On Generalized Sehgal–Guseman-Like Contractions and Their Fixed-Point Results with Applications to Nonlinear Fractional Differential Equations and Boundary Value Problems for Homogeneous Transverse Bars

Abstract

The goal of this study is to describe the class of modified Sehgal–Guseman-like contraction mappings and set up some fixed-point results in $\mathcal{S}$-metric spaces. The class of generalized Sehgal–Guseman-like contraction mappings contains enhancements of Banach contractions, Kannan contractions, Chatterjee contractions, Chatterjee-type contractions, quasi-contractions, Ćirić–Reich–Rus-type contractions, Hardy–Rogers-type contractions, Reich-type contractions, interpolative Kannan contractions, interpolative Chatterjee contractions, among others, with their generalizations in $\mathcal{S}$-metric spaces. We offer significant examples to substantiate the reliability of our results. This study also establishes consequential fixed-point results and applies them to nonlinear fractional differential equations and the boundary value problem for homogeneous transverse bars. At the end of the manuscript, we present an important open problem.

1. Introduction and Preliminaries

A core philosophy in applied mathematical sciences—particularly dynamical systems and nonlinear functional analysis—is fixed-point theory. Since its inception, the Banach principle has been a foundational pillar, providing support for subsequent developments. Sehgal gave it a more comprehensive viewpoint, which greatly aided in its growth. Sehgal investigated fixed points concerning mappings that show a contractive effect at all points within the space [1]. The primary contribution and objective of this study is the development of the following important theorem:

Theorem 1

([1]). In a complete metric space $({\mathring{\mathcal{Y}}}_{ϵ},\rho )$, consider a continuous self-mapping Γ. If, for each $\mathfrak{p}\in {\mathring{\mathcal{Y}}}_{ϵ}$, there exists $\xi \left(\mathfrak{p}\right)\in \mathbb{N}$, such that

$$\rho ({\mathrm{\Gamma }}^{\xi \left(\mathfrak{p}\right)}\mathfrak{p},{\mathrm{\Gamma }}^{\xi \left(\mathfrak{p}\right)}\mathfrak{d})\le \varphi \rho (\mathfrak{p},\mathfrak{d}),\forall \mathfrak{d}\in {\mathring{\mathcal{Y}}}_{ϵ}$$

where $0\le \varphi <1$, then Γ possesses a unique fixed point.

Afterward, Guseman [2] presented his fixed-point theorem extension, expanding upon Theorem 1. Additionally, he demonstrated that this theorem holds even when the assumption of continuity is eliminated.

Numerous real-world dilemmas revolve around determining distances between items, a task often hindered by accuracy challenges. Thus, to model diverse practical scenarios, employing an appropriate metric becomes imperative. Various methodologies exist for precisely gauging these distances, thereby broadening the horizons of fixed-point theory investigations. Researchers have extensively delved into both unique and non-unique fixed-point solutions, exploring them from different angles using diverse metrics. Ćirić [3] demonstrated that fixed points exist uniquely for generalized contraction mappings in a metric space, provided that the mapping exhibits orbital continuity. Ali et al. [4] discussed the dynamic process through integral contractions of the Ćirić kind. Anevska et al. [5] provided proofs for some fixed-point results in 2-Banach spaces. Hammad [6] formulated fixed-point methodologies to address a category of matrix difference equations associated with a novel collection of contractions. Hardy [7] established a generalization of Banach’s theorem in a distinctive manner. Almalki et al. [8] established some Perov-type results with theoretic order. Nazam et al. [9] derived fixed points for a distinct category of contractions within partial b-metric spaces. Zhou et al. [10] broadened the notion of Ulam–Hyers stability to diverse categories of $(\psi ,\varphi )$-Meir–Keeler mappings through the application of fixed-point theory in S-metric spaces. Additionally, Zhou and Liu [11] investigated the study of nonlinear contractions exhibiting the mixed weakly monotone property in partially ordered metric spaces. Joshi et al. [12] conducted an in-depth investigation into fixed points and the corresponding geometry for a specific category of mappings within $\mathcal{S}$-metric spaces. Their work also delved into the intricate problem of satellite web coupling. On partial metric spaces, pioneering the concept of interpolative-type contractions, Karapinar [13] made substantial contributions to the field. Özgür and Taş [14] approached the fixed-circle dilemma within $\mathcal{S}$-metric spaces through a geometric perspective. In the realm of fixed points, Phaneendra and Swamy [15] expanded the study to encompass Chatterjee and Ćirić contractions in S-metric spaces. Additionally, Phaneendra [16] established results concerning Banach and Kannan contractions within $\mathcal{S}$-metric spaces. Reich [17] provided valuable insights and remarks regarding contraction self-mappings. Sedghi et al. [18] took on the task of generalizing fixed-point theorems in S-metric spaces, thereby making noteworthy contributions to the advancement of this research area. Aydi et al. [19] provided an examination of Ri’s result through w-distances, specifically in the context of studying nonlinear fractional integro-differential equations of the Caputo type. Baleanu et al. [20] focused on establishing the existence of solutions for fractional differential equations. Baleanu et al. [21] explored the dynamics of a bead sliding on a wire within the framework of fractional motion.

Recently, there has been a shift in focus toward scrutinizing the geometry of sets that encompass fixed points in different arrangements. These inquiries encompass problems such as fixed circles/discs, fixed ellipses/elliptic discs, and more. The most comprehensive among these is termed the “fixed figure problem”.

In this study, we define the notion of the $\mathcal{D}$-class, which consists of functions satisfying some properties. Using this class of functions, we define the generalized Sehgal–Guseman-like contraction and establish certain findings concerning unique fixed points in $\mathcal{S}$-metric spaces. The investigation establishes significant fixed-point outcomes and employs them in addressing nonlinear fractional equations and the boundary value problem associated with homogeneous transverse bars.

Let us revisit several fundamental definitions that are consistently utilized within the text.

Definition 1

([18]). Let ${\mathring{\mathcal{Y}}}_{ϵ}$ be a nonempty set. Then, the $\mathcal{S}$-metric is a function ${\mathcal{E}}_s:{\mathring{\mathcal{Y}}}_{ϵ}^3\to [0,+\infty )$ such that $\forall \mathfrak{p},\mathfrak{d},\mathfrak{z}\in {\mathring{\mathcal{Y}}}_{ϵ}$:

• $\left({\mathring{\delta }}_1\right)$. ${\mathcal{E}}_s(\mathfrak{p},\mathfrak{d},\mathfrak{z})=0⟺\mathfrak{p}=\mathfrak{d}=\mathfrak{z}$;
• $\left({\mathring{\delta }}_2\right)$. ${\mathcal{E}}_s(\mathfrak{p},\mathfrak{d},\mathfrak{z})\le {\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},\mathfrak{u})+{\mathcal{E}}_s(\mathfrak{d},\mathfrak{d},\mathfrak{u})+{\mathcal{E}}_s(\mathfrak{z},\mathfrak{z},\mathfrak{u}).$

In terms of geometry, when we link three points, $\mathfrak{p},\mathfrak{d}$, and $\mathfrak{z}$, to form a triangle, and if $\mathfrak{u}$ represents a point that mediates within this triangle, then $\left({\mathring{\delta }}_2\right)$ is valid.

Remark 1

([18]). For all $\mathfrak{p},\mathfrak{z}$ in the $\mathcal{S}$-metric space, the following holds:

$${\mathcal{E}}_s(\mathfrak{x},\mathfrak{p},\mathfrak{z})={\mathcal{E}}_s(\mathfrak{z},\mathfrak{z},\mathfrak{p}).$$

Definition 2

([18]). For any sequence ${\left\{{\mathfrak{d}}_i\right\}}_{i=1}^{+\infty }$ in the $\mathcal{S}$-metric space ${\mathring{\mathcal{Y}}}_{ϵ}$, we have the following:

(1) ${\left\{{\mathfrak{d}}_i\right\}}_{i=1}^{+\infty }$ is convergent to some element $\mathfrak{z}\in {\mathring{\mathcal{Y}}}_{ϵ}$ if ${lim}_{i\to \infty }{\mathcal{E}}_s({\mathfrak{d}}_i,{\mathfrak{d}}_i,\mathfrak{z})=0$;

(2) ${\left\{{\mathfrak{d}}_i\right\}}_{i=1}^{+\infty }$ is a Cauchy sequence if ${lim}_{i,j\to \infty }{\mathcal{E}}_s({\mathfrak{d}}_i,{\mathfrak{d}}_i,{\mathfrak{d}}_j)=0,i,j\in \mathbb{N}$;

(3) The $\mathcal{S}$-metric space is complete iff each Cauchy sequence in ${\mathring{\mathcal{Y}}}_{ϵ}$ converges in ${\mathring{\mathcal{Y}}}_{ϵ}$.

2. Main Results

The notion of $\mathcal{D}$-class functions in an $\mathcal{S}$-metric space is first introduced. Contraction maps can be used to identify fixed points and examine their geometric properties with the help of these functions. They also guarantee the existence of fixed points of self-maps. They serve to expand, refine, generalize, and streamline several well-established findings from the literature on $\mathcal{S}$-metric spaces.

Definition 3.

Let $\mathcal{D}$ be the collection of all continuous mappings $\mathfrak{g}:{\mathbb{R}}_{+}^5\to {\mathbb{R}}_{+}$ such that the following holds:

• For all $\mathfrak{p},\mathfrak{d},\mathfrak{z}\in {\mathbb{R}}_{+}$, if $\mathfrak{z}\le \mathfrak{g}[\mathfrak{p},2\mathfrak{p}+\mathfrak{z},\mathfrak{z},\mathfrak{p}+2\mathfrak{d},\mathfrak{d}]$ with $\mathfrak{d}\le 2(\mathfrak{p}+\mathfrak{z})$, then $\mathfrak{z}\le \varphi \mathfrak{p}$ for some $\varphi \in [0,1)$;
• If $\mathbf{w}\le \mathbf{h}$, then $\mathfrak{g}\left(\mathbf{w}\right)\le \mathfrak{g}\left(\mathbf{h}\right)$ for all $\mathbf{w},\mathbf{h}\in {\mathbb{R}}_{+}^5$.

Example 1.

Define ${\mathfrak{g}}_1:{\mathbb{R}}_{+}^5\to {\mathbb{R}}_{+}$ such that ${\mathfrak{g}}_1({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5)=\varphi {\alpha }_1$, where $\varphi \in [0,1)$. Then, ${\mathfrak{g}}_1\in \mathcal{D}$.

Example 2.

Define ${\mathfrak{g}}_2:{\mathbb{R}}_{+}^5\to {\mathbb{R}}_{+}$ such that ${\mathfrak{g}}_2({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5)=\varphi {\alpha }_1+\psi {\alpha }_2+\xi [{\alpha }_3+{\alpha }_4]+\nabla {\alpha }_5$, where $\varphi +\psi +3\xi +\nabla \in [0,1)$. Then, ${\mathfrak{g}}_2\in \mathcal{D}$.

Example 3.

Define ${\mathfrak{g}}_3:{\mathbb{R}}_{+}^5\to {\mathbb{R}}_{+}$ such that ${\mathfrak{g}}_3({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5)=\varphi ·{\alpha }_1^{\xi }·{\alpha }_2^{\psi }·{\alpha }_5^{1-\psi -\xi }$, where $\varphi \in [0,1)$ and $\psi +\xi \in (0,1)$. Then, ${\mathfrak{g}}_3\in \mathcal{D}$.

Definition 4.

Let $({\mathring{\mathcal{Y}}}_{ϵ},{\mathcal{E}}_s)$ be an $\mathcal{S}$-metric space. A mapping $\mathrm{\Gamma }:{\mathring{\mathcal{Y}}}_{ϵ}\to {\mathring{\mathcal{Y}}}_{ϵ}$ is considered as a generalized Sehgal–Guseman-like contraction if there exists a positive integer a and $\mathfrak{g}\in \mathcal{D}$ such that:

$$\begin{array}{cc}\hfill {\mathcal{E}}_s\left({\mathrm{\Gamma }}^a\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{d}\right)\le \mathfrak{g}[{\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},\mathfrak{d}),& {\mathcal{E}}_s({\mathrm{\Gamma }}^a\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{p},\mathfrak{p}),{\mathcal{E}}_s({\mathrm{\Gamma }}^a\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{p},\mathfrak{d}),\hfill \\ & {\mathcal{E}}_s({\mathrm{\Gamma }}^a\mathfrak{d},{\mathrm{\Gamma }}^a\mathfrak{d},\mathfrak{p}),{\mathcal{E}}_s({\mathrm{\Gamma }}^a\mathfrak{d},{\mathrm{\Gamma }}^a\mathfrak{d},\mathfrak{d})],\forall \mathfrak{p},\mathfrak{d}\in {\mathring{\mathcal{Y}}}_{ϵ}\hfill \end{array}$$

provided that ${\mathcal{E}}_s({\mathrm{\Gamma }}^a\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{d})>0$.

Theorem 2.

Let $({\mathring{\mathcal{Y}}}_{ϵ},{\mathcal{E}}_s)$ be a complete $\mathcal{S}$-metric space and $\mathrm{\Gamma }:{\mathring{\mathcal{Y}}}_{ϵ}\to {\mathring{\mathcal{Y}}}_{ϵ}$ a generalized Sehgal–Guseman-type contraction. Then, Γ possesses a unique fixed point.

Proof. 

