{{\mathfrak{b}}^{\prime }\in \Lambda :\underline{\mathcal{G}}({\mathfrak{b}}^{\prime },\mathfrak{b},\mathfrak{b})<\underline{\mathcal{G}}(\mathfrak{b},\mathfrak{b},\mathfrak{b})+\eta \right\}.$$

(4)

Remark 3.

The collection

$${\tau }_{\underline{\mathcal{G}}}:=\{\varnothing \}\cup \left\{\mathcal{V}\subset \Lambda ,\mathcal{V}\ne \varnothing :\mathrm{for}\mathrm{any}\mathfrak{b}\in \mathcal{V},\mathrm{there}\mathrm{exists}\eta >0\mathrm{such}\mathrm{that}\mathcal{B}(\mathfrak{b},\eta )\subset \mathcal{V}\right\}$$

(5)

forms a topology on Λ.

Proposition 5.

Let $(\Lambda ,\underline{\mathcal{G}})$ be an s.g.m. space. If a self-mapping ℜ is continuous at $\mathfrak{b}\in \Lambda$, then for any sequence $\left\{{\mathfrak{b}}_n\right\}\in C_{\mathfrak{b}}(\underline{\mathcal{G}},\Lambda )$, we have

$$\{\Re {\mathfrak{b}}_n\}\in C_{\Re \mathfrak{b}}(\underline{\mathcal{G}},\Lambda ).$$

Proof. 

Since ℜ is continuous at $\mathfrak{b}$, for any $ϵ>0$ there exists ${\delta }_{ϵ}>0$ such that

$$\underline{\mathcal{G}}({\mathfrak{b}}^{″},\mathfrak{b},\mathfrak{b})<\underline{\mathcal{G}}(\mathfrak{b},\mathfrak{b},\mathfrak{b})+{\delta }_{ϵ}\Rightarrow \underline{\mathcal{G}}(\Re {\mathfrak{b}}^{″},\Re \mathfrak{b},\Re \mathfrak{b})<\underline{\mathcal{G}}(\Re \mathfrak{b},\Re \mathfrak{b},\Re \mathfrak{b})+ϵ.$$

Since $\left\{{\mathfrak{b}}_n\right\}\in C_{\mathfrak{b}}(\underline{\mathcal{G}},\Lambda )$, there exists $N\in \mathbb{N}$ such that for all $n\ge N$,

$$\underline{\mathcal{G}}({\mathfrak{b}}_n,\mathfrak{b},\mathfrak{b})<\underline{\mathcal{G}}(\mathfrak{b},\mathfrak{b},\mathfrak{b})+{\delta }_{ϵ}.$$

Hence, for all $n\ge N$,

$$\underline{\mathcal{G}}(\Re {\mathfrak{b}}_n,\Re \mathfrak{b},\Re \mathfrak{b})<\underline{\mathcal{G}}(\Re \mathfrak{b},\Re \mathfrak{b},\Re \mathfrak{b})+ϵ,$$

which shows that $\Re {\mathfrak{b}}_n\to \Re \mathfrak{b}$ as $n\to \infty$. □

4. Some Fixed-Point Theorems in an s.g.m. Space

In this section, we impose further restrictions on the control function $\nu$. Let $\Theta$ denote the set of functions $\nu :[0,\infty )\to [1,\infty )$ satisfying the following conditions:

$\left(\nu 1\right)$

The function $\nu$ is continuous and strictly increasing.

$\left(\nu 2\right)$

For any sequence $\left\{{\mathfrak{b}}_n\right\}\subseteq (0,\infty )$,

$$\underset{n\to \infty }{lim}\nu \left({\mathfrak{b}}_n\right)=1⟺\underset{n\to \infty }{lim}{\mathfrak{b}}_n=0.$$

Definition 8.

Let us denote by $(\Lambda ,\underline{\mathcal{G}})$ an s.g.m. space whose parameter fulfills the condition $\omega \ge 1$. We say that the mapping $\Re :\Lambda \to \Lambda$ is a $\underline{\mathcal{G}}$-$JS$-Hardy–Rogers contraction of type I if

$$\begin{array}{cc}\hfill & \nu \left(\underline{\mathcal{G}}(\Re \mathfrak{b},\Re {\mathfrak{b}}^{\prime },\Re {\mathfrak{b}}^{″})\right)\hfill \\ & \le {\left(\nu \left(\underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{\prime \prime })\right)\right)}^{{\chi }_1}{\left(\nu \left({\displaystyle \frac{\underline{\mathcal{G}}(\mathfrak{b},\Re \mathfrak{b},\Re \mathfrak{b})}{\omega }}\right)\right)}^{{\chi }_2}{\left(\nu \left(\underline{\mathcal{G}}({\mathfrak{b}}^{\prime },\Re {\mathfrak{b}}^{\prime },\Re {\mathfrak{b}}^{\prime })\right)\right)}^{{\chi }_3}\hfill \\ & {\left(\nu \left(\underline{\mathcal{G}}({\mathfrak{b}}^{″},\Re {\mathfrak{b}}^{″},\Re {\mathfrak{b}}^{″})\right)\right)}^{{\chi }_4}\hfill \\ & {\left(\nu \left(\underline{\mathcal{G}}(\mathfrak{b},\Re {\mathfrak{b}}^{\prime },\Re {\mathfrak{b}}^{\prime })\right)\right)}^{{\chi }_5}{\left(\nu \left(\underline{\mathcal{G}}({\mathfrak{b}}^{\prime },\Re {\mathfrak{b}}^{\prime \prime },\Re {\mathfrak{b}}^{\prime \prime })\right)\right)}^{{\chi }_6}{\left(\nu \left(\underline{\mathcal{G}}({\mathfrak{b}}^{\prime \prime },\Re \mathfrak{b},\Re \mathfrak{b})\right)\right)}^{{\chi }_7}\hfill \end{array}$$

(6)

for some ${\chi }_1,{\chi }_2,{\chi }_3,{\chi }_4,{\chi }_5,{\chi }_6,{\chi }_7\ge 0$ with $\sum {\chi }_i<1$ where $\nu \in \Theta$.

For more details on this type of contraction, we refer the reader to [21,31].

Theorem 2.

Suppose $(\Lambda ,\underline{\mathcal{G}})$ is a complete symmetric s.g.m. space. Let $\Re :\Lambda \to \Lambda$ be a mapping that satisfies the following:

(i)

ℜ is a $\underline{\mathcal{G}}$-$JS$-Hardy–Rogers contraction of type I,

(ii)

There exists ${\mathfrak{b}}_0\in \Lambda$ such that

$$\aleph (\underline{\mathcal{G}},\Re ,{\mathfrak{b}}_0):=sup\left\{\underline{\mathcal{G}}\left({\Re }^i{\mathfrak{b}}_0,{\Re }^j{\mathfrak{b}}_0,{\Re }^j{\mathfrak{b}}_0\right):i,j=1,2,\dots \right\}<\infty .$$

Then, Λ contains at least one fixed point of ℜ. If, in addition, every fixed point $\mathfrak{b}$ of ℜ fulfills $\underline{\mathcal{G}}(\mathfrak{b},\mathfrak{b},\mathfrak{b})=0$, then ℜ possesses a unique fixed point.

Proof. 

For each integer $p\ge 1$, define

$$\aleph (\underline{\mathcal{G}},{\Re }^{p+1},{\mathfrak{b}}_0):=sup\left\{\underline{\mathcal{G}}\left({\Re }^{p+i}{\mathfrak{b}}_0,{\Re }^{p+j}{\mathfrak{b}}_0,{\Re }^{p+j}{\mathfrak{b}}_0\right):i,j=1,2,\dots \right\}.$$

It is clear that

$$\aleph (\underline{\mathcal{G}},{\Re }^p,{\mathfrak{b}}_0)\le \aleph (\underline{\mathcal{G}},\Re ,{\mathfrak{b}}_0)<\infty \mathrm{for}\mathrm{all}p\ge 1.$$

