Graph-Theoretic Perspectives on Fixed Points in Double-Composed Metric Spaces

Abstract

This study explores the development of fixed-point results in the setting of the recently proposed double-composed metric spaces. We establish conditions ensuring both existence and uniqueness of fixed points for several types of contractive mappings defined on such spaces. To enrich the analysis, the space is further equipped with a graph structure through the use of concepts from graph theory, leading to the formulation of two novel fixed-point theorems. An illustrative example is also provided to highlight the applicability and relevance of the obtained results.

1. Introduction

Fixed-point (FP) theory constitutes a fundamental pillar of mathematical analysis, serving as a powerful framework for the systematic study of mappings and their structural properties in a wide range of mathematical spaces. Among the classical results in this area, the Banach FP theorem [1]—commonly referred to as the Contraction Mapping Theorem—holds a distinguished position as perhaps the most celebrated result within the theory of metric spaces. The significance of this theorem transcends its original formulation, as it has inspired a vast body of research devoted to its refinement, extension, and application across diverse mathematical and applied contexts. In particular, the last few decades have witnessed an intensification of efforts aimed at exploring FP results within increasingly sophisticated and generalized structures, thereby motivating the development and investigation of FP theorems across various classes of metric spaces [2,3,4,5,6,7,8,9]. For instance, Jiang et al. [10] examined generalized contractive mappings in complete metric spaces, providing new insights into their fixed-point behavior and extending classical results to broader contexts. Similarly, Panda et al. [11] introduced novel approaches for solving nonlinear integral equations through various ${\mathcal{F}}_{B_e}$ contractions, illustrating the practical applicability of FP theory in analytical problems. Additionally, George et al. [12] studied rectangular b-metric spaces, establishing contraction principles that further generalized conventional metric space frameworks and opened new avenues for theoretical exploration.

These developments in generalized metric spaces highlight the theoretical significance of FP theory while opening avenues for applications in geometrically and physically motivated contexts. In fact, by examining how geometric structures influence the existence and properties of fixed points, mathematicians gain deeper insights into both theoretical constructs and practical systems. A notable application is in satellite dynamics, where fixed points are essential for understanding the equilibrium positions of satellites under gravitational forces. The geometric configuration of these points aids in designing stable orbits and predicting satellite behavior. A prominent application was introduced by Joshi et al. [13], where a satellite web coupling was idealized as a thin sheet connecting two cylindrical satellites. The study of radiation from the web coupling between satellites leads to a nonlinear boundary value problem, which can be effectively analyzed using appropriate FP theorems in S-metric spaces. Also, Khan et al. [14] applied fixed-point results in fuzzy S-metric spaces to solve this satellite web coupling problem.

Moreover, the versatility of FP theory extends beyond geometric and physical systems, finding powerful applications in graph theory, where it helps analyze network structures and study mappings on graphs. Indeed, the interplay between FP theory and graph theory has garnered increasing attention, leading to new insights and advancements in both fields. By endowing metric spaces with graph structures, researchers can explore the dynamics of mappings in a more flexible and nuanced manner. For instance, Samreen et al. [15] propose several notions of contraction mappings in b-metric spaces endowed with a graph G, and, using these concepts, they establish corresponding FP results for these classes of contractions. Building on the applicability of their framework, they also derive FP theorems for cyclic operators and prove an existence theorem for the solution of a specific integral equation. In this setting, Souayah et al. [16] introduce the concept of the $G_m$-contraction principle and establish a series of FP theorems within the framework of M-metric spaces endowed with a graph. In addition, the existence and uniqueness of FPs for such contractions are rigorously investigated. Several related contributions and applications have been reported in the following studies [17,18,19,20].

This approach allows for the examination of not only the distances between points but also the relational properties captured by graphs, such as connectivity and directedness. The significance of this intersection is underscored by its diverse applications across various disciplines, including computer science, optimization, and social network analysis. For instance, in network theory, FPs can represent stable states of dynamic systems, while in optimization, iterative methods for finding solutions can be effectively modeled as traversals of graphs.

One of the new metric spaces recently introduced is the double-composed metric spaces introduced in [21]. A new triangle inequality was proposed with the form $D_c(s,t)\le {\alpha }_1\left(D_c(s,r)\right)+{\alpha }_2\left(D_c(r,t)\right)$ for all $r,s,t\in E$, where the control functions ${\alpha }_1,{\alpha }_2:[0,\infty )\to [0,\infty )$ are composed with the metric $D_c$ in the triangle inequality.

This paper is devoted to establishing novel FP theorems within the framework of double-composed metric spaces and to extending these results to settings in which the underlying space is endowed with a graph, in accordance with the framework proposed by Jachymski [22]. By connecting these two domains, we strive to deepen insights into FPs on graphs, paving the way for further theoretical developments and real-world applications.

2. Preliminaries

In 2020, Abdeljawad et al. [23] presented the notion of the double-controlled metric spaces (this serves as an extension of controlled metric spaces) as follows:

Definition 1.