Step 1: If there exists an $n_0\in \mathbb{N}$, such that ${\mathfrak{d}}_{n_0}={\mathfrak{d}}_{n_0+1}$, then ${\mathfrak{d}}_{n_0}$ serves as a fixed point for $\mathrm{\Gamma }$. Since ${\mathfrak{d}}_{n_0}={\mathrm{\Gamma }}^a{\mathfrak{d}}_{n_0},{\mathfrak{d}}_{n_0}$ is a fixed point of ${\mathrm{\Gamma }}^a$. To show that ${\mathfrak{d}}_{n_0}$ is a fixed point of $\mathrm{\Gamma }$, we first show that ${\mathfrak{d}}_{n_0}$ is the only fixed point of ${\mathrm{\Gamma }}^a$. In fact, if ${\mathrm{\Gamma }}^{a}v=v$ for some $v\ne {\mathfrak{d}}_{n_0}$, then we use the definition of $\mathfrak{g}\in \mathcal{D}$ to obtain the following:

$$\begin{array}{cc}\hfill {\mathcal{E}}_s({\mathfrak{d}}_{n_0},{\mathfrak{d}}_{n_0},v)& ={\mathcal{E}}_s({\mathrm{\Gamma }}^a{\mathfrak{d}}_{n_0},{\mathrm{\Gamma }}^a{\mathfrak{d}}_{n_0},{\mathrm{\Gamma }}^{a}v)\hfill \\ & \le \mathfrak{g}[{\mathcal{E}}_s({\mathfrak{d}}_{n_0},{\mathfrak{d}}_{n_0},v),{\mathcal{E}}_s({\mathrm{\Gamma }}^a{\mathfrak{d}}_{n_0},{\mathrm{\Gamma }}^a{\mathfrak{d}}_{n_0},{\mathfrak{d}}_{n_0}),{\mathcal{E}}_s({\mathrm{\Gamma }}^a{\mathfrak{d}}_{n_0},{\mathrm{\Gamma }}^a{\mathfrak{d}}_{n_0},v),\hfill \\ & {\mathcal{E}}_s({\mathrm{\Gamma }}^{a}v,{\mathrm{\Gamma }}^{a}v,{\mathfrak{d}}_{n_0}),{\mathcal{E}}_s({\mathrm{\Gamma }}^{a}v,{\mathrm{\Gamma }}^{a}v,v)]\hfill \\ & =\mathfrak{g}[{\mathcal{E}}_s({\mathfrak{d}}_{n_0},{\mathfrak{d}}_{n_0},v),{\mathcal{E}}_s({\mathfrak{d}}_{n_0},{\mathfrak{d}}_{n_0},{\mathfrak{d}}_{n_0}),{\mathcal{E}}_s({\mathfrak{d}}_{n_0},{\mathfrak{d}}_{n_0},v),{\mathcal{E}}_s(v,v,{\mathfrak{d}}_{n_0}),{\mathcal{E}}_s(v,v,v)]\hfill \\ & =\mathfrak{g}[{\mathcal{E}}_s({\mathfrak{d}}_{n_0},{\mathfrak{d}}_{n_0},v),0,{\mathcal{E}}_s({\mathfrak{d}}_{n_0},{\mathfrak{d}}_{n_0},v),{\mathcal{E}}_s({\mathfrak{d}}_{n_0},{\mathfrak{d}}_{n_0},v),0]\hfill \\ & \le \varphi {\mathcal{E}}_s({\mathfrak{d}}_{n_0},{\mathfrak{d}}_{n_0},v),\hfill \end{array}$$

which is a contradiction, as $0\le \varphi <1$. Then, $\mathrm{\Gamma }{\mathfrak{d}}_{n_0}=\mathrm{\Gamma }{\mathrm{\Gamma }}^a{\mathfrak{d}}_{n_0}={\mathrm{\Gamma }}^a\mathrm{\Gamma }{\mathfrak{d}}_{n_0}$; that is, $\mathrm{\Gamma }{\mathfrak{d}}_{n_0}$ is also a fixed point of ${\mathrm{\Gamma }}^a$. By the uniqueness of the fixed point of ${\mathrm{\Gamma }}^a$, we have $\mathrm{\Gamma }{\mathfrak{d}}_{n_0}={\mathfrak{d}}_{n_0}$, which shows that ${\mathfrak{d}}_{n_0}$ is a fixed point of $\mathrm{\Gamma }$.

Step 2: For every $n\in \mathbb{N}$, we will consider the case where ${\mathfrak{d}}_n$ is not equal to ${\mathfrak{d}}_{n+1}$. Our aim is to establish that when n is not equal to m, then ${\mathfrak{d}}_n$ cannot be equal to ${\mathfrak{d}}_m$. Let us assume, for simplicity, that $m>n$. If ${\mathfrak{d}}_n={\mathfrak{d}}_m$ for $n\ne m$, then

$$\begin{array}{cc}\hfill S\left({\mathfrak{d}}_m,{\mathfrak{d}}_m,{\mathfrak{d}}_{m+1}\right)& =S\left({\mathrm{\Gamma }}^a{\mathfrak{d}}_{m-1},{\mathrm{\Gamma }}^a{\mathfrak{d}}_{m-1},{\mathrm{\Gamma }}^a{\mathfrak{d}}_m\right)\hfill \\ \hfill & \le \mathfrak{g}[S\left({\mathfrak{d}}_{m-1},{\mathfrak{d}}_{m-1},{\mathfrak{d}}_m\right),S({\mathrm{\Gamma }}^a{\mathfrak{d}}_{m-1},{\mathrm{\Gamma }}^a{\mathfrak{d}}_{m-1},{\mathfrak{d}}_{m-1}),S({\mathrm{\Gamma }}^a{\mathfrak{d}}_{m-1},{\mathrm{\Gamma }}^a{\mathfrak{d}}_{m-1},{\mathfrak{d}}_m),\hfill \\ & S({\mathrm{\Gamma }}^a{\mathfrak{d}}_m,{\mathrm{\Gamma }}^a{\mathfrak{d}}_m,{\mathfrak{d}}_{m-1}),S({\mathrm{\Gamma }}^a{\mathfrak{d}}_m,{\mathrm{\Gamma }}^a{\mathfrak{d}}_m,{\mathfrak{d}}_m)]\hfill \\ \hfill & =\mathfrak{g}[S\left({\mathfrak{d}}_{m-1},{\mathfrak{d}}_{m-1},{\mathfrak{d}}_m\right),S\left({\mathfrak{d}}_m,{\mathfrak{d}}_m,{\mathfrak{d}}_{m-1}\right),S\left({\mathfrak{d}}_m,{\mathfrak{d}}_m,{\mathfrak{d}}_m\right),\hfill \\ \hfill & S\left({\mathfrak{d}}_{m+1},{\mathfrak{d}}_{m+1},{\mathfrak{d}}_{m-1}\right),S\left({\mathfrak{d}}_{m+1},{\mathfrak{d}}_{m+1},{\mathfrak{d}}_m\right)]\hfill \\ \hfill & \le \mathfrak{g}[S\left({\mathfrak{d}}_{m-1},{\mathfrak{d}}_{m-1},{\mathfrak{d}}_m\right),S\left({\mathfrak{d}}_{m-1},{\mathfrak{d}}_{m-1},{\mathfrak{d}}_m\right),0,\hfill \\ \hfill & S\left({\mathfrak{d}}_{m-1},{\mathfrak{d}}_{m-1},{\mathfrak{d}}_m\right)+2S\left({\mathfrak{d}}_m,{\mathfrak{d}}_m,{\mathfrak{d}}_{m+1}\right),S\left({\mathfrak{d}}_m,{\mathfrak{d}}_m,{\mathfrak{d}}_{m+1}\right)]\hfill \\ \hfill & \le \varphi S\left({\mathfrak{d}}_{m-1},{\mathfrak{d}}_{m-1},{\mathfrak{d}}_m\right)\hfill \\ \hfill & =\varphi S\left({\mathrm{\Gamma }}^a{\mathfrak{d}}_{m-2},{\mathrm{\Gamma }}^a{\mathfrak{d}}_{m-2},{\mathrm{\Gamma }}^a{\mathfrak{d}}_{m-1}\right)\hfill \\ \hfill & \le {\varphi }^{2}S\left({\mathfrak{d}}_{m-2},{\mathfrak{d}}_{m-2},{\mathfrak{d}}_{m-1}\right)\hfill \\ \hfill & \le \cdots \cdots \hfill \\ \hfill & \le {\varphi }^{m-n}S\left({\mathfrak{d}}_n,{\mathfrak{d}}_n,{\mathfrak{d}}_{n+1}\right)\hfill \\ \hfill & ={\varphi }^{m-n}S\left({\mathfrak{d}}_m,{\mathfrak{d}}_m,{\mathfrak{d}}_{m+1}\right)\hfill \\ \hfill & ={\varphi }^{m-n}S\left({\mathfrak{d}}_m,{\mathfrak{d}}_m,{\mathfrak{d}}_{m+1}\right).\hfill \end{array}$$

This assertion does not hold valid due to the condition $0\le \varphi <1$.

Step 3: At this stage, we demonstrate that for any $\mathfrak{d}\in {\mathring{\mathcal{Y}}}_{ϵ}$, the value of

$r\left(\mathfrak{d}\right)={sup}_n{\mathcal{E}}_s\left({\mathrm{\Gamma }}^n\mathfrak{d},{\mathrm{\Gamma }}^n\mathfrak{d},\mathfrak{d}\right)$ is finite.

Suppose for some $\mathfrak{d}\in {\mathring{\mathcal{Y}}}_{ϵ}$ and

$$z\left(\mathfrak{d}\right)=max\left\{{\mathcal{E}}_s\left({\mathrm{\Gamma }}^k\mathfrak{d},{\mathrm{\Gamma }}^k\mathfrak{d},\mathfrak{d}\right):k=1,2,\cdots ,a,\cdots ,ta,\cdots \right\}.$$

For every positive integer n, there exists a non-negative integer t satisfying

$t·a<n\le (t+1)·a$. We assume distinctness among ${\mathrm{\Gamma }}^n\mathfrak{d}$, ${\mathrm{\Gamma }}^a\mathfrak{d}$, and $\mathfrak{d}$. Otherwise, the conclusion becomes evident without further explanation.

$$\begin{array}{cc}\hfill {\mathcal{E}}_s\left({\mathrm{\Gamma }}^n\mathfrak{d},{\mathrm{\Gamma }}^n\mathfrak{d},\mathfrak{d}\right)& \le {\mathcal{E}}_s\left({\mathrm{\Gamma }}^n\mathfrak{d},{\mathrm{\Gamma }}^n\mathfrak{d},{\mathrm{\Gamma }}^a\mathfrak{d}\right)+2{\mathcal{E}}_s\left({\mathrm{\Gamma }}^a\mathfrak{d},{\mathrm{\Gamma }}^a\mathfrak{d},\mathfrak{d}\right)\hfill \\ \hfill & \le {\mathcal{E}}_s\left({\mathrm{\Gamma }}^a\left({\mathrm{\Gamma }}^{n-a}\mathfrak{d}\right),{\mathrm{\Gamma }}^a\left({\mathrm{\Gamma }}^{n-a}\mathfrak{d}\right),{\mathrm{\Gamma }}^a\mathfrak{d}\right)+2z\left(\mathfrak{d}\right)\hfill \\ \hfill & \le \mathfrak{g}[{\mathcal{E}}_s({\mathrm{\Gamma }}^{n-a}\mathfrak{d},{\mathrm{\Gamma }}^{n-a}\mathfrak{d},\mathfrak{d}),{\mathcal{E}}_s({\mathrm{\Gamma }}^n\mathfrak{d},{\mathrm{\Gamma }}^n\mathfrak{d},{\mathrm{\Gamma }}^{n-a}\mathfrak{d}),{\mathcal{E}}_s({\mathrm{\Gamma }}^n\mathfrak{d},{\mathrm{\Gamma }}^n\mathfrak{d},\mathfrak{d}),\hfill \\ & {\mathcal{E}}_s({\mathrm{\Gamma }}^a\mathfrak{d},{\mathrm{\Gamma }}^a\mathfrak{d},{\mathrm{\Gamma }}^{n-a}\mathfrak{d})),{\mathcal{E}}_s({\mathrm{\Gamma }}^a\mathfrak{d},{\mathrm{\Gamma }}^a\mathfrak{d},\mathfrak{d})]+2z\left(\mathfrak{d}\right)\hfill \\ \hfill & \le \mathfrak{g}[{\mathcal{E}}_s({\mathrm{\Gamma }}^{n-a}\mathfrak{d},{\mathrm{\Gamma }}^{n-a}\mathfrak{d},\mathfrak{d}),2{\mathcal{E}}_s({\mathrm{\Gamma }}^{n-a}\mathfrak{d},{\mathrm{\Gamma }}^{n-a}\mathfrak{d},\mathfrak{d})+{\mathcal{E}}_s\left({\mathrm{\Gamma }}^n\mathfrak{d},{\mathrm{\Gamma }}^n\mathfrak{d},\mathfrak{d}\right),{\mathcal{E}}_s\left({\mathrm{\Gamma }}^n\mathfrak{d},{\mathrm{\Gamma }}^n\mathfrak{d},\mathfrak{d}\right),\hfill \\ & {\mathcal{E}}_s({\mathrm{\Gamma }}^{n-a}\mathfrak{d},{\mathrm{\Gamma }}^{n-a}\mathfrak{d},\mathfrak{d})+2{\mathcal{E}}_s({\mathrm{\Gamma }}^a\mathfrak{d},{\mathrm{\Gamma }}^a\mathfrak{d},\mathfrak{d}),{\mathcal{E}}_s({\mathrm{\Gamma }}^a\mathfrak{d},{\mathrm{\Gamma }}^a\mathfrak{d},\mathfrak{d})]+2z\left(\mathfrak{d}\right)\hfill \\ \hfill & \le \varphi {\mathcal{E}}_s({\mathrm{\Gamma }}^{n-a}\mathfrak{d},{\mathrm{\Gamma }}^{n-a}\mathfrak{d},\mathfrak{d})+2z\left(\mathfrak{d}\right)\hfill \\ \hfill & \le \varphi [{\mathcal{E}}_s({\mathrm{\Gamma }}^{n-a}\mathfrak{d},{\mathrm{\Gamma }}^{n-a}\mathfrak{d},{\mathrm{\Gamma }}^a\mathfrak{d})+2{\mathcal{E}}_s({\mathrm{\Gamma }}^a\mathfrak{d},{\mathrm{\Gamma }}^a\mathfrak{d},\mathfrak{d})]+2z\left(\mathfrak{d}\right)\hfill \\ \hfill & \le {\varphi }^2{\mathcal{E}}_s\left({\mathrm{\Gamma }}^{n-2a}\mathfrak{d},{\mathrm{\Gamma }}^{n-2a}\mathfrak{d},\mathfrak{d}\right)+2z\left(\mathfrak{d}\right)+2\varphi z\left(\mathfrak{d}\right)\hfill \\ \hfill & \le \cdots \cdots \hfill \\ \hfill & \le {\varphi }^t{\mathcal{E}}_s\left({\mathrm{\Gamma }}^{n-ta}\mathfrak{d},{\mathrm{\Gamma }}^{n-ta}\mathfrak{d},\mathfrak{d}\right)+2[z\left(\mathfrak{d}\right)+\varphi z\left(\mathfrak{d}\right)+{\varphi }^{2}z\left(\mathfrak{d}\right)+\cdots ]\hfill \\ \hfill & \le {\varphi }^{t}z\left(\mathfrak{d}\right)+\frac{2z\left(\mathfrak{d}\right)}{1-\varphi }\hfill \\ \hfill & \le z\left(\mathfrak{d}\right)+\frac{2z\left(\mathfrak{d}\right)}{1-\varphi }.\hfill \end{array}$$

Consequently, the finiteness of $r\left(\mathfrak{d}\right)={sup}_n{\mathcal{E}}_s\left({\mathrm{\Gamma }}^n\mathfrak{d},{\mathrm{\Gamma }}^n\mathfrak{d},\mathfrak{d}\right)$ follows.