Then $\forall p\ge 1$ and $\forall i,j=1,2,\cdots$,

$$\begin{array}{cc}\hfill & \nu \left(\underline{\mathcal{G}}({\Re }^{p+i}{\mathfrak{b}}_0,{\Re }^{p+j}{\mathfrak{b}}_0,{\Re }^{p+j}{\mathfrak{b}}_0)\right)\hfill \\ & \le {\left(\nu \left(\underline{\mathcal{G}}({\Re }^{p-1+i}{\mathfrak{b}}_0,{\Re }^{p-1+j}{\mathfrak{b}}_0,{\Re }^{p-1+j}{\mathfrak{b}}_0)\right)\right)}^{{\chi }_1}\hfill \\ \hfill & {\left(\nu \left({\displaystyle \frac{\underline{\mathcal{G}}({\Re }^{p-1+i}{\mathfrak{b}}_0,{\Re }^{p+i}{\mathfrak{b}}_0,{\Re }^{p+i}{\mathfrak{b}}_0)}{\omega }}\right)\right)}^{{\chi }_2}{\left(\nu \left(\underline{\mathcal{G}}({\Re }^{p-1+j}{\mathfrak{b}}_0,{\Re }^{p+j}{\mathfrak{b}}_0,{\Re }^{p+j}{\mathfrak{b}}_0)\right)\right)}^{{\chi }_3}\hfill \\ & {\left(\nu \left(\underline{\mathcal{G}}({\Re }^{p-1+j}{\mathfrak{b}}_0,{\Re }^{p+j}{\mathfrak{b}}_0,{\Re }^{p+j}{\mathfrak{b}}_0)\right)\right)}^{{\chi }_4}\hfill \\ \hfill & {\left(\nu \left(\underline{\mathcal{G}}({\Re }^{p-1+i}{\mathfrak{b}}_0,{\Re }^{p+j}{\mathfrak{b}}_0,{\Re }^{p+j}{\mathfrak{b}}_0)\right)\right)}^{{\chi }_5}{\left(\nu \left(\underline{\mathcal{G}}({\Re }^{p-1+j}{\mathfrak{b}}_0,{\Re }^{p+j}{\mathfrak{b}}_0,{\Re }^{p+j}{\mathfrak{b}}_0)\right)\right)}^{{\chi }_6}\hfill \\ & {\left(\nu \left(\underline{\mathcal{G}}({\Re }^{p-1+j}{\mathfrak{b}}_0,{\Re }^{p+i}{\mathfrak{b}}_0,{\Re }^{p+i}{\mathfrak{b}}_0)\right)\right)}^{{\chi }_7}\hfill \\ \hfill & \le {\left(\nu (\aleph (\underline{\mathcal{G}},{\Re }^p,{\mathfrak{b}}_0))\right)}^{\eta },\hfill \end{array}$$

where $\eta =\sum {\chi }_i$. This implies that, for all $p\ge 1$,

$$\begin{array}{cc}\hfill \nu \left(\aleph (\underline{\mathcal{G}},{\Re }^{p+1},{\mathfrak{b}}_0)\right)& =\nu \left(\underset{i,j\ge 1}{sup}\underline{\mathcal{G}}({\Re }^{p+i}{\mathfrak{b}}_0,{\Re }^{p+j}{\mathfrak{b}}_0,{\Re }^{p+j}{\mathfrak{b}}_0)\right)\hfill \\ & \le {\left(\nu (\aleph (\underline{\mathcal{G}},{\Re }^p,{\mathfrak{b}}_0))\right)}^{\eta }\hfill \\ \hfill & \le {\left(\nu (\aleph (\underline{\mathcal{G}},{\Re }^{p-1},{\mathfrak{b}}_0))\right)}^{{\eta }^2}\hfill \\ \hfill & ⋮\hfill \\ \hfill & \le {\left(\nu (\aleph (\underline{\mathcal{G}},\Re ,{\mathfrak{b}}_0))\right)}^{{\eta }^p}.\hfill \end{array}$$

Let ${\mathfrak{b}}_i=\Re {\mathfrak{b}}_{i-1}={\Re }^i{\mathfrak{b}}_0$, $\forall i\in \mathbb{N}$. For all $m>n\ge 1$ we have

$$\begin{array}{cc}\hfill \nu \left(\underline{\mathcal{G}}({\mathfrak{b}}_n,{\mathfrak{b}}_m,{\mathfrak{b}}_m)\right)& =\nu \left(\underline{\mathcal{G}}({\Re }^n{\mathfrak{b}}_0,{\Re }^m{\mathfrak{b}}_0,{\Re }^m{\mathfrak{b}}_0)\right)\hfill \\ \hfill & =\nu \left(\underline{\mathcal{G}}({\Re }^{n-1+1}{\mathfrak{b}}_0,{\Re }^{n-1+(m-n+1)}{\mathfrak{b}}_0,{\Re }^{n-1+(m-n+1)}{\mathfrak{b}}_0)\right)\hfill \\ \hfill & \le \nu \left(\aleph (\underline{\mathcal{G}},{\Re }^n,{\mathfrak{b}}_0)\right)\hfill \\ \hfill & \le \nu {\left(\aleph (\underline{\mathcal{G}},\Re ,{\mathfrak{b}}_0)\right)}^{{\eta }^{n-1}}⟶1(\mathrm{as}n\to \infty ).\hfill \end{array}$$

Hence, the sequence $\left\{{\mathfrak{b}}_n\right\}$ forms a Cauchy sequence in $\Lambda$. By the completeness of $\Lambda$, it converges; denote its limit by

$$\underset{n\to \infty }{lim}{\mathfrak{b}}_n=\mathfrak{b}\in \Lambda .$$

Next, applying condition (G5) from Definition 6 together with the continuity of ℜ, we obtain

$$\begin{array}{cc}\hfill \underline{\mathcal{G}}(\Re \mathfrak{b},\mathfrak{b},\mathfrak{b})& \le \omega ·\underset{n\to \infty }{lim\; inf}\underline{\mathcal{G}}({\mathfrak{b}}_{n+1},\mathfrak{b},\mathfrak{b})=0.\hfill \end{array}$$

This implies $\Re \mathfrak{b}=\mathfrak{b}$, i.e., $\mathfrak{b}$ is a fixed point of ℜ in $\Lambda$.

If the continuity of ℜ is not assumed, we instead rely on the contractive condition (6) together with Definition 6 to proceed as follows:

$$\begin{array}{cc}\hfill \nu \left(\underline{\mathcal{G}}(\Re \mathfrak{b},{\mathfrak{b}}_{n+1},{\mathfrak{b}}_{n+1})\right)& =\nu \left(\underline{\mathcal{G}}(\Re \mathfrak{b},\Re {\mathfrak{b}}_n,\Re {\mathfrak{b}}_n)\right)\hfill \\ \hfill & \le {\left(\nu \left(\underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}_n,{\mathfrak{b}}_n)\right)\right)}^{{\chi }_1}{\left(\nu \left({\displaystyle \frac{\underline{\mathcal{G}}(\mathfrak{b},\Re \mathfrak{b},\Re \mathfrak{b})}{\omega }}\right)\right)}^{{\chi }_2}{\left(\nu \left(\underline{\mathcal{G}}({\mathfrak{b}}_n,\Re {\mathfrak{b}}_n,\Re {\mathfrak{b}}_n)\right)\right)}^{{\chi }_3}\hfill \\ & {\left(\nu \left(\underline{\mathcal{G}}({\mathfrak{b}}_n,\Re {\mathfrak{b}}_n,\Re {\mathfrak{b}}_n)\right)\right)}^{{\chi }_4}{\left(\nu \left(\underline{\mathcal{G}}(\mathfrak{b},\Re {\mathfrak{b}}_n,\Re {\mathfrak{b}}_n)\right)\right)}^{{\chi }_5}\hfill \\ & {\left(\nu \left(\underline{\mathcal{G}}({\mathfrak{b}}_n,\Re {\mathfrak{b}}_n,\Re {\mathfrak{b}}_n)\right)\right)}^{{\chi }_6}{\left(\nu \left(\underline{\mathcal{G}}({\mathfrak{b}}_n,\Re \mathfrak{b},\Re \mathfrak{b})\right)\right)}^{{\chi }_7}.\hfill \end{array}$$

Taking the liminf and using the symmetry condition on the space $(\Lambda ,\underline{\mathcal{G}})$ and Definition 6 in the above, we obtain that

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim\; inf}\nu \left(\underline{\mathcal{G}}(\Re \mathfrak{b},{\mathfrak{b}}_{n+1},{\mathfrak{b}}_{n+1})\right)& =\underset{n\to \infty }{lim\; inf}\nu \left(\underline{\mathcal{G}}(\Re \mathfrak{b},\Re {\mathfrak{b}}_n,\Re {\mathfrak{b}}_n)\right)\hfill \\ \hfill & \le \underset{n\to \infty }{lim\; inf}{\left[(\nu \left({\displaystyle \frac{\underline{\mathcal{G}}(\mathfrak{b},\Re \mathfrak{b},\Re \mathfrak{b})}{\omega }}\right)\right)}^{{\chi }_2}{\left(\nu \left(\underline{\mathcal{G}}({\mathfrak{b}}_n,\Re \mathfrak{b},\Re \mathfrak{b})\right)\right)}^{{\chi }_7}]\hfill \\ \hfill & ={\left(\nu \left({\displaystyle \frac{\underline{\mathcal{G}}(\mathfrak{b},\Re \mathfrak{b},\Re \mathfrak{b})}{\omega }}\right)\right)}^{{\chi }_2}\underset{n\to \infty }{lim\; inf}{\left(\nu \left(\underline{\mathcal{G}}({\mathfrak{b}}_n,\Re \mathfrak{b},\Re \mathfrak{b})\right)\right)}^{{\chi }_7}.\hfill \end{array}$$

Consequently, we obtain that

$$\begin{array}{c}\hfill {\left[\nu \left(\underset{n\to \infty }{lim\; inf}\underline{\mathcal{G}}({\mathfrak{b}}_{n+1},\Re \mathfrak{b},\Re \mathfrak{b})\right)\right]}^{1-{\chi }_7}\le {\left(\nu \left({\displaystyle \frac{\underline{\mathcal{G}}(\mathfrak{b},\Re \mathfrak{b},\Re \mathfrak{b})}{\omega }}\right)\right)}^{{\chi }_2}\end{array}$$

which further implies that

$$\begin{array}{cc}\hfill & \nu \left(\underset{n\to \infty }{lim\; inf}\underline{\mathcal{G}}({\mathfrak{b}}_{n+1},\Re \mathfrak{b},\Re \mathfrak{b})\right)\le {\left(\nu \left({\displaystyle \frac{\underline{\mathcal{G}}(\mathfrak{b},\Re \mathfrak{b},\Re \mathfrak{b})}{\omega }}\right)\right)}^{{\displaystyle \frac{{\chi }_2}{1-{\chi }_7}}}.\hfill \end{array}$$