Step 4: We proceed to establish that $\left({\mathfrak{d}}_n\right)$ forms a Cauchy sequence. To begin, we need to demonstrate that ${lim}_{n\to \infty }{\mathcal{E}}_s\left({\mathfrak{d}}_n,{\mathfrak{d}}_n,{\mathfrak{d}}_{n+1}\right)=0$. We utilize the definition of $\mathfrak{g}$ within $\mathcal{D}$, and from Step 2, we obtain the following:

$$\begin{array}{cc}\hfill {\mathcal{E}}_s({\mathfrak{d}}_n,{\mathfrak{d}}_n,{\mathfrak{d}}_{n+1})& ={\mathcal{E}}_s({\mathrm{\Gamma }}^a{\mathfrak{d}}_{n-1},{\mathrm{\Gamma }}^a{\mathfrak{d}}_{n-1},{\mathrm{\Gamma }}^a{\mathfrak{d}}_n)\hfill \\ & \le \varphi {\mathcal{E}}_s\left({\mathfrak{d}}_{n-1},{\mathfrak{d}}_{n-1},{\mathfrak{d}}_n\right)\hfill \\ & \le \cdots \cdots \hfill \\ & \le {\varphi }^n{\mathcal{E}}_s\left({\mathfrak{d}}_0,{\mathfrak{d}}_0,{\mathfrak{d}}_1\right)\hfill \\ & \le {\varphi }^{n}r\left({\mathfrak{d}}_0\right).\hfill \end{array}$$

Then,

$$\underset{n\to \infty }{lim}{\mathcal{E}}_s\left({\mathfrak{d}}_n,{\mathfrak{d}}_n,{\mathfrak{d}}_{n+1}\right)=0.$$

Now, for $m,n\in \mathbb{N}$ with $m>n$, we obtain the following:

$$\begin{array}{cc}\hfill {\mathcal{E}}_s({\mathfrak{d}}_n,{\mathfrak{d}}_n,{\mathfrak{d}}_m)& \le 2{\mathcal{E}}_s({\mathfrak{d}}_n,{\mathfrak{d}}_n,{\mathfrak{d}}_{n+1})+{\mathcal{E}}_s({\mathfrak{d}}_{n+1},{\mathfrak{d}}_{n+1},{\mathfrak{d}}_m)\hfill \\ & \le {\mathcal{E}}_s({\mathfrak{d}}_n,{\mathfrak{d}}_n,{\mathfrak{d}}_{n+1})+2{\mathcal{E}}_s({\mathfrak{d}}_{n+1},{\mathfrak{d}}_{n+1},{\mathfrak{d}}_{n+2})+{\mathcal{E}}_s({\mathfrak{d}}_{n+2},{\mathfrak{d}}_{n+2},{\mathfrak{d}}_m)\hfill \\ & ⋮\hfill \\ & \le 2[{\mathcal{E}}_s({\mathfrak{d}}_n,{\mathfrak{d}}_n,{\mathfrak{d}}_{n+1})+{\mathcal{E}}_s({\mathfrak{d}}_{n+1},{\mathfrak{d}}_{n+1},{\mathfrak{d}}_{n+2})+\cdots +{\mathcal{E}}_s({\mathfrak{d}}_{m-2},{\mathfrak{d}}_{m-2},{\mathfrak{d}}_{m-1})+\hfill \\ & {\mathcal{E}}_s({\mathfrak{d}}_{m-1},{\mathfrak{d}}_{m-1},{\mathfrak{d}}_m)]\hfill \\ & \le 2[{\varphi }^n+{\varphi }^{n+1}+\cdots +{\varphi }^{m-1}]{\mathcal{E}}_s({\mathfrak{d}}_0,{\mathfrak{d}}_0,{\mathfrak{d}}_1)\hfill \\ & \le 2{\varphi }^n{\mathcal{E}}_s({\mathfrak{d}}_0,{\mathfrak{d}}_0,{\mathfrak{d}}_1)[1+\varphi +{\varphi }^2+\cdots ]\hfill \\ & =\frac{2{\varphi }^n{\mathcal{E}}_s({\mathfrak{d}}_0,{\mathfrak{d}}_0,{\mathfrak{d}}_1)}{1-\varphi }\hfill \\ & <\frac{2{\mathcal{E}}_s({\mathfrak{d}}_0,{\mathfrak{d}}_0,{\mathfrak{d}}_1)}{1-\varphi }.\hfill \end{array}$$

This shows that $\left({\mathfrak{d}}_n\right)$ is a Cauchy sequence. Since ${\mathring{\mathcal{Y}}}_{ϵ}$ is complete, there exists $u\in {\mathring{\mathcal{Y}}}_{ϵ}$ such that ${\mathfrak{d}}_n\to u$ as $n\to \infty$.

Subsequently, in establishing u, as a fixed point of $\mathrm{\Gamma }$, we initially demonstrate its status as a fixed point of ${\mathrm{\Gamma }}^a$. To accomplish this, we refer to the definition of $\mathfrak{g}\in \mathcal{D}$:

$$\begin{array}{cc}\hfill {\mathcal{E}}_s({\mathrm{\Gamma }}^{a}u,{\mathrm{\Gamma }}^{a}u,{\mathfrak{d}}_{n+1})& ={\mathcal{E}}_s({\mathrm{\Gamma }}^{a}u,{\mathrm{\Gamma }}^{a}u,{\mathrm{\Gamma }}^a{\mathfrak{d}}_n)\hfill \\ & \le \mathfrak{g}[{\mathcal{E}}_s(u,u,{\mathfrak{d}}_n),{\mathcal{E}}_s({\mathrm{\Gamma }}^{a}u,{\mathrm{\Gamma }}^{a}u,u),{\mathcal{E}}_s({\mathrm{\Gamma }}^{a}u,{\mathrm{\Gamma }}^{a}u,{\mathfrak{d}}_n),\hfill \\ & {\mathcal{E}}_s({\mathfrak{d}}_{n+1},{\mathfrak{d}}_{n+1},u),{\mathcal{E}}_s({\mathfrak{d}}_{n+1},{\mathfrak{d}}_{n+1},{\mathfrak{d}}_n)].\hfill \end{array}$$

As $n\to \infty$, we obtain the following:

$$\begin{array}{cc}\hfill {\mathcal{E}}_s({\mathrm{\Gamma }}^{a}u,{\mathrm{\Gamma }}^{a}u,u)& \le \mathfrak{g}[0,{\mathcal{E}}_s({\mathrm{\Gamma }}^{a}u,{\mathrm{\Gamma }}^{a}u,u),{\mathcal{E}}_s({\mathrm{\Gamma }}^{a}u,{\mathrm{\Gamma }}^{a}u,u),0,0].\hfill \end{array}$$

According to the definition of $\mathfrak{g}\in \mathcal{D}$, this implies that ${\mathcal{E}}_s({\mathrm{\Gamma }}^{a}u,{\mathrm{\Gamma }}^{a}u,u)=0$ leads to ${\mathrm{\Gamma }}^{a}u=u.$ Now,

$$\begin{array}{cc}\hfill {\mathcal{E}}_s(\mathrm{\Gamma }u,\mathrm{\Gamma }u,u)& ={\mathcal{E}}_s(\mathrm{\Gamma }\left({\mathrm{\Gamma }}^{a}u\right),\mathrm{\Gamma }\left({\mathrm{\Gamma }}^{a}u\right),{\mathrm{\Gamma }}^{a}u)\hfill \\ & ={\mathcal{E}}_s({\mathrm{\Gamma }}^a\left(\mathrm{\Gamma }u\right),{\mathrm{\Gamma }}^a\left(\mathrm{\Gamma }u\right),{\mathrm{\Gamma }}^{a}u)\hfill \\ & \le \mathfrak{g}[{\mathcal{E}}_s(\mathrm{\Gamma }u,\mathrm{\Gamma }u,u),{\mathcal{E}}_s({\mathrm{\Gamma }}^a\left(\mathrm{\Gamma }u\right),{\mathrm{\Gamma }}^a\left(\mathrm{\Gamma }u\right),\mathrm{\Gamma }u),{\mathcal{E}}_s({\mathrm{\Gamma }}^a\left(\mathrm{\Gamma }u\right),{\mathrm{\Gamma }}^a\left(\mathrm{\Gamma }u\right),u),\hfill \\ & {\mathcal{E}}_s({\mathrm{\Gamma }}^{a}u,{\mathrm{\Gamma }}^{a}u,{\mathrm{\Gamma }}^{a}u),{\mathcal{E}}_s({\mathrm{\Gamma }}^{a}u,{\mathrm{\Gamma }}^{a}u,u)]\hfill \\ & =\mathfrak{g}[{\mathcal{E}}_s(\mathrm{\Gamma }u,\mathrm{\Gamma }u,u),{\mathcal{E}}_s(\mathrm{\Gamma }u,\mathrm{\Gamma }u,\mathrm{\Gamma }u),{\mathcal{E}}_s(\mathrm{\Gamma }u,\mathrm{\Gamma }u,u),0,{\mathcal{E}}_s(u,u,u)]\hfill \\ & =\mathfrak{g}[{\mathcal{E}}_s(\mathrm{\Gamma }u,\mathrm{\Gamma }u,u),0,{\mathcal{E}}_s(\mathrm{\Gamma }u,\mathrm{\Gamma }u,u),0,0].\hfill \end{array}$$

Again, by the definition of $\mathfrak{g}\in \mathcal{D}$, this implies that ${\mathcal{E}}_s(\mathrm{\Gamma }u,\mathrm{\Gamma }u,u)=0$. This leads to the conclusion that $\mathrm{\Gamma }u=u$.

To ascertain uniqueness, let us consider the opposite scenario: suppose there exists $v\in {\mathring{\mathcal{Y}}}_{ϵ}$ such that $\mathrm{\Gamma }v=v$ and $u\ne v$. This assumption brings us to the following subsequent inference:

$$\begin{array}{cc}\hfill {\mathcal{E}}_s(u,u,v)& ={\mathcal{E}}_s(\mathrm{\Gamma }u,\mathrm{\Gamma }u,\mathrm{\Gamma }v)\hfill \\ & \le \mathfrak{g}[{\mathcal{E}}_s(u,u,v),{\mathcal{E}}_s(\mathrm{\Gamma }u,\mathrm{\Gamma }u,u),{\mathcal{E}}_s(\mathrm{\Gamma }u,\mathrm{\Gamma }u,v),{\mathcal{E}}_s(\mathrm{\Gamma }v,\mathrm{\Gamma }v,u),{\mathcal{E}}_s(\mathrm{\Gamma }v,\mathrm{\Gamma }v,v)]\hfill \\ & =\mathfrak{g}[{\mathcal{E}}_s(u,u,v),0,{\mathcal{E}}_s(u,u,v),{\mathcal{E}}_s(u,u,v),0].\hfill \end{array}$$

Once more, following the definition of $\mathfrak{g}\in \mathcal{D}$, it results in ${\mathcal{E}}_s(u,u,v)=0$, inferring that $u=v$. With this, the proof is complete. □

Example 4.

Let ${\mathring{\mathcal{Y}}}_{ϵ}=[0,1]$ and an $\mathcal{S}$-metric is ${\mathcal{E}}_s:{[0,1]}^3\to [0,+\infty )$ by

$${\mathcal{E}}_s(\mathfrak{p},\mathfrak{d},\mathfrak{z})=|\mathfrak{p}-\mathfrak{d}|+|\mathfrak{p}+2\mathfrak{d}-3\mathfrak{z}|,$$

then $({\mathring{\mathcal{Y}}}_{ϵ},{\mathcal{E}}_s)$ is a complete $\mathcal{S}$-metric space. Next, we define the self-mapping $\mathrm{\Gamma }:[0,1]\to [0,1]$ given by $\mathrm{\Gamma }\left(\mathfrak{d}\right)=\frac{{\mathfrak{d}}^2}{2}+\frac{1}{8}$ and $\mathfrak{g}:{\mathbb{R}}_{+}^5\to {\mathbb{R}}_{+}$ by $\mathfrak{g}({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5)=\varphi {\alpha }_1+\psi {\alpha }_2+\xi [{\alpha }_3+{\alpha }_4]+\nabla {\alpha }_5,$ for $\varphi +\psi +3\xi +\nabla \in [0,1).$ Clearly, $\mathfrak{g}\in \mathcal{D}$. Since for any $a\in \mathbb{N}$, we have ${\mathcal{E}}_s({\mathrm{\Gamma }}^a\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{d})>0$ and ${\mathcal{E}}_s(\mathrm{\Gamma }\mathfrak{p},\mathrm{\Gamma }\mathfrak{p},\mathrm{\Gamma }\mathfrak{d})>0$.

To underscore the novelty of our findings, we demonstrate that

$${\mathcal{E}}_s(\mathrm{\Gamma }\mathfrak{p},\mathrm{\Gamma }\mathfrak{p},\mathrm{\Gamma }\mathfrak{d})\le \varphi {\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},\mathfrak{d})+\psi {\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},\mathrm{\Gamma }\mathfrak{p})+\zeta [{\mathcal{E}}_s(\mathfrak{d},\mathfrak{d},\mathrm{\Gamma }\mathfrak{p})+{\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},\mathrm{\Gamma }\mathfrak{d})]+\nabla {\mathcal{E}}_s(\mathfrak{d},\mathfrak{d},\mathrm{\Gamma }\mathfrak{d})$$

does not hold, yet for some $a\in \mathbb{N},a>1$,

$${\mathcal{E}}_s({\mathrm{\Gamma }}^a\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{d})\le \varphi {\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},\mathfrak{d})+\psi {\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{p})+\zeta [{\mathcal{E}}_s(\mathfrak{d},\mathfrak{d},{\mathrm{\Gamma }}^a\mathfrak{p})+{\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{d})]+\nabla {\mathcal{E}}_s(\mathfrak{d},\mathfrak{d},{\mathrm{\Gamma }}^a\mathfrak{d})$$

holds true for $\varphi +\psi +3\xi +\nabla \in [0,1).$ As

$$\begin{array}{cc}\hfill {\mathcal{E}}_s(\mathrm{\Gamma }\mathfrak{p},\mathrm{\Gamma }\mathfrak{p},\mathrm{\Gamma }\mathfrak{d})& =3|(\frac{{\mathfrak{p}}^2}{2}+\frac{1}{8})-(\frac{{\mathfrak{d}}^2}{2}+\frac{1}{8})|\hfill \\ \hfill & =\frac{3}{2}|{\mathfrak{p}}^2-{\mathfrak{d}}^2|\hfill \\ \hfill & =\frac{3}{2}(\mathfrak{p}+\mathfrak{d})|\mathfrak{p}-\mathfrak{d}|\hfill \\ \hfill & \le 3\left(|\mathfrak{p}-\mathfrak{d}|\right)\hfill \\ \hfill & \le {\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},\mathfrak{d})+\psi {\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},\mathrm{\Gamma }\mathfrak{p})+\zeta [{\mathcal{E}}_s(\mathfrak{d},\mathfrak{d},\mathrm{\Gamma }\mathfrak{p})+{\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},\mathrm{\Gamma }\mathfrak{d})]+\nabla {\mathcal{E}}_s(\mathfrak{d},\mathfrak{d},\mathrm{\Gamma }\mathfrak{d}),\hfill \end{array}$$

but the expression $1+\psi +3\xi +\nabla$ is outside the range $[0,1)$ for any combination of $\psi ,\xi ,\nabla \in [0,1)$. Consequently, the application of the main theorem by Joshi et al. [12] to this particular example, ensuring the existence of a fixed point for Γ in ${\mathring{\mathcal{Y}}}_{ϵ}$, is not viable.

Next, for $a=2\in \mathbb{N}$, with the above notions, we obtain

$$\begin{array}{cc}\hfill {\mathcal{E}}_s({\mathrm{\Gamma }}^2\mathfrak{p},{\mathrm{\Gamma }}^2\mathfrak{p},{\mathrm{\Gamma }}^2\mathfrak{d})& =3|\frac{1}{16}(2{\mathfrak{p}}^4-2{\mathfrak{d}}^4+{\mathfrak{p}}^2-{\mathfrak{d}}^2)|\hfill \\ \hfill & =\frac{3}{16}|2({\mathfrak{p}}^4-{\mathfrak{d}}^4)+({\mathfrak{p}}^2-{\mathfrak{d}}^2)|\hfill \\ \hfill & \le \frac{3}{8}|{\mathfrak{p}}^4-{\mathfrak{d}}^4|+\frac{3}{16}|{\mathfrak{p}}^2-{\mathfrak{d}}^2|\hfill \\ \hfill & \le \frac{3}{4}|{\mathfrak{p}}^2-{\mathfrak{d}}^2|+\frac{3}{8}|\mathfrak{p}-\mathfrak{d}|\hfill \\ \hfill & =3\left[\frac{5}{8}|\mathfrak{p}-\mathfrak{d}|\right]\hfill \\ \hfill & \le \varphi {\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},\mathfrak{d})+\psi {\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},{\mathrm{\Gamma }}^2\mathfrak{p})+\zeta [{\mathcal{E}}_s(\mathfrak{d},\mathfrak{d},{\mathrm{\Gamma }}^2\mathfrak{p})+{\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},{\mathrm{\Gamma }}^2\mathfrak{d})]+\nabla {\mathcal{E}}_s(\mathfrak{d},\mathfrak{d},{\mathrm{\Gamma }}^2\mathfrak{d}),\hfill \end{array}$$

(1)

where $\varphi =\frac{1}{2},\psi =\xi =\nabla =\frac{1}{16}$ with $\varphi +\psi +3\xi +\nabla \in [0,1)$. Hence, all assumptions of Theorem 2 are satisfied, so Γ has a unique fixed point in ${\mathring{\mathcal{Y}}}_{ϵ}$, which is $\mathfrak{d}=1-\frac{\sqrt{3}}{2}$.

Example 5.

Let $({\mathring{\mathcal{Y}}}_{ϵ}=[1,4],{\mathcal{E}}_s)$ be a complete $\mathcal{S}$-metric space with ${\mathcal{E}}_s(\mathfrak{p},\mathfrak{d},\mathfrak{w})=|\mathfrak{p}-\mathfrak{d}|+|\mathfrak{p}+3\mathfrak{d}-4\mathfrak{w}|$. We define the mapping $\mathrm{\Gamma }:{\mathring{\mathcal{Y}}}_{ϵ}\to {\mathring{\mathcal{Y}}}_{ϵ}$ by $\mathrm{\Gamma }\left(\mathfrak{w}\right)=2\sqrt{\mathfrak{w}}$ and $\mathfrak{g}:{[0,\infty )}^5\to [0,\infty )$ by $\mathfrak{g}({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5)=\varphi {\alpha }_1$ for some $\varphi \in [0,1)$. Then, $\mathfrak{g}\in \mathcal{D}$.

Since ${\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},\mathfrak{d})=4|\mathfrak{p}-\mathfrak{y}|$. Therefore, $\forall \mathfrak{p},\mathfrak{d}\in {\mathring{\mathcal{Y}}}_{ϵ}$,

$${\mathcal{E}}_s(\mathrm{\Gamma }\mathfrak{p},\mathrm{\Gamma }\mathfrak{p},\mathrm{\Gamma }\mathfrak{d})\nleqq \varphi {\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},\mathfrak{d})$$

for any $\varphi \in [0,1)$. Therefore, Γ is not a contraction, and the application of the main theorem by Joshi et al. [12] and Sedghi et al. [18] to this particular example, ensuring the existence of a fixed point for Γ in ${\mathring{\mathcal{Y}}}_{ϵ}$, is not viable. But,

$$\begin{array}{c}\hfill {\mathcal{E}}_s({\mathrm{\Gamma }}^2\mathfrak{p},{\mathrm{\Gamma }}^2\mathfrak{p},{\mathrm{\Gamma }}^2\mathfrak{d})\le \frac{1}{\sqrt{2}}{\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},\mathfrak{d}),\forall \mathfrak{p},\mathfrak{d}\in {\mathring{\mathcal{Y}}}_{ϵ}.\end{array}$$

Therefore, by Theorem 2, there exists a unique fixed point of Γ in ${\mathring{\mathcal{Y}}}_{ϵ}$ (that is $\mathfrak{w}=4$).

Example 6.

The space ${\mathring{\mathcal{Y}}}_{ϵ}=l^1=\{\left({\mathfrak{d}}_n\right)\subset \mathbb{R}:{\sum }_{n=1}^{\infty }|{\mathfrak{d}}_n|<\infty \}$, with $\mathcal{S}$-metric

$${\mathcal{E}}_s(\mathfrak{p},\mathfrak{d},\mathfrak{w})=\sum _{n=1}^{\infty }|{\mathfrak{p}}_n-{\mathfrak{d}}_n|+\sum _{n=1}^{\infty }|{\mathfrak{p}}_n+{\mathfrak{d}}_n-2{\mathfrak{w}}_n|,$$

where $\mathfrak{p}=\left({\mathfrak{p}}_n\right),\mathfrak{d}=\left({\mathfrak{d}}_n\right),\mathfrak{w}=\left({\mathfrak{w}}_n\right)\in l^1$, is a complete $\mathcal{S}$-metric space. Define the mappings $\mathrm{\Gamma }:{\mathring{\mathcal{Y}}}_{ϵ}\to {\mathring{\mathcal{Y}}}_{ϵ}$ by

$$\mathrm{\Gamma }\left(\mathfrak{p}\right)=(0,{\mathfrak{p}}_1,\frac{{\mathfrak{p}}_2}{3},\frac{{\mathfrak{p}}_3}{3},\frac{{\mathfrak{p}}_4}{3},\frac{{\mathfrak{p}}_5}{3},\cdots ),\forall \mathfrak{p}=\left({\mathfrak{p}}_n\right)\in l^1$$

and $\mathfrak{g}:{[0,\infty )}^5\to [0,\infty )$ by $\mathfrak{g}({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5)=\varphi {\alpha }_1$ for some $\varphi \in [0,1)$. Then, $\mathfrak{g}\in \mathcal{D}$ and note that the only fixed point of Γ is $(0,0,0,0,\cdots )$. Next,

$$\begin{gathered} {\mathrm{\Gamma }}^2\left(\mathfrak{p}\right)=(0,0,\frac{{\mathfrak{p}}_1}{3},\frac{{\mathfrak{p}}_2}{3^2},\frac{{\mathfrak{p}}_3}{3^2},\frac{{\mathfrak{p}}_4}{3^2},\frac{{\mathfrak{p}}_5}{3^2},\cdots ), \\ {\mathrm{\Gamma }}^3\left(\mathfrak{p}\right)=(0,0,0,\frac{{\mathfrak{p}}_1}{3^2},\frac{{\mathfrak{p}}_2}{3^3},\frac{{\mathfrak{p}}_3}{3^3},\frac{{\mathfrak{p}}_4}{3^3},\frac{{\mathfrak{p}}_5}{3^3},\cdots ), \\ ⋮ \\ {\mathrm{\Gamma }}^a\left(\mathfrak{p}\right)=(\underset{a}{\underbrace{0,\cdots ,0}},\frac{{\mathfrak{p}}_1}{3^{a-1}},\frac{{\mathfrak{p}}_2}{3^a},\frac{{\mathfrak{p}}_3}{3^a},\frac{{\mathfrak{p}}_4}{3^a},\frac{{\mathfrak{p}}_5}{3^a},\cdots ). \end{gathered}$$

Now, for $\mathfrak{p}=\left({\mathfrak{p}}_n\right),\mathfrak{d}=\left({\mathfrak{d}}_n\right)\in {\mathring{\mathcal{Y}}}_{ϵ}=l^1$, we obtain the following:

$$\begin{array}{cc}\hfill {\mathcal{E}}_s({\mathrm{\Gamma }}^a\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{d})& =2\left[\frac{|{\mathfrak{p}}_1-{\mathfrak{d}}_1|}{3^{a-1}}+\frac{|{\mathfrak{p}}_2-{\mathfrak{d}}_2|}{3^a}+\frac{|{\mathfrak{p}}_3-{\mathfrak{d}}_3|}{3^a}+\frac{|{\mathfrak{p}}_4-{\mathfrak{d}}_4|}{3^a}+\cdots \right]\hfill \\ \hfill & \le 2\left[\frac{|{\mathfrak{p}}_1-{\mathfrak{d}}_1|}{3^{a-1}}+\frac{|{\mathfrak{p}}_2-{\mathfrak{d}}_2|}{3^{a-1}}+\frac{|{\mathfrak{p}}_3-{\mathfrak{d}}_3|}{3^{a-1}}+\frac{|{\mathfrak{p}}_4-{\mathfrak{d}}_4|}{3^{a-1}}+\cdots \right]\hfill \\ \hfill & =2\left[\sum _{n=1}^{\infty }\frac{|{\mathfrak{p}}_n-{\mathfrak{d}}_n|}{3^{a-1}}\right]\hfill \\ \hfill & =\frac{1}{3^{a-1}}{\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},\mathfrak{d}).\hfill \end{array}$$

So, the mapping Γ satisfies all assumptions of Theorem 2. On the other hand, Γ is not a contraction as defined in [12,18]. As for $\mathfrak{p}=(3,0,0,\cdots ),\mathfrak{d}=(4,0,0,\cdots )\in {\mathring{\mathcal{Y}}}_{ϵ}=l^1$, we obtain $\mathrm{\Gamma }\left(\mathfrak{p}\right)=(0,3,0,0,\cdots ),\mathrm{\Gamma }\left(\mathfrak{d}\right)=(0,4,0,0,\cdots )$. Also, ${\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},\mathfrak{d})=1$ and ${\mathcal{E}}_s(\mathrm{\Gamma }\mathfrak{p},\mathrm{\Gamma }\mathfrak{p},\mathrm{\Gamma }\mathfrak{d})=1$. Thus,

$${\mathcal{E}}_s(\mathrm{\Gamma }\mathfrak{p},\mathrm{\Gamma }\mathfrak{p},\mathrm{\Gamma }\mathfrak{d})>\varphi {\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},\mathfrak{d}),$$

for every $\varphi \in [0,1)$.