On the other hand,

$$\underline{\mathcal{G}}(\mathfrak{b},\Re \mathfrak{b},\Re \mathfrak{b})\le \omega \left(\underset{n\to \infty }{lim\; inf}\underline{\mathcal{G}}({\mathfrak{b}}_n,\Re \mathfrak{b},\Re \mathfrak{b})\right).$$

Therefore,

$$1\le \nu \left({\displaystyle \frac{\underline{\mathcal{G}}(\mathfrak{b},\Re \mathfrak{b},\Re \mathfrak{b})}{\omega }}\right)\le \nu \left(\underset{n\to \infty }{lim\; inf}\underline{\mathcal{G}}({\mathfrak{b}}_n,\Re \mathfrak{b},\Re \mathfrak{b})\right)\le {\left(\nu \left({\displaystyle \frac{\underline{\mathcal{G}}(\mathfrak{b},\Re \mathfrak{b},\Re \mathfrak{b})}{\omega }}\right)\right)}^{{\displaystyle \frac{{\chi }_2}{1-{\chi }_7}}}$$

which is impossible unless that ${\displaystyle \frac{\underline{\mathcal{G}}(\mathfrak{b},\Re \mathfrak{b},\Re \mathfrak{b})}{\omega }}=0$. Hence, $\mathfrak{b}=\Re \mathfrak{b}$.

Now, if $\mathfrak{b}$ and ${\mathfrak{b}}^{\prime }$ are two fixed points of ℜ in $\Lambda$ with $\underline{\mathcal{G}}(\mathfrak{b},\mathfrak{b},\mathfrak{b})=0$ and $\underline{\mathcal{G}}({\mathfrak{b}}^{\prime },{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{\prime })=0$, then

$$\begin{array}{cc}\hfill \nu \left(\underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{\prime })\right)& =\nu \left(\underline{\mathcal{G}}(\Re \mathfrak{b},\Re {\mathfrak{b}}^{\prime },\Re {\mathfrak{b}}^{\prime })\right)\hfill \\ \hfill & \le {\left(\nu \left(\underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{\prime })\right)\right)}^{{\chi }_1}{\left(\nu \left({\displaystyle \frac{\underline{\mathcal{G}}(\mathfrak{b},\Re \mathfrak{b},\Re \mathfrak{b})}{\omega }}\right)\right)}^{{\chi }_2}{\left(\nu \left(\underline{\mathcal{G}}({\mathfrak{b}}^{\prime },\Re {\mathfrak{b}}^{\prime },\Re {\mathfrak{b}}^{\prime })\right)\right)}^{{\chi }_3}{\left(\nu \left(\underline{\mathcal{G}}({\mathfrak{b}}^{\prime },\Re {\mathfrak{b}}^{\prime },\Re {\mathfrak{b}}^{\prime })\right)\right)}^{{\chi }_4}\hfill \\ \hfill & {\left(\nu \left(\underline{\mathcal{G}}(\mathfrak{b},\Re {\mathfrak{b}}^{\prime },\Re {\mathfrak{b}}^{\prime })\right)\right)}^{{\chi }_5}{\left(\nu \left(\underline{\mathcal{G}}({\mathfrak{b}}^{\prime },\Re {\mathfrak{b}}^{\prime },\Re {\mathfrak{b}}^{\prime })\right)\right)}^{{\chi }_6}{\left(\nu \left(\underline{\mathcal{G}}({\mathfrak{b}}^{\prime },\Re \mathfrak{b},\Re \mathfrak{b})\right)\right)}^{{\chi }_7}\hfill \\ \hfill & ={\left(\nu \left(\underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{\prime })\right)\right)}^{{\chi }_1+{\chi }_5+{\chi }_7},\hfill \end{array}$$

where ${\chi }_1+{\chi }_5+{\chi }_7<1$. This implies that $\underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{\prime })=0$. Hence, $\mathfrak{b}={\mathfrak{b}}^{\prime }$. □

In the following we work with contractions that act on two members instead of three. Such theorems lead to easier calculations.

Definition 9.

Let $(\Lambda ,\underline{\mathcal{G}})$ represent an s.g.m. space with $\omega \ge 1$. A mapping $\Re :\Lambda \to \Lambda$ is called a $\underline{\mathcal{G}}$-$JS$-Hardy–Rogers contraction of type II if

$$\begin{array}{cc}\hfill & \nu \left(\underline{\mathcal{G}}(\Re \mathfrak{b},\Re {\mathfrak{b}}^{\prime },\Re {\mathfrak{b}}^{\prime })\right)\hfill \\ & \le {\left(\nu \left(\underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{\prime })\right)\right)}^{{\chi }_1}{\left(\nu \left({\displaystyle \frac{\underline{\mathcal{G}}(\mathfrak{b},\Re \mathfrak{b},\Re \mathfrak{b})}{\omega }}\right)\right)}^{{\chi }_2}{\left(\nu \left(\underline{\mathcal{G}}({\mathfrak{b}}^{\prime },\Re {\mathfrak{b}}^{\prime },\Re {\mathfrak{b}}^{\prime })\right)\right)}^{{\chi }_3}\hfill \\ & {\left(\nu \left(\underline{\mathcal{G}}(\mathfrak{b},\Re {\mathfrak{b}}^{\prime },\Re {\mathfrak{b}}^{\prime })\right)\right)}^{{\chi }_4}{\left(\nu \left(\underline{\mathcal{G}}({\mathfrak{b}}^{\prime },\Re \mathfrak{b},\Re \mathfrak{b})\right)\right)}^{{\chi }_5}\hfill \end{array}$$

(7)

for some ${\chi }_1,{\chi }_2,{\chi }_3,{\chi }_4,{\chi }_5\ge 0$ with $\sum {\chi }_i<1$ where $\nu \in \Theta$.

Theorem 3.

Let $(\Lambda ,\underline{\mathcal{G}})$ be a complete symmetric s.g.m. space, and let $\Re :\Lambda \to \Lambda$ be a mapping satisfying the following:

(i)

ℜ is a $\underline{\mathcal{G}}$-$JS$-Hardy–Rogers contraction of type II,

(ii)

there exists ${\mathfrak{b}}_0\in \Lambda$ such that

$$\aleph (\underline{\mathcal{G}},\Re ,{\mathfrak{b}}_0):=sup\left\{\underline{\mathcal{G}}\left({\Re }^i{\mathfrak{b}}_0,{\Re }^j{\mathfrak{b}}_0,{\Re }^j{\mathfrak{b}}_0\right):i,j=1,2,\dots \right\}<\infty .$$

Then, the space Λ contains at least one fixed point of ℜ.

In addition, if all fixed points $\mathfrak{b}$ of ℜ in Λ fulfill

$$\underline{\mathcal{G}}(\mathfrak{b},\mathfrak{b},\mathfrak{b})=0,$$

then ℜ admits a unique fixed point.

Proof. 

It is sufficient to take ${\mathfrak{b}}^{\prime }={\mathfrak{b}}^{″}$ in Theorem 2. □

The subsequent corollary is noteworthy in that it holds without imposing the strong symmetry condition on $(\Lambda ,\underline{\mathcal{G}})$ which was assumed in the earlier theorem.

Corollary 1.

Let $(\Lambda ,\underline{\mathcal{G}})$ be a complete s.g.m. space and $\Re :\Lambda \to \Lambda$ be a mapping such that:

(i)

ℜ is a $\underline{\mathcal{G}}$-$JS$-Reich type contraction, that is,

$$\nu \left(\underline{\mathcal{G}}(\Re \mathfrak{b},\Re {\mathfrak{b}}^{\prime },\Re {\mathfrak{b}}^{\prime })\right)\le {\left(\nu \left(\underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{\prime })\right)\right)}^{{\chi }_1}{\left(\nu \left({\displaystyle \frac{\underline{\mathcal{G}}(\mathfrak{b},\Re \mathfrak{b},\Re \mathfrak{b})}{\omega }}\right)\right)}^{{\chi }_2}{\left(\nu \left(\underline{\mathcal{G}}({\mathfrak{b}}^{\prime },\Re {\mathfrak{b}}^{\prime },\Re {\mathfrak{b}}^{\prime })\right)\right)}^{{\chi }_3},$$

for some ${\chi }_1,{\chi }_2,{\chi }_3\ge 0$ with ${\chi }_1+{\chi }_2+{\chi }_3<1$ where $\nu \in \Theta$.

(ii)

there is ${\mathfrak{b}}_0\in \Lambda$ so that

$$\aleph (\underline{\mathcal{G}},\Re ,{\mathfrak{b}}_0):=sup\left\{\underline{\mathcal{G}}\left({\Re }^i{\mathfrak{b}}_0,{\Re }^j{\mathfrak{b}}_0,{\Re }^j{\mathfrak{b}}_0\right):i,j=1,2,\dots \right\}<\infty .$$

It follows that ℜ possesses a fixed point in Λ. Furthermore, the condition $\underline{\mathcal{G}}(\mathfrak{b},\mathfrak{b},\mathfrak{b})=0$ for every fixed point $\mathfrak{b}$ ensures the uniqueness of the fixed point of ℜ.