After our main theorem, a sequence of corollaries unfolds, observable in the subsequent content:

Corollary 1.

Consider a complete $\mathcal{S}$-metric space $({\mathring{\mathcal{Y}}}_{ϵ},{\mathcal{E}}_s)$ and a self-mapping $\mathrm{\Gamma }:{\mathring{\mathcal{Y}}}_{ϵ}\to {\mathring{\mathcal{Y}}}_{ϵ}$. Suppose there exists $a\in \mathbb{N}$ such that for all $\mathfrak{p},\mathfrak{d}\in {\mathring{\mathcal{Y}}}_{ϵ}$:

$${\mathcal{E}}_s({\mathrm{\Gamma }}^a\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{d})\le \varphi {\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},\mathfrak{d}),\varphi \in [0,1).$$

Then, Γ possesses only one fixed point.

Proof. 

Take $\mathfrak{g}\in \mathcal{D}$ defined by $\mathfrak{g}({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5)=\varphi {\alpha }_1$, where $\varphi \in [0,1)$. Then, the conclusion readily arises from Theorem 2. □

Remark 2.

Corollary 1 extends the findings of Sedghi et al. [18] by providing a refinement specifically tailored for Banach contractions within the framework of Sehgal–Guseman-like contractions.

Corollary 2.

Let $({\mathring{\mathcal{Y}}}_{ϵ},{\mathcal{E}}_s)$ be a complete $\mathcal{S}$-metric space and $\mathrm{\Gamma }:{\mathring{\mathcal{Y}}}_{ϵ}\to {\mathring{\mathcal{Y}}}_{ϵ}$ be a self-mapping such that there exists $a\in \mathbb{N}$ for which $\forall \mathfrak{p},\mathfrak{d}\in {\mathring{\mathcal{Y}}}_{ϵ}$:

$${\mathcal{E}}_s({\mathrm{\Gamma }}^a\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{d})\le \varphi [{\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{p})+{\mathcal{E}}_s(\mathfrak{d},\mathfrak{d},{\mathrm{\Gamma }}^a\mathfrak{d})],\varphi \in [0,\frac{1}{3}).$$

Then, Γ possesses only one fixed point.

Proof. 

Take $\mathfrak{g}\in \mathcal{D}$ defined by $\mathfrak{g}({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5)=\varphi [{\alpha }_2+{\alpha }_5]$, where $\varphi \in [0,\frac{1}{3})$. Then, the conclusion readily arises from Theorem 2. □

Remark 3.

Corollary 2 extends the findings of Phaneendra [16] by providing a refinement specifically tailored for Kannan contractions within the framework of Sehgal–Guseman-like contractions.

Corollary 3.

Let $({\mathring{\mathcal{Y}}}_{ϵ},{\mathcal{E}}_s)$ be a complete $\mathcal{S}$-metric space and $\mathrm{\Gamma }:{\mathring{\mathcal{Y}}}_{ϵ}\to {\mathring{\mathcal{Y}}}_{ϵ}$ be a self-mapping such that there exists $a\in \mathbb{N}$ for which $\forall \mathfrak{p},\mathfrak{d}\in {\mathring{\mathcal{Y}}}_{ϵ}$:

$${\mathcal{E}}_s({\mathrm{\Gamma }}^a\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{d})\le \varphi [{\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{d})+{\mathcal{E}}_s(\mathfrak{d},\mathfrak{d},{\mathrm{\Gamma }}^a\mathfrak{p})],\varphi \in [0,\frac{1}{3}).$$

Then, Γ possesses only one fixed point.

Proof. 

Take $\mathfrak{g}\in \mathcal{D}$ defined by $\mathfrak{g}({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5)=\varphi [{\alpha }_3+{\alpha }_4]$, where $\varphi \in [0,\frac{1}{3})$. Then, the conclusion readily arises from Theorem 2. □

Remark 4.

Corollary 3 extends the findings of Phaneendra and Swamy [15] by providing a refinement specifically tailored to Chatterjee contractions within the framework of Sehgal–Guseman-like contractions.

Corollary 4.

Let $({\mathring{\mathcal{Y}}}_{ϵ},{\mathcal{E}}_s)$ be a complete $\mathcal{S}$-metric space and $\mathrm{\Gamma }:{\mathring{\mathcal{Y}}}_{ϵ}\to {\mathring{\mathcal{Y}}}_{ϵ}$ be a self-mapping such that there exists $a\in \mathbb{N}$ for which $\forall \mathfrak{p},\mathfrak{d}\in {\mathring{\mathcal{Y}}}_{ϵ}$:

$${\mathcal{E}}_s({\mathrm{\Gamma }}^a\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{d})\le \varphi max\{{\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},\mathfrak{d}),{\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{p}),{\mathcal{E}}_s(\mathfrak{d},\mathfrak{d},{\mathrm{\Gamma }}^a\mathfrak{p}),{\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{d}),{\mathcal{E}}_s(\mathfrak{d},\mathfrak{d},{\mathrm{\Gamma }}^a\mathfrak{d})\},$$

where $\varphi \in [0,\frac{1}{3})$. Then, Γ possesses only one fixed point.

Proof. 

Take $\mathfrak{g}\in \mathcal{D}$ defined by $\mathfrak{g}({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5)=\varphi max\{{\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5\}$, where $\varphi \in [0,\frac{1}{3})$. Then, the conclusion readily arises from Theorem 2. □

Remark 5.

Corollary 4 extends the findings of Phaneendra and Swamy [15] by providing a refinement specifically tailored to Ćirić-type contractions within the framework of Sehgal–Guseman-like contractions.

Corollary 5.

Let $({\mathring{\mathcal{Y}}}_{ϵ},{\mathcal{E}}_s)$ be a complete $\mathcal{S}$-metric space and $\mathrm{\Gamma }:{\mathring{\mathcal{Y}}}_{ϵ}\to {\mathring{\mathcal{Y}}}_{ϵ}$ be a self-mapping such that there exists $a\in \mathbb{N}$ for which $\forall \mathfrak{p},\mathfrak{d}\in {\mathring{\mathcal{Y}}}_{ϵ}$:

$${\mathcal{E}}_s({\mathrm{\Gamma }}^a\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{d})\le \varphi {\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},\mathfrak{d})+\psi {\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{p})+\zeta [{\mathcal{E}}_s(\mathfrak{d},\mathfrak{d},{\mathrm{\Gamma }}^a\mathfrak{p})+{\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{d})]+\nabla {\mathcal{E}}_s(\mathfrak{d},\mathfrak{d},{\mathrm{\Gamma }}^a\mathfrak{d})\},$$

where $\varphi +\psi +3\zeta +\nabla \in [0,1)$. Then, Γ possesses only one fixed point.

Proof. 

Take $\mathfrak{g}\in \mathcal{D}$ defined by $\mathfrak{g}({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5)=\varphi {\alpha }_1+\psi {\alpha }_2+\zeta [{\alpha }_3+{\alpha }_4]+\nabla {\alpha }_5$, where $\varphi +\psi +3\zeta +\nabla \in [0,1)$. Then, the conclusion readily arises from Theorem 2. □

Remark 6.

Corollary 5 improves upon the results put forth by Hardy and Roger [7] by providing a more refined viewpoint in the context of Sehgal–Guseman-like contractions in $\mathcal{S}$-metric spaces, consequently broadening the scope of their findings.

Corollary 6.

Let $({\mathring{\mathcal{Y}}}_{ϵ},{\mathcal{E}}_s)$ be a complete $\mathcal{S}$-metric space and $\mathrm{\Gamma }:{\mathring{\mathcal{Y}}}_{ϵ}\to {\mathring{\mathcal{Y}}}_{ϵ}$ be a self-mapping such that there exists $a\in \mathbb{N}$ for which $\forall \mathfrak{p},\mathfrak{d}\in {\mathring{\mathcal{Y}}}_{ϵ}$:

$${\mathcal{E}}_s({\mathrm{\Gamma }}^a\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{d})\le \varphi {\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},\mathfrak{d})+\psi {\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{p})+\nabla {\mathcal{E}}_s(\mathfrak{d},\mathfrak{d},{\mathrm{\Gamma }}^a\mathfrak{d})\},\varphi +\psi +\nabla \in [0,1).$$

Then, Γ possesses only one fixed point.

Proof. 

Take $\mathfrak{g}\in \mathcal{D}$ defined by $\mathfrak{g}({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5)=\varphi {\alpha }_1+\psi {\alpha }_2+\nabla {\alpha }_5$, where $\varphi +\psi +\nabla \in [0,1)$. Then, the conclusion readily arises from Theorem 2. □

Remark 7.

Corollary 6 improves upon the results put forth by Reich [17] by providing a more refined viewpoint in the context of Sehgal–Guseman-like contractions in $\mathcal{S}$-metric spaces, consequently broadening the scope of their findings.

Corollary 7.

Let $({\mathring{\mathcal{Y}}}_{ϵ},{\mathcal{E}}_s)$ be a complete $\mathcal{S}$-metric space and $\mathrm{\Gamma }:{\mathring{\mathcal{Y}}}_{ϵ}\to {\mathring{\mathcal{Y}}}_{ϵ}$ be a self-mapping such that there exists $a\in \mathbb{N}$ for which $\forall \mathfrak{p},\mathfrak{d}\in {\mathring{\mathcal{Y}}}_{ϵ}$ with $\mathfrak{p}\ne \mathrm{\Gamma }\mathfrak{p},\mathfrak{d}\ne \mathrm{\Gamma }\mathfrak{d}$:

$${\mathcal{E}}_s({\mathrm{\Gamma }}^a\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{d})\le \varphi ·{\mathcal{E}}_s{(\mathfrak{p},\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{p})}^{\psi }·{\mathcal{E}}_s{(\mathfrak{d},\mathfrak{d},{\mathrm{\Gamma }}^a\mathfrak{d})}^{1-\psi },\varphi \in [0,1)\&\psi \in (0,1).$$

Then, Γ has a fixed point.

Proof. 

Take $\mathfrak{g}\in \mathcal{D}$ defined by $\mathfrak{g}({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5)=\varphi ·{\alpha }_2^{\psi }·{\alpha }_5^{1-\psi }$, where $\varphi \in [0,1)$ and $\psi \in (0,1)$. Then, the conclusion readily arises from Theorem 2. □

Remark 8.

Corollary 7 improves upon the results put forth by Karapınar [13] by providing a more refined viewpoint in the context of Sehgal–Guseman-like contractions in $\mathcal{S}$-metric spaces, consequently broadening the scope of their findings.

Corollary 8.

Let $({\mathring{\mathcal{Y}}}_{ϵ},{\mathcal{E}}_s)$ be a complete $\mathcal{S}$-metric space and $\mathrm{\Gamma }:{\mathring{\mathcal{Y}}}_{ϵ}\to {\mathring{\mathcal{Y}}}_{ϵ}$ be a self-mapping such that there exists $a\in \mathbb{N}$ for which $\forall \mathfrak{p},\mathfrak{d}\in {\mathring{\mathcal{Y}}}_{ϵ}$ with $\mathfrak{p}\ne \mathrm{\Gamma }\mathfrak{p},\mathfrak{d}\ne \mathrm{\Gamma }\mathfrak{d}$:

$${\mathcal{E}}_s({\mathrm{\Gamma }}^a\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{d})\le \varphi ·{\mathcal{E}}_s{(\mathfrak{d},\mathfrak{d},{\mathrm{\Gamma }}^a\mathfrak{p})}^{\psi }·{\mathcal{E}}_s{(\mathfrak{p},\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{d})}^{1-\psi },\varphi \in [0,1)\&\psi \in (0,1).$$

Then, Γ has a fixed point.

Proof. 

Take $\mathfrak{g}\in \mathcal{D}$ defined by $\mathfrak{g}({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5)=\varphi ·{\alpha }_3^{\psi }·{\alpha }_4^{1-\psi }$, where $\varphi \in [0,1)$ and $\psi \in (0,1)$. Then, the conclusion readily arises from Theorem 2. □

Remark 9.

Corollary 8 enhances the results concerning interpolative Chatterjee operators in the context of Sehgal–Guseman-like contractions within $\mathcal{S}$-metric spaces, consequently broadening the scope of their findings.

Corollary 9.

Let $({\mathring{\mathcal{Y}}}_{ϵ},{\mathcal{E}}_s)$ be a complete $\mathcal{S}$-metric space and $\mathrm{\Gamma }:{\mathring{\mathcal{Y}}}_{ϵ}\to {\mathring{\mathcal{Y}}}_{ϵ}$ be a self-mapping such that there exists $a\in \mathbb{N}$ for which $\forall \mathfrak{p},\mathfrak{d}\in {\mathring{\mathcal{Y}}}_{ϵ}$ with $\mathfrak{p}\ne \mathrm{\Gamma }\mathfrak{p},\mathfrak{d}\ne \mathrm{\Gamma }\mathfrak{d}$:

$${\mathcal{E}}_s({\mathrm{\Gamma }}^a\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{d})\le \varphi ·{\mathcal{E}}_s{(\mathfrak{p},\mathfrak{p},\mathfrak{d})}^{\xi }{\mathcal{E}}_s{(\mathfrak{p},\mathfrak{p},{\mathrm{\Gamma }}^a\mathfrak{p})}^{\psi }·{\mathcal{E}}_s{(\mathfrak{d},\mathfrak{d},{\mathrm{\Gamma }}^a\mathfrak{d})}^{1-\xi -\psi },\varphi \in [0,1)\&\psi +\xi \in (0,1).$$

Then, Γ has a fixed point.

Proof. 

Take $\mathfrak{g}\in \mathcal{D}$ defined by $\mathfrak{g}({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5)=\varphi ·{\alpha }_1^{\xi }·{\alpha }_2^{\psi }·{\alpha }_5^{1-\psi -\xi }$, where $\varphi \in [0,1)$ and $\psi +\xi \in (0,1)$. Then, the conclusion readily arises from Theorem 2. □

Remark 10.