Proof. 

When we use ${\chi }_4={\chi }_5=0$ in Theorem 3, we have the above result which is a contraction of the Reich kind. □

Remark 4.

By setting ${\chi }_1=0$ in Corollary 1, the classical Kannan fixed-point theorem extends naturally to $\underline{\mathcal{G}}$-$JS$-Kannan type contractions within the context of s.g.m. spaces.

Corollary 2.

Take $(\Lambda ,\underline{\mathcal{G}})$ as a complete s.g.m. space. Let $\Re :\Lambda \to \Lambda$ denote a mapping subject to the following requirements:

(i)

ℜ is a $\underline{\mathcal{G}}$-$JS$-contraction, meaning that

$$\nu \left(\underline{\mathcal{G}}(\Re \mathfrak{b},\Re {\mathfrak{b}}^{\prime },\Re {\mathfrak{b}}^{\prime })\right)\le {\left(\nu \left(\underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{\prime })\right)\right)}^{{\chi }_1},$$

for some ${\chi }_1\in [0,1)$;

(ii)

there exists ${\mathfrak{b}}_0\in \Lambda$ such that

$$\aleph (\underline{\mathcal{G}},\Re ,{\mathfrak{b}}_0):=sup\left\{\underline{\mathcal{G}}\left({\Re }^i{\mathfrak{b}}_0,{\Re }^j{\mathfrak{b}}_0,{\Re }^j{\mathfrak{b}}_0\right):i,j=1,2,\dots \right\}<\infty .$$

Under these assumptions, ℜ possesses one unique fixed point in Λ.

Proof. 

Considering ${\chi }_2={\chi }_3={\chi }_4={\chi }_5=0$ as an exceptional instance of Theorem 3, we arrive at the above outcome. □

Corollary 3.

Take $(\Lambda ,\underline{\mathcal{G}})$ as a complete symmetric s.g.m. space, and let $\Re :\Lambda \to \Lambda$ be a mapping that fulfills:

(i)

ℜ is a $\underline{\mathcal{G}}$-Hardy–Rogers-type contraction, i.e.,

$$\begin{array}{cc}\hfill \underline{\mathcal{G}}(\Re \mathfrak{b},\Re {\mathfrak{b}}^{\prime },\Re {\mathfrak{b}}^{\prime })& \le {\chi }_1\underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{\prime })+{\chi }_2{\displaystyle \frac{\underline{\mathcal{G}}(\mathfrak{b},\Re \mathfrak{b},\Re \mathfrak{b})}{\omega }}+{\chi }_3\underline{\mathcal{G}}({\mathfrak{b}}^{\prime },\Re {\mathfrak{b}}^{\prime },\Re {\mathfrak{b}}^{\prime })\hfill \\ \hfill & +{\chi }_4\underline{\mathcal{G}}(\mathfrak{b},\Re {\mathfrak{b}}^{\prime },\Re {\mathfrak{b}}^{\prime })+{\chi }_5\underline{\mathcal{G}}({\mathfrak{b}}^{\prime },\Re \mathfrak{b},\Re \mathfrak{b}),\hfill \end{array}$$

(8)

for some non-negative constants ${\chi }_1,\dots ,{\chi }_5$ with ${\sum }_{i=1}^5{\chi }_i<1$;

(ii)

there exists ${\mathfrak{b}}_0\in \Lambda$ such that

$$\aleph (\underline{\mathcal{G}},\Re ,{\mathfrak{b}}_0):=sup\left\{\underline{\mathcal{G}}({\Re }^i{\mathfrak{b}}_0,{\Re }^j{\mathfrak{b}}_0,{\Re }^j{\mathfrak{b}}_0):i,j=1,2,\dots \right\}<\infty .$$

It follows that ℜ possesses at least one fixed point in Λ. In addition, if each fixed point $\mathfrak{b}$ satisfies

$$\underline{\mathcal{G}}(\mathfrak{b},\mathfrak{b},\mathfrak{b})=0,$$

then such a fixed point is necessarily unique.

Proof. 

By setting $\nu \left(t\right)=e^t$ in Theorem 3, we have the above result. □

Remark 5.

$\underline{\mathcal{G}}$-Banach type contractions, $\underline{\mathcal{G}}$-Kannan type contractions, $\underline{\mathcal{G}}$-Chatterjea type contractions, $\underline{\mathcal{G}}$-Reich type contractions and $\underline{\mathcal{G}}$-Ćirić type contractions are special cases of the above contraction.

Example 5.

Consider $\Lambda =[0,\infty )$ and $\underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″})=e^{\left(\right|\mathfrak{b}-{\mathfrak{b}}^{\prime }|+|{\mathfrak{b}}^{\prime }-{\mathfrak{b}}^{″}|+|{\mathfrak{b}}^{″}-\mathfrak{b}{\left|\right)}^2}-1$, $\forall \mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″}\in \Lambda$. Then $\underline{\mathcal{G}}$ forms an s.g.m. on Λ.

Define $\Re :\Lambda \to \Lambda$ by $\Re \mathfrak{b}={\displaystyle \frac{1}{4}}({sinh}^{-1}\mathfrak{b}-{tan}^{-1}\mathfrak{b})$ for all $\mathfrak{b}\in \Lambda$. Then ℜ satisfies all the conditions of Corollary 2 for ${\chi }_1={\displaystyle \frac{1}{2}}$, $\nu \left(t\right)=e^{\sqrt{t}}$ and clearly, ℜ has a unique fixed point $0\in \Lambda$, because

$$\begin{array}{cc}\hfill \nu \left(\underline{\mathcal{G}}(\Re \mathfrak{b},\Re {\mathfrak{b}}^{\prime },\Re {\mathfrak{b}}^{\prime })\right)& =e^{\left(\sqrt{e^{{\displaystyle \frac{1}{8}}{|{sinh}^{-1}\mathfrak{b}-{tan}^{-1}\mathfrak{b}-{sinh}^{-1}{\mathfrak{b}}^{\prime }+{tan}^{-1}{\mathfrak{b}}^{\prime }|}^2}-1}\right)}\hfill \\ \hfill & \le e^{\left(\sqrt{e^{{\displaystyle \frac{1}{4}}\left(\right|{sinh}^{-1}\mathfrak{b}-{sinh}^{-1}{\mathfrak{b}}^{\prime }{|}^2+|{tan}^{-1}\mathfrak{b}-{tan}^{-1}{\mathfrak{b}}^{\prime }{|}^2)}-1}\right)}\hfill \\ \hfill & \le e^{\left(\sqrt{e^{{\displaystyle \frac{1}{4}}\left(\right|\mathfrak{b}-{\mathfrak{b}}^{\prime }{|}^2+|\mathfrak{b}-{\mathfrak{b}}^{\prime }{|}^2)}-1}\right)}\hfill \\ \hfill & =e^{\left(\sqrt{e^{{\displaystyle \frac{1}{2}}\left(\right|\mathfrak{b}-{\mathfrak{b}}^{\prime }{\left|\right)}^2}-1}\right)}\hfill \\ \hfill & \le {\left(e^{\left(\sqrt{e^{2|\mathfrak{b}-{\mathfrak{b}}^{\prime }{|}^2+{|{\mathfrak{b}}^{\prime }-{\mathfrak{b}}^{\prime }|}^2}-1}\right)}\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & ={\left(\nu \left(\underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{\prime })\right)\right)}^{{\chi }_1}.\hfill \end{array}$$

Keep in mind that $e^{\left(\sqrt{e^{{\displaystyle \frac{1}{2}}\mathfrak{b}}-1}\right)}\le \sqrt{e^{\left(\sqrt{e^{2\mathfrak{b}}-1}\right)}}$ for any positive real number $\mathfrak{b}$.

5. Prešić Type Fixed Point Results

This part unifies the ideas presented by Jleli-Samet, Hardy–Rogers, and Prešić.

Theorem 4

([32]). Let’s assume $(\Lambda ,d)$ is a complete metric space and let $\Re :\Lambda \to \Lambda$ so that

$$d(\Re 𝚥,\Re \varsigma )\le {\chi }_{1}d(𝚥,\varsigma )\mathrm{for}\mathrm{all}𝚥,\varsigma \in \Lambda ,$$

where ${\chi }_1\in [0,1).$ Then, $\mathfrak{b}=\Re \mathfrak{b}$ where $\mathfrak{b}$ is unique in Λ. Additionally, the sequence ${\mathfrak{b}}_{n+1}=\Re {\mathfrak{b}}_n$ converges to $\mathfrak{b}$ for each ${\mathfrak{b}}_0\in \Lambda$.

Numerous extensions and generalizations of the Banach contraction principle (BCP) have been made (see, for instance, [17]). The results of Prešić [33] were as follows:

Theorem 5

([33]). Let’s assume $(\Lambda ,d)$ is a complete metric space and let $\Re :{\Lambda }^k\to \Lambda$ (k is a positive integer). Suppose that

$$d(\Re ({\mathfrak{b}}_1\dots ,{\mathfrak{b}}_k),\Re ({\mathfrak{b}}_2,\dots ,{\mathfrak{b}}_{k+1}))\le \sum _{i=1}^k{\lambda }_{i}d({\mathfrak{b}}_i,{\mathfrak{b}}_{i+1})$$

(9)

for all ${\mathfrak{b}}_1,\dots ,{\mathfrak{b}}_{k+1}$ in Λ, where ${\lambda }_i\ge 0$ and ${\displaystyle \sum _{i=1}^k}{\lambda }_i\in [0,1)$. Then ℜ has a unique fixed point ${\mathfrak{b}}^{\ast }$ (that is, $\Re ({\mathfrak{b}}^{\ast },\dots ,{\mathfrak{b}}^{\ast })={\mathfrak{b}}^{\ast }$). Moreover, for all arbitrary points ${\mathfrak{b}}_1,\dots {\mathfrak{b}}_{k+1}$ in $\Lambda ,$ the sequence $\left\{{\mathfrak{b}}_n\right\}$ defined by ${\mathfrak{b}}_{n+k}=\Re ({\mathfrak{b}}_n,{\mathfrak{b}}_{n+1},\dots ,{\mathfrak{b}}_{n+k-1})$, converges to ${\mathfrak{b}}^{\ast }$.

The BCP and Theorem 5 clearly coincide for $k=1$. Ćirić and Prešić [34] generalized the aforementioned theorem as follows:

Theorem 6

([34]). Let’s assume that $(\Lambda ,d)$ is a complete metric space and that ℜ is a mapping from ${\Lambda }^k$ to Λ, where k is a positive integer.

If ℜ satisfies the following contractive condition:

$$d(\Re ({\mathfrak{b}}_1,\dots ,{\mathfrak{b}}_k),\Re ({\mathfrak{b}}_2,\dots ,{\mathfrak{b}}_{k+1}))\le \lambda max\{d({\mathfrak{b}}_i,{\mathfrak{b}}_{i+1}):1\le i\le k\},$$

for all ${\mathfrak{b}}_1,\dots ,{\mathfrak{b}}_{k+1}$ in Λ, where $\lambda \in [0,1)$, then ℜ possesses a fixed point ${\mathfrak{b}}^{\ast }\in \Lambda$.

Furthermore, for any initial points ${\mathfrak{b}}_1,\dots ,{\mathfrak{b}}_{k+1}\in \Lambda$, the sequence $\left\{{\mathfrak{b}}_n\right\}$ defined by ${\mathfrak{b}}_{n+k}=\Re ({\mathfrak{b}}_n,{\mathfrak{b}}_{n+1},\dots ,{\mathfrak{b}}_{n+k-1})$ converges to ${\mathfrak{b}}^{\ast }$.

The fixed point of ℜ is unique if the following additional condition holds:

$$d(\Re (\rho ,\dots ,\rho ),\Re (\varrho ,\dots ,\varrho \left)\right)<d(\rho ,\varrho ),$$

for all $\rho ,\varrho \in \Lambda$ with $\rho \ne \varrho$.

However, according to [35] the above theorem is a simple consequence of the results in [36].

Readers interested in learning more about Prešić type contractions are directed to [33]. In 2013, Shukla et al. [37] proposed significant generalizations of Prešić fixed-point theorem by combining the classical Prešić contraction with other well-known contraction types, such as those introduced by Kannan, Chatterjea, Reich, and Hardy–Rogers. In their work, they introduced new classes of hybrid contractions, namely the Prešić–Kannan, Prešić–Chatterjea, Prešić–Reich, Prešić–Ćirić and Prešić–Hardy–Rogers types. These generalized contractive conditions offer a unified approach that encompasses several classical fixed-point results as particular cases, specially in the case $k=1$.

Encouraged by [38]’s Theorem 6, we prove the following theorem, which will aid us in Prešić type results.

Theorem 7.

Suppose that $G_1,G_2,\dots ,G_n$ be s.g.m.s on nonempty sets ${\Lambda }_1,{\Lambda }_2,\dots ,{\Lambda }_n$ with constants $p_1,p_2,\dots ,p_n$, respectively, and let ${\rm Y}:{[0,\infty )}^n⟶[0,\infty )$ be increasing with respect to all variables so that ${\rm Y}({\mathfrak{b}}_1,\dots ,{\mathfrak{b}}_n)=0$ if and only if ${\mathfrak{b}}_i=0$ for all $i=1,2,3,\dots ,n$ and

1.

$$\begin{array}{cc}& {\rm Y}(G_1({𝚥}^{11},{𝚥}^{11},{𝚥}^{21}),G_2({𝚥}^{12},{𝚥}^{12},{𝚥}^{22}),\dots ,G_n({𝚥}^{1n},{𝚥}^{1n},{𝚥}^{2n}))>0,\hfill \end{array}$$

where ${𝚥}^{1j}\ne {𝚥}^{2j}$ for all $1\le j\le n$.

2.

$$\begin{array}{cc}& {\rm Y}(G_1({𝚥}^{11},{𝚥}^{11},{𝚥}^{21}),G_2({𝚥}^{12},{𝚥}^{12},{𝚥}^{22}),\dots ,G_n({𝚥}^{1n},{𝚥}^{1n},{𝚥}^{2n}))\hfill \\ & \le {\rm Y}(G_1({𝚥}^{11},{𝚥}^{21},{𝚥}^{31}),G_2({𝚥}^{12},{𝚥}^{22},{𝚥}^{32}),\dots ,G_n({𝚥}^{1n},{𝚥}^{2n},{𝚥}^{3n})),\hfill \end{array}$$

where ${𝚥}^{2j}\ne {𝚥}^{3j}$ for all $1\le j\le n$.

3.

$$\begin{array}{c}\hfill {\rm Y}(plim\; inf{𝚥}_m^1,plim\; inf{𝚥}_m^2,\dots ,plim\; inf{𝚥}_m^n)\le plim\; inf{\rm Y}({𝚥}_m^1,{𝚥}_m^2,\dots ,{𝚥}_m^n)\end{array}$$

where $\left\{{𝚥}_m^j\right\}$ for all $1\le j\le n$ is a sequence of real numbers.

Then

$$\begin{array}{cc}\hfill & \tilde{G}(({𝚥}^{11},{𝚥}^{12},\dots ,{𝚥}^{1n}),({𝚥}^{21},{𝚥}^{22},\dots ,{𝚥}^{2n}),({𝚥}^{31},{𝚥}^{32},\dots ,{𝚥}^{3n}))\hfill \\ & ={\rm Y}(G_1({𝚥}^{11},{𝚥}^{21},{𝚥}^{31}),G_2({𝚥}^{12},{𝚥}^{22},{𝚥}^{32}),\dots ,G_n({𝚥}^{1n},{𝚥}^{2n},{𝚥}^{3n})),\hfill \end{array}$$

is a s.g.m. in ${[{\Lambda }_1\times {\Lambda }_2\times \dots \times {\Lambda }_n]}^3$.

Proof. 

We only show the inequality G5 in a s.g.m. space. Let $p=\underset{1\le i\le n}{max}{p}_i$ and ${𝚥}^{ij}\in {\Lambda }_j$ for all $1\le i\le 3$ and $1\le j\le n$. Also, let $\left\{{𝚥}_m^{1j}\right\}$ be sequences in ${\Lambda }_j$ for $1\le j\le n$ such that ${𝚥}_m^{1j}\to {𝚥}^{1j}$ for all $1\le j\le n$.

So,

$$\begin{array}{cc}\hfill & \tilde{G}(({𝚥}^{11},{𝚥}^{12},\dots ,{𝚥}^{1n}),({𝚥}^{21},{𝚥}^{22},\dots ,{𝚥}^{2n}),({𝚥}^{31},{𝚥}^{32},\dots ,{𝚥}^{3n}))\hfill \\ \hfill & ={\rm Y}(G_1({𝚥}^{11},{𝚥}^{21},{𝚥}^{31}),G_2({𝚥}^{12},{𝚥}^{22},{𝚥}^{32}),\dots ,G_n({𝚥}^{1n},{𝚥}^{2n},{𝚥}^{3n}))\hfill \\ \hfill & \le {\rm Y}(p\underset{m\to \infty }{lim\; inf}{G}_1({𝚥}_m^{11},{𝚥}^{21},{𝚥}^{31}),p\underset{m\to \infty }{lim\; inf}{G}_2({𝚥}_m^{12},{𝚥}^{22},{𝚥}^{32}),\dots ,p\underset{m\to \infty }{lim\; inf}{G}_n({𝚥}_m^{1n},{𝚥}^{2n},{𝚥}^{3n}))\hfill \\ \hfill & \le p\underset{m\to \infty }{lim\; inf}{\rm Y}(G_1({𝚥}_m^{11},{𝚥}^{21},{𝚥}^{31}),G_2({𝚥}_m^{12},{𝚥}^{22},{𝚥}^{32}),\dots ,G_n({𝚥}_m^{1n},{𝚥}^{2n},{𝚥}^{3n}))\hfill \\ \hfill & =p\underset{m\to \infty }{lim\; inf}\tilde{G}(({𝚥}_m^{11},{𝚥}_m^{12},\dots ,{𝚥}_m^{1n}),({𝚥}^{21},{𝚥}^{22},\dots ,{𝚥}^{2n}),({𝚥}^{31},{𝚥}^{32},\dots ,{𝚥}^{3n})).\hfill \end{array}$$

□

Definition 10.