Corollary 9 improves upon the results put forth by Karapınar [13] by providing a more refined viewpoint in the context of Sehgal–Guseman-like contractions in $\mathcal{S}$-metric spaces, consequently broadening the scope of their findings.

In particular, setting a to 1 reveals that we derive the following corollaries:

Corollary 10.

Let $({\mathring{\mathcal{Y}}}_{ϵ},{\mathcal{E}}_s)$ be a complete $\mathcal{S}$-metric space and $\mathrm{\Gamma }:{\mathring{\mathcal{Y}}}_{ϵ}\to {\mathring{\mathcal{Y}}}_{ϵ}$ be a self-mapping such that $\forall \mathfrak{p},\mathfrak{d}\in {\mathring{\mathcal{Y}}}_{ϵ}$ with $\mathfrak{p}\ne \mathrm{\Gamma }\mathfrak{p},\mathfrak{d}\ne \mathrm{\Gamma }\mathfrak{d}$:

$${\mathcal{E}}_s(\mathrm{\Gamma }\mathfrak{p},\mathrm{\Gamma }\mathfrak{p},\mathrm{\Gamma }\mathfrak{d})\le \varphi ·{\mathcal{E}}_s{(\mathfrak{p},\mathfrak{p},\mathfrak{d})}^{\xi }{\mathcal{E}}_s{(\mathfrak{p},\mathfrak{p},\mathrm{\Gamma }\mathfrak{p})}^{\psi }·{\mathcal{E}}_s{(\mathfrak{d},\mathfrak{d},\mathrm{\Gamma }\mathfrak{d})}^{1-\xi -\psi },\varphi \in [0,1)\&\psi +\xi \in (0,1).$$

Then, Γ has a fixed point.

Remark 11.

Corollary 10 refines the findings presented by Karapınar [13] for the interpolative Reich–Rus–Ćirić operator within the context of $\mathcal{S}$-metric spaces, thereby extending and enhancing the scope of their results.

Corollary 11.

Let $({\mathring{\mathcal{Y}}}_{ϵ},{\mathcal{E}}_s)$ be a complete $\mathcal{S}$-metric space and $\mathrm{\Gamma }:{\mathring{\mathcal{Y}}}_{ϵ}\to {\mathring{\mathcal{Y}}}_{ϵ}$ be a self-mapping such that $\forall \mathfrak{p},\mathfrak{d}\in {\mathring{\mathcal{Y}}}_{ϵ}$ with $\mathfrak{p}\ne \mathrm{\Gamma }\mathfrak{p},\mathfrak{d}\ne \mathrm{\Gamma }\mathfrak{d}$:

$${\mathcal{E}}_s(\mathrm{\Gamma }\mathfrak{p},\mathrm{\Gamma }\mathfrak{p},\mathrm{\Gamma }\mathfrak{d})\le \varphi ·{\mathcal{E}}_s{(\mathfrak{p},\mathfrak{p},\mathrm{\Gamma }\mathfrak{p})}^{\psi }·{\mathcal{E}}_s{(\mathfrak{d},\mathfrak{d},\mathrm{\Gamma }\mathfrak{d})}^{1-\psi },\varphi \in [0,1)\&\psi \in (0,1).$$

Then, Γ has a fixed point.

Remark 12.

Corollary 11 refines the findings presented by Karapınar [13] for the interpolative Kannan operator within the context of $\mathcal{S}$-metric spaces, thereby extending and enhancing the scope of their results.

Corollary 12.

Let $({\mathring{\mathcal{Y}}}_{ϵ},{\mathcal{E}}_s)$ be a complete $\mathcal{S}$-metric space and $\mathrm{\Gamma }:{\mathring{\mathcal{Y}}}_{ϵ}\to {\mathring{\mathcal{Y}}}_{ϵ}$ be a self-mapping such that $\forall \mathfrak{p},\mathfrak{d}\in {\mathring{\mathcal{Y}}}_{ϵ}$ with $\mathfrak{p}\ne \mathrm{\Gamma }\mathfrak{p},\mathfrak{d}\ne \mathrm{\Gamma }\mathfrak{d}$:

$${\mathcal{E}}_s(\mathrm{\Gamma }\mathfrak{p},\mathrm{\Gamma }\mathfrak{p},\mathrm{\Gamma }\mathfrak{d})\le \varphi ·{\mathcal{E}}_s{(\mathfrak{d},\mathfrak{d},\mathrm{\Gamma }\mathfrak{p})}^{\psi }·{\mathcal{E}}_s{(\mathfrak{p},\mathfrak{p},\mathrm{\Gamma }\mathfrak{d})}^{1-\psi },\varphi \in [0,1)\&\psi \in (0,1).$$

Then, Γ has a fixed point.

Remark 13.

Corollary 12 elevates the outcomes related to interpolative Chatterjee operators within the realm of $\mathcal{S}$-metric spaces, thereby expanding and enriching the breadth of their discoveries.

Corollary 13.

Let $({\mathring{\mathcal{Y}}}_{ϵ},{\mathcal{E}}_s)$ be a complete $\mathcal{S}$-metric space and $\mathrm{\Gamma }:{\mathring{\mathcal{Y}}}_{ϵ}\to {\mathring{\mathcal{Y}}}_{ϵ}$ be a self-mapping such that $\forall \mathfrak{p},\mathfrak{d}\in {\mathring{\mathcal{Y}}}_{ϵ}$:

$${\mathcal{E}}_s(\mathrm{\Gamma }\mathfrak{p},\mathrm{\Gamma }\mathfrak{p},\mathrm{\Gamma }\mathfrak{d})\le \varphi {\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},\mathfrak{d}),\varphi \in [0,1).$$

Then, Γ possesses only one fixed point.

Remark 14.

Corollary 13 stands as the primary outcome introduced by Sedghi et al. [18], representing an improvement of the BCP in the context of $\mathcal{S}$-metric spaces.

Corollary 14.

Let $({\mathring{\mathcal{Y}}}_{ϵ},{\mathcal{E}}_s)$ be a complete $\mathcal{S}$-metric space and $\mathrm{\Gamma }:{\mathring{\mathcal{Y}}}_{ϵ}\to {\mathring{\mathcal{Y}}}_{ϵ}$ be a self-mapping such that $\forall \mathfrak{p},\mathfrak{d}\in {\mathring{\mathcal{Y}}}_{ϵ}$:

$${\mathcal{E}}_s(\mathrm{\Gamma }\mathfrak{p},\mathrm{\Gamma }\mathfrak{p},\mathrm{\Gamma }\mathfrak{d})\le \varphi [{\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},\mathrm{\Gamma }\mathfrak{p})+{\mathcal{E}}_s(\mathfrak{d},\mathfrak{d},\mathrm{\Gamma }\mathfrak{d})],\varphi \in [0,\frac{1}{3}).$$

Then, Γ possesses only one fixed point.

Remark 15.

Corollary 14 stands as the primary outcome introduced by Phaneendra [16], representing an improvement of the Kannan FPT in the context of $\mathcal{S}$-metric spaces.

Corollary 15.

Let $({\mathring{\mathcal{Y}}}_{ϵ},{\mathcal{E}}_s)$ be a complete $\mathcal{S}$-metric space and $\mathrm{\Gamma }:{\mathring{\mathcal{Y}}}_{ϵ}\to {\mathring{\mathcal{Y}}}_{ϵ}$ be a self-mapping such that $\forall \mathfrak{p},\mathfrak{d}\in {\mathring{\mathcal{Y}}}_{ϵ}$:

$${\mathcal{E}}_s(\mathrm{\Gamma }\mathfrak{p},\mathrm{\Gamma }\mathfrak{p},\mathrm{\Gamma }\mathfrak{d})\le \varphi [{\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},\mathrm{\Gamma }\mathfrak{d})+{\mathcal{E}}_s(\mathfrak{d},\mathfrak{d},\mathrm{\Gamma }\mathfrak{p})],\varphi \in [0,\frac{1}{3}).$$

Then, Γ possesses only one fixed point.

Remark 16.

Corollary 15 stands as the primary outcome introduced by Phaneendra and Swamy [15], representing an improvement of the Chatterjee FPT in the context of $\mathcal{S}$-metric spaces.

Corollary 16.

Let $({\mathring{\mathcal{Y}}}_{ϵ},{\mathcal{E}}_s)$ be a complete $\mathcal{S}$-metric space and $\mathrm{\Gamma }:{\mathring{\mathcal{Y}}}_{ϵ}\to {\mathring{\mathcal{Y}}}_{ϵ}$ be a self-mapping such that $\forall \mathfrak{p},\mathfrak{d}\in {\mathring{\mathcal{Y}}}_{ϵ}$:

$${\mathcal{E}}_s(\mathrm{\Gamma }\mathfrak{p},\mathrm{\Gamma }\mathfrak{p},\mathrm{\Gamma }\mathfrak{d})\le \varphi max\{{\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},\mathfrak{d}),{\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},\mathrm{\Gamma }\mathfrak{p}),{\mathcal{E}}_s(\mathfrak{d},\mathfrak{d},\mathrm{\Gamma }\mathfrak{p}),{\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},\mathrm{\Gamma }\mathfrak{d}),{\mathcal{E}}_s(\mathfrak{d},\mathfrak{d},\mathrm{\Gamma }\mathfrak{d})\},\varphi \in [0,\frac{1}{3}).$$

Then, Γ possesses only one fixed point.

Remark 17.

Corollary 16 stands as the primary outcome introduced by Phaneendra and Swamy [15], representing an improvement of the Ćirić-type FPT in the context of $\mathcal{S}$-metric spaces.

Corollary 17.

Let $({\mathring{\mathcal{Y}}}_{ϵ},{\mathcal{E}}_s)$ be a complete $\mathcal{S}$-metric space and $\mathrm{\Gamma }:{\mathring{\mathcal{Y}}}_{ϵ}\to {\mathring{\mathcal{Y}}}_{ϵ}$ be a self-mapping such that $\forall \mathfrak{p},\mathfrak{d}\in {\mathring{\mathcal{Y}}}_{ϵ}$:

$${\mathcal{E}}_s(\mathrm{\Gamma }\mathfrak{p},\mathrm{\Gamma }\mathfrak{p},\mathrm{\Gamma }\mathfrak{d})\le \varphi {\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},\mathfrak{d})+\psi {\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},\mathrm{\Gamma }\mathfrak{p})+\zeta [{\mathcal{E}}_s(\mathfrak{d},\mathfrak{d},\mathrm{\Gamma }\mathfrak{p})+{\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},\mathrm{\Gamma }\mathfrak{d})]+\nabla {\mathcal{E}}_s(\mathfrak{d},\mathfrak{d},\mathrm{\Gamma }\mathfrak{d})\},$$

where $\varphi +\psi +3\zeta +\nabla \in [0,1)$.

Then, Γ possesses only one fixed point.

Remark 18.

Corollary 17 stands as the primary outcome introduced by Hardy and Roger [7], representing an improvement in the context of $\mathcal{S}$-metric spaces.

Corollary 18.

Let $({\mathring{\mathcal{Y}}}_{ϵ},{\mathcal{E}}_s)$ be a complete $\mathcal{S}$-metric space and $\mathrm{\Gamma }:{\mathring{\mathcal{Y}}}_{ϵ}\to {\mathring{\mathcal{Y}}}_{ϵ}$ be a self-mapping such that $\forall \mathfrak{p},\mathfrak{d}\in {\mathring{\mathcal{Y}}}_{ϵ}$:

$${\mathcal{E}}_s(\mathrm{\Gamma }\mathfrak{p},\mathrm{\Gamma }\mathfrak{p},\mathrm{\Gamma }\mathfrak{d})\le \varphi {\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},\mathfrak{d})+\psi {\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},\mathrm{\Gamma }\mathfrak{p})+\nabla {\mathcal{E}}_s(\mathfrak{d},\mathfrak{d},\mathrm{\Gamma }\mathfrak{d})\},\varphi +\psi +\nabla \in [0,1).$$

Then, Γ possesses only one fixed point.

Remark 19.

Corollary 18 stands as the primary outcome introduced by Reich [17], representing an improvement in the context of $\mathcal{S}$-metric spaces.

3. Applications of Transverse Oscillations of a Homogeneous Bar

This section focuses on utilizing the established results to explore the existence of a solution to the boundary value problem that dictates the transverse oscillations of a homogeneous bar.

Let $\mathcal{I}=[0,1]$, ${\mathring{\mathcal{Y}}}_{ϵ}=\mathcal{C}[\mathcal{I},\mathbb{R}]$ be the set of all continuous functions from $\mathcal{I}$ to $\mathbb{R}$, and the $\mathcal{S}$-metric is given by:

$${\mathcal{E}}_s(\mathfrak{g},\mathfrak{h},\mathfrak{f})=\underset{t\in \mathcal{I}}{sup}|\mathfrak{g}\left(t\right)-\mathfrak{h}\left(t\right)|+\underset{t\in \mathcal{I}}{sup}|\mathfrak{g}\left(t\right)+\mathfrak{h}\left(t\right)-2\mathfrak{f}\left(t\right)|.$$

Then, the pair $({\mathring{\mathcal{Y}}}_{ϵ},{\mathcal{E}}_s)$ form a complete $\mathcal{S}$-metric space.