Let $(\Lambda ,G)$ be an s.g.m. space with parameter $\omega \ge 1$. A mapping $\Re :{\Lambda }^n\to \Lambda$ is called a G-$JS$-Prešić-Hardy–Rogers-type contraction if

$$\begin{array}{cc}\hfill & \nu \left(G(\Re ({𝚥}_1,{𝚥}_2,\dots ,{𝚥}_n),\Re ({𝚥}_2,{𝚥}_3,\dots ,{𝚥}_{n+1}),\Re ({𝚥}_2,{𝚥}_3,\dots ,{𝚥}_{n+1}))\right)\hfill \\ \hfill & \le {\left(\nu \left({\displaystyle \frac{{\sum }_{i=1}^{n}G({𝚥}_i,{𝚥}_{i+1},{𝚥}_{i+1})}{n}}\right)\right)}^{{\chi }_1}\hfill \\ & {\left(\nu \left({\displaystyle \frac{{\sum }_{i=1}^{n}G({𝚥}_i,\Re ({𝚥}_1,{𝚥}_2,\dots ,{𝚥}_n),\Re ({𝚥}_1,{𝚥}_2,\dots ,{𝚥}_n))}{n\omega }}\right)\right)}^{{\chi }_2}\hfill \\ & {\left(\nu \left({\displaystyle \frac{{\sum }_{i=1}^{n}G({𝚥}_{i+1},\Re ({𝚥}_2,{𝚥}_3,\dots ,{𝚥}_{n+1}),\Re ({𝚥}_2,{𝚥}_3,\dots ,{𝚥}_{n+1}))}{n}}\right)\right)}^{{\chi }_3}\hfill \\ \hfill & {\left(\nu \left({\displaystyle \frac{{\sum }_{i=1}^{n}G({𝚥}_i,\Re ({𝚥}_2,{𝚥}_3,\dots ,{𝚥}_{n+1}),\Re ({𝚥}_2,{𝚥}_3,\dots ,{𝚥}_{n+1}))}{n}}\right)\right)}^{{\chi }_4}\hfill \\ \hfill & {\left(\nu \left({\displaystyle \frac{{\sum }_{i=1}^{n}G({𝚥}_{i+1},\Re ({𝚥}_1,{𝚥}_2,\dots ,{𝚥}_n),\Re ({𝚥}_2,{𝚥}_3,\dots ,{𝚥}_{n+1}))}{n}}\right)\right)}^{{\chi }_5}\hfill \end{array}$$

(10)

for all ${𝚥}_1,{𝚥}_2,\dots ,J_{n+1}\in \Lambda$ and for some ${\chi }_1,{\chi }_2,{\chi }_3,{\chi }_4,{\chi }_5\ge 0$ with $\sum {\chi }_i<1$ where $\nu \in \Theta$.

Theorem 8.

Let $(\Lambda ,G)$ be a complete symmetry s.g.m. space and $\Re :{\Lambda }^n\to \Lambda$ be a mapping such that:

(i)

ℜ is a G-$JS$-Prešić-Hardy–Rogers-type contraction,

(ii)

There is ${\mathfrak{b}}_0\in \Lambda$ so that

$$\aleph (G,\Re ,\tilde{\mathfrak{b}}):=sup\left\{G\left({\tilde{\Re }}^i\tilde{\mathfrak{b}},{\tilde{\Re }}^j\tilde{\mathfrak{b}},{\tilde{\Re }}^j\tilde{\mathfrak{b}}\right):i,j=1,2,\dots \right\}<\infty$$

in which

$$\tilde{\Re }\tilde{\mathfrak{b}}=\left(\underset{ntimes}{\underbrace{\Re ({\mathfrak{b}}_0,\dots ,{\mathfrak{b}}_0),\dots ,\Re ({\mathfrak{b}}_0,\dots ,{\mathfrak{b}}_0)}}\right)$$

and $\tilde{\mathfrak{b}}=({\mathfrak{b}}_0,\dots ,{\mathfrak{b}}_0)$.

This implies that there is at least one fixed point for ℜ in ${\Lambda }^n$. Additionally, if $G(\mathfrak{b},\mathfrak{b},\mathfrak{b})=0$ is true for every fixed point $\mathfrak{b}$ of ℜ in ${\Lambda }^n$, then the fixed point of ℜ is unique.

Proof. 

We define the mapping $\tilde{\Re }:{\Lambda }^n\to {\Lambda }^n$ by

$$\tilde{\Re }({\mathfrak{b}}_1,\dots ,{\mathfrak{b}}_n)=(\Re ({\mathfrak{b}}_1,\dots ,{\mathfrak{b}}_n),\dots ,\Re ({\mathfrak{b}}_1,\dots ,{\mathfrak{b}}_n)).$$

We show that all of the requirements of Theorem 3 are satisfied by $\tilde{\Re }$.

From Theorem 7, we know that

$$\begin{array}{cc}\hfill & \tilde{G}(({𝚥}_1,{𝚥}_2,\dots ,{𝚥}_n),({\varsigma }_1,{\varsigma }_2,\dots ,{\varsigma }_n),({\mathfrak{b}}_1,{\mathfrak{b}}_2,\dots ,{\mathfrak{b}}_n))\hfill \\ \hfill & ={\displaystyle \frac{G({𝚥}_1,{\varsigma }_1,{\mathfrak{b}}_1)+G({𝚥}_2,{\varsigma }_2,{\mathfrak{b}}_2)+\cdots +G({𝚥}_n,{\varsigma }_n,{\mathfrak{b}}_n)}{n}}\hfill \end{array}$$

is an s.g.m. on ${\Lambda }^n$.

From (10), we have

$$\begin{array}{c}\hfill \begin{array}{cc}& \nu (\tilde{G}(\tilde{\Re }({𝚥}_1,{𝚥}_2,\dots ,{𝚥}_n),\tilde{\Re }({𝚥}_2,{𝚥}_3,\dots ,{𝚥}_{n+1}),\tilde{\Re }({𝚥}_2,{𝚥}_3,\dots ,{𝚥}_{n+1}))\hfill \\ & =\nu \left(\tilde{G}\right((\Re ({𝚥}_1,{𝚥}_2,\dots ,{𝚥}_n),\Re ({𝚥}_1,{𝚥}_2,\dots ,{𝚥}_n),\dots ,\Re ({𝚥}_1,{𝚥}_2,\dots ,{𝚥}_n)),\hfill \\ & (\Re ({𝚥}_2,{𝚥}_3,\dots ,{𝚥}_{n+1}),\Re ({𝚥}_2,{𝚥}_3,\dots ,{𝚥}_{n+1}),\dots ,\Re ({𝚥}_2,{𝚥}_3,\dots ,{𝚥}_{n+1})),\hfill \\ & (\Re ({𝚥}_2,{𝚥}_3,\dots ,{𝚥}_{n+1}),\Re ({𝚥}_2,{𝚥}_3,\dots ,{𝚥}_{n+1}),\dots ,\Re ({𝚥}_2,{𝚥}_3,\dots ,{𝚥}_{n+1}))\left)\right)\hfill \\ & =\nu \left(G(\Re ({𝚥}_1,{𝚥}_2,\dots ,{𝚥}_n),\Re ({𝚥}_2,{𝚥}_3,\dots ,{𝚥}_{n+1}),\dots ,\Re ({𝚥}_2,{𝚥}_3,\dots ,{𝚥}_{n+1}))\right)\hfill \\ & \le {\left(\nu \left({\displaystyle \frac{{\sum }_{i=1}^{n}G({𝚥}_i,{𝚥}_{i+1},{𝚥}_{i+1})}{n}}\right)\right)}^{{\chi }_1}\hfill \\ & {\left(\nu \left({\displaystyle \frac{{\sum }_{i=1}^{n}G({𝚥}_i,\Re ({𝚥}_1,{𝚥}_2,\dots ,{𝚥}_n),\Re ({𝚥}_1,{𝚥}_2,\dots ,{𝚥}_n))}{n\omega }}\right)\right)}^{{\chi }_2}\hfill \\ & {\left(\nu \left({\displaystyle \frac{{\sum }_{i=1}^{n}G({𝚥}_{i+1},\Re ({𝚥}_2,{𝚥}_3,\dots ,{𝚥}_{n+1}),\Re ({𝚥}_2,{𝚥}_3,\dots ,{𝚥}_{n+1}))}{n}}\right)\right)}^{{\chi }_3}\hfill \\ & {\left(\nu \left({\displaystyle \frac{{\sum }_{i=1}^{n}G({𝚥}_i,\Re ({𝚥}_2,{𝚥}_3,\dots ,{𝚥}_{n+1}),\Re ({𝚥}_2,{𝚥}_3,\dots ,{𝚥}_{n+1}))}{n}}\right)\right)}^{{\chi }_4}\hfill \\ & {\left(\nu \left({\displaystyle \frac{{\sum }_{i=1}^{n}G({𝚥}_{i+1},\Re ({𝚥}_1,{𝚥}_2,\dots ,{𝚥}_n),\Re ({𝚥}_1,{𝚥}_2,\dots ,{𝚥}_n))}{n}}\right)\right)}^{{\chi }_5}\hfill \\ & \le {\left(\nu \left(\tilde{G}(({𝚥}_1,{𝚥}_2,\dots ,{𝚥}_n),({𝚥}_2,{𝚥}_3,\dots ,{𝚥}_{n+1}),({𝚥}_2,{𝚥}_3,\dots ,{𝚥}_{n+1}))\right)\right)}^{{\chi }_1}\hfill \\ & {\left(\nu \left({\displaystyle \frac{\tilde{G}(({𝚥}_1,{𝚥}_2,\dots ,{𝚥}_n),\tilde{\Re }({𝚥}_1,{𝚥}_2,\dots ,{𝚥}_n),\tilde{\Re }({𝚥}_1,{𝚥}_2,\dots ,{𝚥}_n))}{\omega }}\right)\right)}^{{\chi }_2}\hfill \\ & {\left(\nu \left(\tilde{G}(({𝚥}_2,{𝚥}_3,\dots ,{𝚥}_{n+1}),\tilde{\Re }({𝚥}_2,{𝚥}_3,\dots ,{𝚥}_{n+1}),\tilde{\Re }({𝚥}_2,{𝚥}_3,\dots ,{𝚥}_{n+1}))\right)\right)}^{{\chi }_3}\hfill \\ & {\left(\nu \left(\tilde{G}(({𝚥}_1,{𝚥}_2,\dots ,{𝚥}_n),\tilde{\Re }({𝚥}_2,{𝚥}_3,\dots ,{𝚥}_{n+1}),\tilde{\Re }({𝚥}_2,{𝚥}_3,\dots ,{𝚥}_{n+1}))\right)\right)}^{{\chi }_4}\hfill \\ & {\left(\nu \left(\tilde{G}(({𝚥}_2,{𝚥}_3,\dots ,{𝚥}_{n+1}),\tilde{\Re }({𝚥}_1,{𝚥}_2,\dots ,{𝚥}_n),\tilde{\Re }({𝚥}_1,{𝚥}_2,\dots ,{𝚥}_n))\right)\right)}^{{\chi }_5},\hfill \end{array}\end{array}$$