The phenomenon of transverse oscillations of a homogeneous bar holds practical significance. Let us examine a scenario involving a homogeneous bar fixed at one end and free at the other, where the axis of the bar aligns with the segment $(0,1)$ of the x-axis. Considering the deflection parallel to the z-axis at a specific point, the subsequent ordinary differential equation describes the transverse oscillations of the homogeneous bar.

$$\left\{\begin{array}{c}\frac{d_4\mathfrak{w}\left(t\right)}{d{t}^4}={\varphi }^4\mathcal{L}(t,\mathfrak{w}\left(t\right));t\in \mathcal{I}\hfill \\ \mathfrak{w}\left(0\right)={\mathfrak{w}}^{\prime }\left(0\right)={\mathfrak{w}}^{\prime \prime }\left(1\right)={\mathfrak{w}}^{\prime \prime \prime }\left(1\right)=0\hfill \end{array}\right.$$

(2)

where $\mathcal{L}:\mathcal{I}\times \mathbb{R}\to \mathbb{R}$ is a continuous function and $\varphi >0$ is a constant. By utilizing the routine calculus, it is simple to show that the above problem (2) is equivalent to the following Fredholm integral equation:

$$\mathfrak{w}\left(t\right)={\varphi }^4{\int }_0^1\mathcal{R}(t,s)\mathcal{L}(s,\mathfrak{w}\left(s\right))ds,$$

(3)

whose Green function is given by

$$\mathcal{R}=\left\{\begin{array}{c}\frac{3s^{2}t-s^3}{6},0\le s\le t\le 1;\hfill \\ \frac{3t^{2}s-t^3}{6},0\le t\le s\le 1.\hfill \end{array}\right.$$

(4)

Next, suppose the following conditions $\left({\mathcal{B}}_1\right)$ and $\left({\mathcal{B}}_2\right)$ on account of the self-mapping $\mathrm{\Gamma }:{\mathring{\mathcal{Y}}}_{ϵ}\to {\mathring{\mathcal{Y}}}_{ϵ}$ defined as follows:

$$\mathrm{\Gamma }\mathfrak{w}\left(t\right)={\varphi }^4{\int }_0^1\mathcal{R}(t,s)\mathcal{L}(s,\mathfrak{w}\left(s\right))ds,\forall t\in \mathcal{I}.$$

(5)

• $\left({\mathcal{B}}_1\right)$ Suppose a constant $\varphi$ as

  $$0\le \underset{t\in \mathcal{I}}{sup}\nabla (t,\varphi )<1,$$

  where,

  $$\nabla (t,\varphi )={\varphi }^4\left(\frac{t^4-4t^3+6t^2}{24}\right);$$
• $\left({\mathcal{B}}_2\right)$ for each $t\in [0,1]$ and $\mathfrak{p},\mathfrak{d}\in {\mathring{\mathcal{Y}}}_{ϵ}$, the following holds:

  $$|\mathcal{L}(t,\mathrm{\Gamma }\mathfrak{p}\left(t\right))-\mathcal{L}(t,\mathrm{\Gamma }\mathfrak{d}\left(t\right))|\le |\mathfrak{p}(t)-\mathfrak{d}(t)|$$

  (6)

Theorem 3.

Given the specified assumptions $\left({\mathcal{B}}_1\right)$ and $\left({\mathcal{B}}_2\right)$ for self-mapping Γ as defined in (5), a unique solution exists for the ordinary differential Equation (2) that governs the transverse oscillations of the homogeneous bar.

Proof. 

Following the self-mapping $\mathrm{\Gamma }$, and assumptions $\left({\mathcal{B}}_1\right)$ and $\left({\mathcal{B}}_2\right)$, we obtain

$$\begin{array}{cc}\hfill |{\mathrm{\Gamma }}^2\mathfrak{p}\left(t\right)-{\mathrm{\Gamma }}^2\mathfrak{d}\left(t\right)|& =\left|{\varphi }^4{\int }_0^1\mathcal{R}(t,s)\mathcal{L}(s,\mathrm{\Gamma }\mathfrak{p}\left(s\right))ds-{\varphi }^4{\int }_0^1\mathcal{R}(t,s)\mathcal{L}(s,\mathrm{\Gamma }\mathfrak{d}\left(s\right))ds\right|\hfill \\ \hfill & \le {\varphi }^4{\int }_0^1\mathcal{R}(t,s)\left|\mathcal{L}(t,\mathrm{\Gamma }\mathfrak{p}(t\left)\right)-\mathcal{L}(t,\mathrm{\Gamma }\mathfrak{d}(t\left)\right)\right|ds\hfill \\ \hfill & \le {\varphi }^4{\int }_0^1\mathcal{R}(t,s)|\mathfrak{p}\left(t\right)-\mathfrak{d}\left(t\right)|ds\hfill \\ \hfill & \le {\varphi }^4\underset{\xi \in \mathcal{I}}{sup}|\mathfrak{p}\left(\xi \right)-\mathfrak{d}\left(\xi \right)|{\int }_0^1\mathcal{R}(t,s)ds\hfill \\ \hfill & ={\varphi }^4\left(\frac{t^4-4t^3+6t^2}{24}\right)\underset{\xi \in \mathcal{I}}{sup}|\mathfrak{p}\left(\xi \right)-\mathfrak{d}\left(\xi \right)|\hfill \end{array}$$

or this implies the following:

$$\underset{\xi \in \mathcal{I}}{sup}|{\mathrm{\Gamma }}^2\mathfrak{p}\left(t\right)-{\mathrm{\Gamma }}^2\mathfrak{d}\left(t\right)|\le \underset{\xi \in \mathcal{I}}{sup}\nabla (\xi ,\varphi )·\underset{\xi \in \mathcal{I}}{sup}|\mathfrak{p}\left(\xi \right)-\mathfrak{d}\left(\xi \right)|.$$

Equivalently,

$${\mathcal{E}}_s({\mathrm{\Gamma }}^2\mathfrak{p},{\mathrm{\Gamma }}^2\mathfrak{p},{\mathrm{\Gamma }}^2\mathfrak{d})\le \eta {\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},\mathfrak{d}),$$

for $0\le \eta ={sup}_{\xi \in \mathcal{I}}\nabla (\xi ,\varphi )<1$. Hence, all assumptions of Corollary 1 hold for $a=2$. So, there exists a unique solution for (2) in ${\mathring{\mathcal{Y}}}_{ϵ}$. □

4. Application to Nonlinear Fractional Integro-Differential Equations

In this context, we provide a solution for a nonlinear fractional integro-differential equation of the Caputo type. Aydi [19] provided an examination of Ri’s result through w-distances, specifically in the context of studying nonlinear fractional integro-differential equations of the Caputo type. Baleanu [20] focused on establishing the existence of solutions for fractional differential equations. Additionally, Baleanu [21] addressed the fractional motion of a bead sliding on a wire.

The Caputo derivative of a continuous function $g:[0,\infty )\to \mathbb{R}$ (with a positive order $\delta$) is defined in the following manner:

$${}_{}{}^{cf}{\mathcal{D}}^{\delta }\left(\mathfrak{w}\left(t\right)\right)_{}^{}:=\frac{1}{\mathfrak{D}\left(n-\delta \right)}{\int }_0^t{(t-\tau )}^{n-\delta -1}{\mathfrak{w}}^{\left(n\right)}\left(\tau \right)d\tau (n-1<\delta <n,n=\left[\delta \right]+1)$$

In this context, $\mathfrak{D}\left(n-\delta \right)$ denotes the gamma function of $n-\delta$, and $\left[\delta \right]$ signifies the integer part of the positive real number $\delta$.

In this section, we explore the integro-differential equation of the Caputo type with a nonlinear component:

$${}_{}{}^{cf}{\mathcal{D}}^{\delta }\left(\mathfrak{w}\left(t\right)\right)_{}^{}=\mathcal{G}(t,\mathfrak{w}\left(t\right))$$

(7)

subject to the specified boundary conditions, $\mathfrak{w}\left(0\right)=0,\mathfrak{w}\left(1\right)={\int }_0^{\mathrm{\Theta }}\mathfrak{w}\left(\tau \right)d\tau .$

In this scenario, $\mathcal{G}:\mathcal{I}\times \mathbb{R}\to \mathbb{R}$ is a continuous function, and $\mathfrak{w}\in \mathcal{C}(\mathcal{I},[0,\infty \left)\right)$ satisfies $0<\mathrm{\Theta }<1$ and $1<\delta \le 2$ (refer to [20] for more details). It is apparent that a solution to Equation (7) corresponds to a fixed point of the following integral equation:

$$\begin{array}{cc}\hfill \mathrm{\Gamma }\mathfrak{w}\left(t\right)=& \frac{1}{\mathfrak{D}\left(\delta \right)}{\int }_0^t{(t-\tau )}^{\delta -1}\mathcal{G}(\tau ,\mathfrak{w}\left(\tau \right))d\tau \hfill \\ \hfill & -\frac{2t}{\left(2-{\mathrm{\Theta }}^2\right)\mathfrak{D}\left(\delta \right)}{\int }_0^1{(1-\tau )}^{\delta -1}\mathcal{G}(\tau ,\mathfrak{w}\left(\tau \right))d\tau \hfill \\ \hfill & +\frac{2t}{\left(2-{\mathrm{\Theta }}^2\right)\mathfrak{D}\left(\delta \right)}{\int }_0^{\mathrm{\Theta }}\left({\int }_0^{\tau }{(\tau -m)}^{\delta -1}\mathcal{G}(m,\mathfrak{w}\left(m\right))dm\right)d\tau \hfill \end{array}$$

(8)

Theorem 4.

In light of the nonlinear fractional differential Equation (7), we posit the following assumption:

$$|\mathcal{G}(\tau ,\mathfrak{w}\left(\tau \right))-\mathcal{G}(\tau ,\mathfrak{u}\left(\tau \right))|\le \frac{\mathfrak{D}\left(\delta +1\right)}{5}{e}^{-\xi }\mid \sqrt{|\mathfrak{w}(\tau )|}-\sqrt{|\mathfrak{u}(\tau )|\mid }$$

and

$$|\mathcal{G}(\tau ,\mathfrak{w}\left(\tau \right))+\mathcal{G}(\tau ,\mathfrak{u}\left(\tau \right))|\le \frac{\mathfrak{D}\left(\delta +1\right)}{5}{e}^{-\xi }\mid \sqrt{|\mathfrak{w}(\tau )|}+\sqrt{|\mathfrak{u}(\tau )|\mid }$$

for any given τ within the interval $\mathcal{I}$, with $\xi >0$, and for all functions, $\mathfrak{w},\mathfrak{u}\in \mathcal{C}[\mathcal{I},\mathbb{R}]$. If the specified conditions are met, then there exists a unique solution to Equation (7).

Proof. 

As $\mathcal{I}=[0,1]$, and ${\mathring{\mathcal{Y}}}_{ϵ}=\mathcal{C}[\mathcal{I},let\mathbb{R}]$ be the set of all continuous functions from $\mathcal{I}$ to $\mathbb{R}$ with the $\mathcal{S}$-metric given by the following:

$${\mathcal{E}}_{\tau }(\mathfrak{g},\mathfrak{h},\mathfrak{f})=\underset{t\in \mathcal{I}}{sup}|\mathfrak{g}\left(t\right)-\mathfrak{h}\left(t\right)|+\underset{t\in \mathcal{I}}{sup}|\mathfrak{h}\left(t\right)-\mathfrak{f}\left(t\right)|.$$

Then, the pair $({\mathring{\mathcal{Y}}}_{ϵ},{\mathcal{E}}_{\tau })$ form a complete $\mathcal{S}$-metric space. Next, we show that $\mathrm{\Gamma }$ satisfies all assumptions of Corollary 1 for $a=2$. In this context, we consider $\mathfrak{w},\mathfrak{u}\in {\mathring{\mathcal{Y}}}_{ϵ}$, with t belonging to the interval $\mathcal{I}$. The result is as follows:

$$\begin{array}{cc}\hfill |\mathrm{\Gamma }\mathfrak{w}(t)-\mathrm{\Gamma }\mathfrak{u}(t)|=& \mid \frac{1}{\mathfrak{D}\left(\delta \right)}{\int }_0^t{(t-\tau )}^{\delta -1}\mathcal{G}(\tau ,\mathfrak{w}\left(\tau \right))d\tau \hfill \\ & -\frac{2t}{\left(2-{\mathrm{\Theta }}^2\right)\mathfrak{D}\left(\delta \right)}{\int }_0^1{(1-\tau )}^{\delta -1}\mathcal{G}(\tau ,\mathfrak{w}\left(\tau \right))d\tau \hfill \\ & +\frac{2t}{\left(2-{\mathrm{\Theta }}^2\right)\mathfrak{D}\left(\delta \right)}{\int }_0^{\mathrm{\Theta }}\left({\int }_0^{\tau }{(\tau -m)}^{\delta -1}\mathcal{G}(m,\mathfrak{w}\left(m\right))dm\right)d\tau \hfill \\ & -\frac{1}{\mathfrak{D}\left(\delta \right)}{\int }_0^t{(t-\tau )}^{\delta -1}\mathcal{G}(\tau ,\mathfrak{u}\left(\tau \right))d\tau \hfill \\ & +\frac{2t}{\left(2-{\mathrm{\Theta }}^2\right)\mathfrak{D}\left(\delta \right)}{\int }_0^1{(1-\tau )}^{\delta -1}\mathcal{G}(\tau ,\mathfrak{u}\left(\tau \right))d\tau \hfill \\ & -\frac{2t}{\left(2-{\mathrm{\Theta }}^2\right)\mathfrak{D}\left(\delta \right)}{\int }_0^{\mathrm{\Theta }}\left({\int }_0^{\tau }{(\tau -m)}^{\delta -1}\mathcal{G}(m,\mathfrak{u}\left(m\right))dm\right)d\tau \mid .\hfill \end{array}$$

Consequently,

$$\begin{array}{cc}\hfill |\mathrm{\Gamma }\mathfrak{w}(t)-\mathrm{\Gamma }\mathfrak{u}(t)|\le & \frac{1}{\mathfrak{D}\left(\delta \right)}{\int }_0^t{|t-\tau |}^{\delta -1}\frac{\mathfrak{D}\left(\delta +1\right)}{5}\left[e^{-\xi }\underset{\tau \in \mathcal{I}}{sup}\mid \sqrt{|\mathfrak{w}(\tau )|}-\sqrt{|\mathfrak{u}(\tau )|\mid }\right]d\tau \hfill \\ & +\frac{2t}{\left(2-{\mathrm{\Theta }}^2\right)\mathfrak{D}\left(\delta \right)}{\int }_0^1{(1-\tau )}^{\delta -1}\frac{\mathfrak{D}\left(\delta +1\right)}{5}\left[e^{-\xi }\underset{\tau \in \mathcal{I}}{sup}\mid \sqrt{|\mathfrak{w}(\tau )|}-\sqrt{|\mathfrak{u}(\tau )|\mid }\right]d\tau \hfill \\ & +\frac{2t}{\left(2-{\mathrm{\Theta }}^2\right)\mathfrak{D}\left(\delta \right)}{\int }_0^{\mathrm{\Theta }}\left|{\int }_0^{\tau }{(\tau -m)}^{\delta -1}\frac{\mathfrak{D}\left(\delta +1\right)}{5}\left[e^{-\xi }\underset{\tau \in \mathcal{I}}{sup}\mid \sqrt{|\mathfrak{w}(\tau )|}-\sqrt{|\mathfrak{u}(\tau )|\mid }\right]dm\right|d\tau \hfill \\ \hfill \le & \frac{\mathfrak{D}\left(\delta +1\right)}{5}\left[e^{-\xi }\underset{\tau \in \mathcal{I}}{sup}\mid \sqrt{\mathfrak{w}\left(\tau \right)}-\sqrt{\mathfrak{u}\left(\tau \right)\mid }\right]\times \underset{t\in \mathcal{I}}{sup}\left(\frac{1}{\mathfrak{D}\left(\delta \right)}{\int }_0^1{|t-\tau |}^{\delta -1}d\tau \right.\hfill \\ & \left.+\frac{2t}{\left(2-{\mathrm{\Theta }}^2\right)\mathfrak{D}\left(\delta \right)}{\int }_0^1{(1-\tau )}^{\delta -1}d\tau +\frac{2t}{\left(2-{\mathrm{\Theta }}^2\right)\mathfrak{D}\left(\delta \right)}{\int }_0^{\mathrm{\Theta }}{\int }_0^{\tau }{|\tau -m|}^{\delta -1}dmd\tau \right)\hfill \\ \hfill \le & e^{-\xi }\underset{\tau \in \mathcal{I}}{sup}\mid \sqrt{|\mathfrak{w}(\tau )|}-\sqrt{|\mathfrak{u}(\tau )|\mid }.\hfill \end{array}$$

Similarly,

$$\begin{array}{cc}\hfill |\mathrm{\Gamma }\mathfrak{w}(t)+\mathrm{\Gamma }\mathfrak{u}(t)|=& \mid \frac{1}{\mathfrak{D}\left(\delta \right)}{\int }_0^t{(t-\tau )}^{\delta -1}\mathcal{G}(\tau ,\mathfrak{w}\left(\tau \right))d\tau \hfill \\ & -\frac{2t}{\left(2-{\mathrm{\Theta }}^2\right)\mathfrak{D}\left(\delta \right)}{\int }_0^1{(1-\tau )}^{\delta -1}\mathcal{G}(\tau ,\mathfrak{w}\left(\tau \right))d\tau \hfill \\ & +\frac{2t}{\left(2-{\mathrm{\Theta }}^2\right)\mathfrak{D}\left(\delta \right)}{\int }_0^{\mathrm{\Theta }}\left({\int }_0^{\tau }{(\tau -m)}^{\delta -1}\mathcal{G}(m,\mathfrak{w}\left(m\right))dm\right)d\tau \mid \hfill \\ & +\mid \frac{1}{\mathfrak{D}\left(\delta \right)}{\int }_0^t{(t-\tau )}^{\delta -1}\mathcal{G}(\tau ,\mathfrak{u}\left(\tau \right))d\tau \hfill \\ & -\frac{2t}{\left(2-{\mathrm{\Theta }}^2\right)\mathfrak{D}\left(\delta \right)}{\int }_0^1{(1-\tau )}^{\delta -1}\mathcal{G}(\tau ,\mathfrak{u}\left(\tau \right))d\tau \hfill \\ & +\frac{2t}{\left(2-{\mathrm{\Theta }}^2\right)\mathfrak{D}\left(\delta \right)}{\int }_0^{\mathrm{\Theta }}\left({\int }_0^{\tau }{(\tau -m)}^{\delta -1}\mathcal{G}(m,\mathfrak{u}\left(m\right))dm\right)d\tau \mid ,\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill |\mathrm{\Gamma }\mathfrak{w}(t)|+|\mathrm{\Gamma }\mathfrak{u}(t)|\le & \frac{1}{\mathfrak{D}\left(\delta \right)}{\int }_0^t{|t-\tau |}^{\delta -1}\left(\frac{\mathfrak{D}\left(\delta +1\right)}{5}{e}^{-\xi }|\sqrt{|\mathfrak{w}(\tau )|}+\sqrt{|\mathfrak{u}(\tau )|}|\right)d\tau \hfill \\ & +\frac{2t}{\left(2-{\mathrm{\Theta }}^2\right)\mathfrak{D}\left(\delta \right)}{\int }_0^1{(1-\tau )}^{\delta -1}\left(\left|\frac{\mathfrak{D}\left(\delta +1\right)}{5}{e}^{-\xi }\right|\sqrt{\mathfrak{w}\left(\tau \right)}+\sqrt{\mathfrak{u}\left(\tau \right)\mid }\right)d\tau \hfill \\ & +\frac{2t}{\left(2-{\mathrm{\Theta }}^2\right)\mathfrak{D}\left(\delta \right)}{\int }_0^{\mathrm{\Theta }}\left|{\int }_0^{\tau }{(\tau -m)}^{\delta -1}\frac{\mathfrak{D}\left(\delta +1\right)}{5}{e}^{-\xi }\right|\sqrt{\mathfrak{w}\left(m\right)}+\sqrt{\mathfrak{u}\left(m\right)\mid }dm\mid d\tau \hfill \\ \hfill \le & \frac{\mathfrak{D}\left(\delta +1\right)}{5}\underset{\tau \in \mathcal{I}}{sup}\mid \sqrt{|\mathfrak{w}(\tau )|}+\sqrt{|\mathfrak{u}(\tau )|}\hfill \\ & \underset{t\in \mathcal{I}}{sup}\left(\frac{1}{\mathfrak{D}\left(\delta \right)}{\int }_0^t{|t-\tau |}^{\delta -1}d\tau +\frac{2t}{\left(2-{\mathrm{\Theta }}^2\right)\mathfrak{D}\left(\delta \right)}{\int }_0^1{(1-\tau )}^{\delta -1}d\tau \right.\hfill \\ & \left.+\frac{2t}{\left(2-{\mathrm{\Theta }}^2\right)\mathfrak{D}\left(\delta \right)}{\int }_0^{\mathrm{\Theta }}\left|{\int }_0^{\tau }{(\tau -m)}^{\delta -1}dm\right|d\tau \right)\hfill \\ \hfill \le & \underset{\tau \in \mathcal{I}}{sup}|\sqrt{|\mathfrak{w}(\tau )|}+\sqrt{|\mathfrak{u}(\tau )|}|.\hfill \end{array}$$

This gives rise to the following:

$$\underset{t\in \mathcal{I}}{sup}(|\mathrm{\Gamma }\mathfrak{w}\left(t\right)|+|\mathrm{\Gamma }\mathfrak{u}\left(t\right)|)\le \underset{\tau \in \mathcal{I}}{sup}\mid \sqrt{|\mathfrak{w}(\tau )|}+\sqrt{|\mathfrak{u}(\tau )|}.$$

In conclusion,

$$\begin{array}{cc}\hfill {\mathcal{E}}_{\tau }\left({\mathrm{\Gamma }}^2\mathfrak{w},{\mathrm{\Gamma }}^2\mathfrak{w},{\mathrm{\Gamma }}^2\mathfrak{u}\right)& =\underset{t\in \mathcal{I}}{sup}|{\mathrm{\Gamma }}^2\mathfrak{w}-{\mathrm{\Gamma }}^2\mathfrak{u}|\hfill \\ & =\underset{t\in \mathcal{I}}{sup}(|\mathrm{\Gamma }\mathfrak{w}\left(t\right)-\mathrm{\Gamma }\mathfrak{u}\left(t\right)|\times |\mathrm{\Gamma }\mathfrak{w}\left(t\right)+\mathrm{\Gamma }\mathfrak{u}\left(t\right)|)\hfill \\ & =\underset{t\in \mathcal{I}}{sup}(|\mathrm{\Gamma }\mathfrak{w}\left(t\right)-\mathrm{\Gamma }\mathfrak{u}\left(t\right)|)\times \underset{t\in \mathcal{I}}{sup}(|\mathrm{\Gamma }\mathfrak{w}\left(t\right)+\mathrm{\Gamma }\mathfrak{u}\left(t\right)|)\hfill \\ & \le \underset{t\in \mathcal{I}}{sup}(|\mathrm{\Gamma }\mathfrak{w}\left(t\right)-\mathrm{\Gamma }\mathfrak{u}\left(t\right)|)\times \underset{t\in \mathcal{I}}{sup}(|\mathrm{\Gamma }\mathfrak{w}\left(t\right)|+|\mathrm{\Gamma }\mathfrak{u}\left(t\right)|)\hfill \\ & \le e^{-\xi }\underset{\tau \in \mathcal{I}}{sup}\left|\sqrt{|\mathfrak{w}(\tau )|}-\sqrt{|\mathfrak{u}(\tau )|\mid }\times \underset{\tau \in \mathcal{I}}{sup}\right|\sqrt{|\mathfrak{w}(\tau )|}+\sqrt{|\mathfrak{u}(\tau )|}\hfill \\ & =e^{-\xi }\underset{\tau \in \mathcal{I}}{sup}|\mathfrak{w}\left(\tau \right)|-|\mathfrak{u}\left(\tau \right)|\mid \hfill \\ & \le e^{-\xi }\underset{\tau \in \mathcal{I}}{sup}|\mathfrak{w}\left(\tau \right)-\mathfrak{u}\left(\tau \right)|\hfill \\ & =\Phi {\mathcal{E}}_{\tau }(\mathfrak{w},\mathfrak{w},\mathfrak{u}),\hfill \end{array}$$

All conditions are met for $\mathfrak{w},\mathfrak{u}\in {\mathring{\mathcal{Y}}}_{ϵ}$ and $\Phi =e^{-\xi }$. This ensures the fulfillment of all criteria outlined in Corollary 1, specifically for $a=2$. Consequently, a unique fixed point is guaranteed for $\mathrm{\Gamma }$. Hence, a unique solution exists for the Caputo-type nonlinear fractional differential Equation (7). □

5. Conclusions and Future Directions

In summary, our investigation introduces the novel class of generalized Sehgal–Guseman-like contraction mappings in the context of $\mathcal{S}$-metric spaces. These mappings encompass extensions of well-known contraction types, such as Banach contractions, Kannan contractions, Chatterjee contractions, and others, within the framework of $\mathcal{S}$-metric spaces. This study establishes consequential fixed-point results and extends their application to address nonlinear fractional equations and the boundary value problem for homogeneous transverse bars. To illustrate the practical implications of our results, we present non-trivial examples that showcase the applicability and versatility of the proposed generalized contraction mappings. These examples not only serve as concrete instances but also highlight the broader scope and significance of our findings in diverse mathematical contexts.

In essence, our research contributes both theoretical advancements in $\mathcal{S}$-metric spaces and practical insights by demonstrating the existence and uniqueness of solutions, particularly in the domains of nonlinear fractional equations and transverse oscillations of homogeneous bars.

Open problem: Investigate the specific conditions on $\mathfrak{g}\in \mathcal{D}$ and $\mathrm{\Gamma }$ such that the self-mapping $\mathrm{\Gamma }$ possesses a unique solution satisfying the following: For every $\mathfrak{p}\in {\mathring{\mathcal{Y}}}_{ϵ}$, there exists $a\left(\mathfrak{p}\right)\in \mathbb{N}$ such that, for all $\mathfrak{d}\in {\mathring{\mathcal{Y}}}_{ϵ}$, the following inequality holds:

$$\begin{array}{cc}\hfill {\mathcal{E}}_s\left({\mathrm{\Gamma }}^{a\left(\mathfrak{p}\right)}\mathfrak{p},{\mathrm{\Gamma }}^{a\left(\mathfrak{p}\right)}\mathfrak{p},{\mathrm{\Gamma }}^{a\left(\mathfrak{p}\right)}\mathfrak{d}\right)& \le \mathfrak{g}[{\mathcal{E}}_s(\mathfrak{p},\mathfrak{p},\mathfrak{d}),{\mathcal{E}}_s({\mathrm{\Gamma }}^{a\left(\mathfrak{p}\right)}\mathfrak{p},{\mathrm{\Gamma }}^{a\left(\mathfrak{p}\right)}\mathfrak{p},\mathfrak{p}),{\mathcal{E}}_s({\mathrm{\Gamma }}^{a\left(\mathfrak{p}\right)}\mathfrak{p},{\mathrm{\Gamma }}^{a\left(\mathfrak{p}\right)}\mathfrak{p},\mathfrak{d}),\hfill \\ \hfill & {\mathcal{E}}_s({\mathrm{\Gamma }}^{a\left(\mathfrak{p}\right)}\mathfrak{d},{\mathrm{\Gamma }}^{a\left(\mathfrak{p}\right)}\mathfrak{d},\mathfrak{p}),{\mathcal{E}}_s({\mathrm{\Gamma }}^{a\left(\mathfrak{p}\right)}\mathfrak{d},{\mathrm{\Gamma }}^{a\left(\mathfrak{p}\right)}\mathfrak{d},\mathfrak{d})].\hfill \end{array}$$

Explore the specific conditions required for $\mathrm{\Gamma }$ and $\mathfrak{g}$ to ensure the uniqueness of the fixed point of $\mathrm{\Gamma }$. This question invites further exploration and investigation to establish the necessary conditions for uniqueness in the context of the given self-mapping.