We now conclude that $\tilde{\Re }$ admits at least one fixed point by Theorem 3. Consequently, there exist elements ${\mathfrak{b}}_1,\dots ,{\mathfrak{b}}_n\in \Lambda$ such that

$$\Re ({\mathfrak{b}}_1,\dots ,{\mathfrak{b}}_n)={\mathfrak{b}}_1=\dots ={\mathfrak{b}}_n,$$

which shows that ℜ has at least one Prešić-type fixed point. □

6. Application

The following system of Fredholm integral equations can be considered:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mathcal{Q}}_1\left(\mathfrak{b}\right)={\int }_0^{\kappa }K(\mathfrak{b},\varsigma )f(\varsigma ,{\mathcal{Q}}_1\left(\varsigma \right),{\mathcal{Q}}_2\left(\varsigma \right),\dots ,{\mathcal{Q}}_n\left(\varsigma \right))d\varsigma \\ {\mathcal{Q}}_2\left(\mathfrak{b}\right)={\int }_0^{\kappa }K(\mathfrak{b},\varsigma )f(\varsigma ,{\mathcal{Q}}_2\left(\varsigma \right),\dots ,{\mathcal{Q}}_n\left(\varsigma \right),{\mathcal{Q}}_1\left(\varsigma \right))d\varsigma \\ ⋮\\ {\mathcal{Q}}_n\left(\mathfrak{b}\right)={\int }_0^{\kappa }K(\mathfrak{b},\varsigma )f(\varsigma ,{\mathcal{Q}}_n\left(\varsigma \right),{\mathcal{Q}}_1\left(\varsigma \right),\dots ,{\mathcal{Q}}_{n-1}\left(\varsigma \right))d\varsigma .\end{array}\right.\end{array}$$

(11)

Here, $f:[0,\kappa ]\times {\mathbb{R}}^n\to \mathbb{R}$ is a continuous function.

Assume that the set of continuous real functions defined on $I=[0,\kappa ]$ is $\Lambda =C(I,\mathbb{R})$. An existence theorem for a solution of (11) in $\Lambda$ is now provided.

For all $\mathcal{Q}\in \Lambda$, define

$${\parallel \mathcal{Q}\parallel }_{\infty }=\underset{\mathfrak{b}\in I}{sup}\left|\mathcal{Q}\left(\mathfrak{b}\right)\right|.$$

Here, $(\Lambda ,\parallel ·{\parallel }_{\infty })$ is a Banach space.

Evidently, $\Lambda$ is a s.g.m. space if we define

$$\underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″})=\left|\right|\mathfrak{b}-{\mathfrak{b}}^{\prime }{\left|\right|}_{\infty }+\left|\right|{\mathfrak{b}}^{\prime }-{\mathfrak{b}}^{″}{\left|\right|}_{\infty }+\left|\right|{\mathfrak{b}}^{″}-\mathfrak{b}{\left|\right|}_{\infty },\forall \mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″}\in \Lambda .$$

Define $\Re :{\Lambda }^n\to \Lambda$ by

$$\Re (u_i,\dots ,u_n,\dots ,u_{i-1})\left(\mathfrak{b}\right)={\int }_0^{\kappa }K(\mathfrak{b},\varsigma )f(\varsigma ,u_i\left(\varsigma \right),\dots ,u_n\left(\varsigma \right),\dots ,u_{i-1}\left(\varsigma \right))d\varsigma ,u_i\in \Lambda ,\mathfrak{b}\in I.$$

A function $u\in \Lambda$ is, obviously, a solution of (11) if and only if it is a fixed point of ℜ, that is,

$$\Re (u,\dots ,u)=u.$$

Consider the following assumptions:

(1)

For all $u_1,\dots ,u_n,u_1^{\prime },\dots ,u_n^{\prime }\in \Lambda$ and $\forall \varsigma \in I$,

$$|f(\varsigma ,u_i\left(\varsigma \right),\dots ,u_n\left(\varsigma \right),\dots ,u_{i-1}\left(\varsigma \right))-f(\varsigma ,u_i^{\prime }\left(\varsigma \right),\dots ,u_n^{\prime }\left(\varsigma \right),\dots ,u_{i-1}^{\prime }\left(\varsigma \right))|\le \left({\displaystyle \frac{{\sum }_{j=1}^n|u_j\left(\varsigma \right)-u_j^{\prime }\left(\varsigma \right)|}{n}}\right).$$

(2)

$\underset{\mathfrak{b}\in I}{max}{\int }_0^{\kappa }\left|K(\mathfrak{b},\varsigma )\right|d\varsigma <1$.

(3)

There is $(u^0,\dots ,u^0)\in {\Lambda }^n$ so that

$$sup\left\{\underline{\mathcal{G}}\left({\tilde{\Re }}^i\tilde{\mathfrak{b}},{\tilde{\Re }}^j\tilde{\mathfrak{b}},{\tilde{\Re }}^j\tilde{\mathfrak{b}}\right):i,j=1,2,\dots \right\}<\infty$$

in which

$$\tilde{\Re }\tilde{\mathfrak{b}}=\left(\underset{ntimes}{\underbrace{\Re (u^0,\dots ,u^0),\dots ,\Re (u^0,\dots ,u^0)}}\right)$$

and $\tilde{\mathfrak{b}}=(u^0,\dots ,u^0)$.

Theorem 9.

Let us assume that (1)–(3) above are true. The solution to Equation (11) is thus found in Λ.

Proof. 

The verification that ℜ is a $\underline{\mathcal{G}}$-$JS$ contraction constitutes the final step in confirming that all the assumptions of Theorem 3 are satisfied.

We have

$$\begin{array}{cc}\hfill & |\Re (u_i,\dots ,u_n,\dots ,u_{i-1})\left(\mathfrak{b}\right)-\Re (u_i^{\prime },\dots ,u_n^{\prime },\dots ,u_{i-1}^{\prime })\left(\mathfrak{b}\right)|\hfill \\ & \le \underset{\mathfrak{b}\in I}{max}|{\int }_0^{\kappa }K(\mathfrak{b},\varsigma )f(\varsigma ,u_i\left(\varsigma \right),\dots ,u_n\left(\varsigma \right),\dots ,u_{i-1}\left(\varsigma \right))d\varsigma \hfill \\ \hfill & -{\int }_0^{\kappa }K(\mathfrak{b},\varsigma )f(\varsigma ,u_i^{\prime }\left(\varsigma \right),\dots ,u_n^{\prime }\left(\varsigma \right),\dots ,u_{i-1}^{\prime }\left(\varsigma \right))d\varsigma |\hfill \\ \hfill & \le \underset{\mathfrak{b}\in I}{max}{\int }_0^{\kappa }\left|K(\mathfrak{b},\varsigma )\right||f(\varsigma ,u_i\left(\varsigma \right),\dots ,u_n\left(\varsigma \right),\dots ,u_{i-1}\left(\varsigma \right))\hfill \\ \hfill & -f(\varsigma ,u_i^{\prime }\left(\varsigma \right),\dots ,u_n^{\prime }\left(\varsigma \right),\dots ,u_{i-1}^{\prime }\left(\varsigma \right))|d\varsigma \hfill \\ \hfill & \le \underset{\mathfrak{b}\in I}{max}{\int }_0^{\kappa }\left|K(\mathfrak{b},\varsigma )\right|\left({\displaystyle \frac{{\sum }_{j=1}^n|u_j\left(\varsigma \right)-u_j^{\prime }\left(\varsigma \right)|}{n}}\right)d\varsigma \hfill \\ \hfill & \le \underset{\mathfrak{b}\in I}{max}{\int }_0^{\kappa }\left|K(\mathfrak{b},\varsigma )\right|d\varsigma \left({\displaystyle \frac{{\sum }_{j=1}^n\left|\right|u_j-u_j^{\prime }\left|\right|}{n}}\right).\hfill \end{array}$$

So,

$$\begin{array}{cc}\hfill & \underline{\mathcal{G}}(\Re (u_i,\dots ,u_n,\dots ,u_{i-1}),\Re (u_i^{\prime },\dots ,u_n^{\prime },\dots ,u_{i-1}^{\prime }),\Re (u_i^{\prime },\dots ,u_n^{\prime },\dots ,u_{i-1}^{\prime }))\hfill \\ & \le 2\underset{\mathfrak{b}\in I}{max}{\int }_0^{\kappa }\left|K(\mathfrak{b},\varsigma )\right|d\varsigma \left({\displaystyle \frac{{\sum }_{j=1}^n\left|\right|u_j-u_j^{\prime }\left|\right|}{n}}\right)\hfill \\ & =\underset{\mathfrak{b}\in I}{max}{\int }_0^{\kappa }\left|K(\mathfrak{b},\varsigma )\right|d\varsigma \left({\displaystyle \frac{{\sum }_{j=1}^{n}2\left|\right|u_j-u_j^{\prime }\left|\right|}{n}}\right)\hfill \\ & \le \underset{\mathfrak{b}\in I}{max}{\int }_0^{\kappa }\left|K(\mathfrak{b},\varsigma )\right|d\varsigma \left({\displaystyle \frac{{\sum }_{j=1}^n\underline{\mathcal{G}}(u_j,u_j^{\prime },u_j^{\prime })}{n}}\right).\hfill \end{array}$$

From the above inequality, we get

$$\begin{array}{cc}\hfill & e^{\underline{\mathcal{G}}(\Re (u_i,\dots ,u_n,\dots ,u_{i-1}),\Re (u_i^{\prime },\dots ,u_n^{\prime },\dots ,u_{i-1}^{\prime }),\Re (u_i^{\prime },\dots ,u_n^{\prime },\dots ,u_{i-1}^{\prime }))}\hfill \\ \hfill & \le {\left(e^{\left({\displaystyle \frac{{\sum }_{j=1}^n\underline{\mathcal{G}}(u_i,u_i^{\prime },u_i^{\prime })}{n}}\right)}\right)}^{\underset{\mathfrak{b}\in I}{max}{\int }_0^{\kappa }\left|K(\mathfrak{b},\varsigma )\right|d\varsigma }\hfill \end{array}$$

Taking $\nu \left(\varsigma \right)=e^{\varsigma }$ and ${\chi }_1=\underset{\mathfrak{b}\in I}{max}{\int }_0^{\kappa }\left|K(\mathfrak{b},\varsigma )\right|d\varsigma$ and ${\chi }_2={\chi }_3={\chi }_4={\chi }_5=0$, we get

$$\begin{array}{cc}\hfill \nu (\underline{\mathcal{G}}(\Re (u_i,\dots ,u_n,\dots ,u_{i-1}),\Re (u_i^{\prime },\dots ,u_n^{\prime },\dots ,u_{i-1}^{\prime }),& \Re (u_i^{\prime },\dots ,u_n^{\prime },\dots ,u_{i-1}^{\prime })\left)\right)\hfill \\ \hfill \le & {\left(\nu \left({\displaystyle \frac{{\sum }_{j=1}^n\underline{\mathcal{G}}(u_i,u_i^{\prime },u_i^{\prime })}{n}}\right)\right)}^{{\chi }_1}.\hfill \end{array}$$

As a consequence, it is evident that the mapping ℜ satisfies all the assumptions of Theorem 3. Therefore, the Fredholm integral Equation (11) admits a unique solution, since ℜ possesses a unique fixed point. □

Remark 6.

We can prove that $(\Lambda ,\underline{\mathcal{G}})$ is a complete s.g.m. space if we choose

$$\underline{\mathcal{G}}(\mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″})=sinh\left(\left|\right|\mathfrak{b}-{\mathfrak{b}}^{\prime }{\left|\right|}_{\infty }+\left|\right|{\mathfrak{b}}^{\prime }-{\mathfrak{b}}^{″}{\left|\right|}_{\infty }+\left|\right|{\mathfrak{b}}^{″}-\mathfrak{b}{\left|\right|}_{\infty }\right),\forall \mathfrak{b},{\mathfrak{b}}^{\prime },{\mathfrak{b}}^{″}\in \Lambda .$$

This mapping cannot be a usual G-metric.

7. Example

The following system of Fredholm integral equations is an example of system (11):

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mathcal{Q}}_1\left(\mathfrak{b}\right)={\int }_0^1{\displaystyle \frac{e^{-\mathfrak{b}\varsigma }}{2}}({sinh}^{-1}{\displaystyle \frac{{\mathcal{Q}}_1\left(\varsigma \right)}{2}}+{sinh}^{-1}{\displaystyle \frac{{\mathcal{Q}}_2\left(\varsigma \right)}{3}})d\varsigma \\ {\mathcal{Q}}_2\left(\mathfrak{b}\right)={\int }_0^1{\displaystyle \frac{e^{-\mathfrak{b}\varsigma }}{2}}({sinh}^{-1}{\displaystyle \frac{{\mathcal{Q}}_2\left(\varsigma \right)}{2}}+{sinh}^{-1}{\displaystyle \frac{{\mathcal{Q}}_1\left(\varsigma \right)}{3}})d\varsigma \end{array}\right.\end{array}$$

(12)

where $f:[0,1]\times {\mathbb{R}}^2\to \mathbb{R}$ is defined as follows:

$$f(\varsigma ,u_1\left(\varsigma \right),u_2\left(\varsigma \right))={sinh}^{-1}{\displaystyle \frac{u_1\left(\varsigma \right)}{2}}+{sinh}^{-1}{\displaystyle \frac{u_2\left(\varsigma \right)}{3}}.$$

Also, we know that $K(\mathfrak{b},\varsigma )={\displaystyle \frac{e^{-\mathfrak{b}\varsigma }}{2}}.$

We consider the set of continuous real functions defined on $I=[0,1]$.

Now, $\Re :{\Lambda }^2\to \Lambda$ is defined by

$$\Re (u_1,u_2)\left(\mathfrak{b}\right)={\int }_0^1{\displaystyle \frac{e^{-\mathfrak{b}\varsigma }}{2}}[{sinh}^{-1}{\displaystyle \frac{u_1\left(\varsigma \right)}{2}}+{sinh}^{-1}{\displaystyle \frac{u_2\left(\varsigma \right)}{3}}]d\varsigma ,u_i\in \Lambda ,\mathfrak{b}\in I.$$

Now we check the satisfaction of the presumptions of Theorem 9.

(1)

For all $u_1,u_2,u_1^{\prime },u_2^{\prime }\in \Lambda$ and $\forall \varsigma \in I$, according to the mean value theorem we can see that

$$|f(\varsigma ,u_1\left(\varsigma \right),u_2\left(\varsigma \right))-f(\varsigma ,u_1^{\prime }\left(\varsigma \right),u_2^{\prime }\left(\varsigma \right))|\le \left({\displaystyle \frac{{\sum }_{j=1}^2|u_j\left(\varsigma \right)-u_j^{\prime }\left(\varsigma \right)|}{2}}\right).$$

(2)

$\underset{\mathfrak{b}\in I}{max}{\int }_0^1\left|K(\mathfrak{b},\varsigma )\right|d\varsigma =\underset{\mathfrak{b}\in I}{max}{\int }_0^1{\displaystyle \frac{e^{-\mathfrak{b}\varsigma }}{2}}d\varsigma =\underset{\mathfrak{b}\in I}{max}{\displaystyle \frac{1-e^{-\mathfrak{b}}}{2\mathfrak{b}}}<1$.

(3)

There is $(u^0,u^0)=(0,0)\in {\Lambda }^2$ so that

$$sup\left\{\underline{\mathcal{G}}\left({\tilde{\Re }}^i\tilde{\mathfrak{b}},{\tilde{\Re }}^j\tilde{\mathfrak{b}},{\tilde{\Re }}^j\tilde{\mathfrak{b}}\right):i,j=1,2,\dots \right\}=0<\infty$$

in which

$$\tilde{\Re }\tilde{\mathfrak{b}}=\left(\underbrace{\Re (0,0),\Re (0,0)}\right)$$

and $\tilde{\mathfrak{b}}=(0,0)$.

As a result, it is clear that every one of the Theorem 9 requirements is satisfied. So, the system of Fredholm integral Equation (12) has a unique solution.

8. Conclusions

In this paper, we presented several fixed-point results in the framework of s.g.m. spaces. In fact, we obtained some new fixed-point results in the context of s.g.m. spaces by combining the concepts of Jleli-Samet, Hardy–Rogers, and Prešić contractions. In our novel framework, we finally solved a system of Fredholm-type integral equations. Now some questions come to mind, which are as follows.

Is it possible to prove the results of this paper about Prešić mappings by replacing the liminf with the limsup in an s.g.m. space?

Is it possible to remove the symmetry condition of the space from the results obtained in this paper?

Is it possible to prove the existence of a fixed point for mappings that satisfy other contraction conditions such as Wardowski contractions, weak-Wardowski contractions, etc.?