Let $\mathcal{S}$ be a nonempty set, and ${\alpha }_1,{\alpha }_2:\mathcal{W}\times \mathcal{W}\to [1,\infty )$. A function ${\rm Y}:\mathcal{W}\times \mathcal{W}\to [0,\infty )$ is called a double-controlled metric-type space if it satisfies

1. 

${\rm Y}(s,t)=0$ if and only if $s=t$ for all $s,t\in \mathcal{W}$;

2. 

${\rm Y}(s,t)={\rm Y}(t,s)$ for all $s,t\in \mathcal{W}$;

3. 

${\rm Y}(s,t)\le {\alpha }_1(s,\mu ){\rm Y}(s,\mu )+{\alpha }_2(\mu ,t){\rm Y}(\mu ,t)$ for all $s,t,\mu \in \mathcal{W}$.

The pair $\left(\mathcal{S},{\rm Y}\right)$ is a said to be double-controlled metric-type space.

Inspired from the double-controlled metric spaces, recently, Irshad et al. [21] introduced double-composed metric spaces with a new triangle inequality using control functions as follows:

Definition 2

(DCM) [21]). Let $\mathcal{S}$ be a nonempty set, and consider the two non-constant functions ${\alpha }_1,{\alpha }_2:[0,\infty )\to [0,\infty )$. A function ${\rm Y}:\mathcal{S}\times \mathcal{S}\to [0,\infty )$ is said to be a double-composed metric if it satisfies

1. 

${\rm Y}(s,t)=0$⟺$s=t$ for all $s,t\in \mathcal{S}$;

2. 

${\rm Y}(s,t)={\rm Y}(t,s)$ for all $s,t\in \mathcal{S}$;

3. 

${\rm Y}(s,t)\le {\alpha }_1\left({\rm Y}(s,\mu )\right)+{\alpha }_2\left({\rm Y}(\mu ,t)\right)$ for all $\mu ,t,s\in \mathcal{S}$.

The pair $\left(\mathcal{S},{\rm Y}\right)$ is a called double-composed metric space ($DCM$).

Remark 1.

Every metric space can be considered a DCM with the control functions ${\alpha }_1\left(s\right)={\alpha }_2\left(s\right)=s$. However, the converse does not always hold, as illustrated by the following example.

Example 1

([21]). Let $\mathcal{S}=\mathbb{R}$ and ${\alpha }_1,{\alpha }_2:[0,\infty )\to [0,\infty )$ are defined by ${\alpha }_1\left(s\right)={\alpha }_2\left(s\right)=e^s$. Define a function ${\rm Y}:\mathcal{S}\times \mathcal{S}\to [0,\infty )$ by

$${\rm Y}(s,t)={(s-t)}^2.$$

Then, $(\mathcal{S},{\rm Y})$ is a $DCM$ with control functions ${\alpha }_1$ and ${\alpha }_2$, but it is not a metric space.

From Example 1, DCMs need not be metric spaces; however, every DCM with sub-additive control functions gives rise to a metric space in the following manner.

Proposition 1

([21]). Let $(\mathcal{S},{\rm Y})$ be a DCM, with ${\alpha }_1,{\alpha }_2:[0,\infty )\to [0,\infty )$ being two non-constant sub-additive control functions. Define a function ${\mathcal{D}}_m:\mathcal{S}\times \mathcal{S}\to [0,\infty )$ by

$${\mathcal{D}}_m(s,t)=\left\{\begin{array}{cc}0,\hfill & ⟺s=t\hfill \\ {\alpha }_1\left({\rm Y}(s,t)\right)+{\alpha }_2\left({\rm Y}(s,t)\right),\hfill & \mathrm{if}s\ne t\hfill \end{array}\right.$$

Then, $(\mathcal{S},{\mathcal{D}}_m)$ is a metric space.

Example 2.

Let $\mathcal{S}=[-{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}]$. Define function ${\alpha }_1,{\alpha }_2:[0,\infty )\to [0,\infty )$ as follows:

$${\alpha }_1\left(r\right)={\alpha }_2\left(r\right)=\left\{\begin{array}{cc}sin\left(3{sin}^{-1}\left(r\right)\right)\hfill & if0\le r\le {\displaystyle \frac{1}{2}}\hfill \\ 1\hfill & if{\displaystyle \frac{1}{2}}<r<\infty .\hfill \end{array}\right.$$

Define a function ${\rm Y}:\mathcal{S}\to [0,\infty )$ by

$${\rm Y}(r,s)={(r-s)}^2.$$

Then, $(\mathcal{S},{\rm Y})$ is a DCM with the control functions ${\alpha }_1$ and ${\alpha }_2$.

Proof. 

The first and second assumptions in Definition 2 are straightforward to verify. Let us prove the triangle inequality. For all $r,s,t\in \mathcal{S}$, we have

$${(r-s)}^2\le {(r-t+t-s)}^2\le 2{(r-t)}^2+2{(t-s)}^2.$$

(1)

On the other hand, consider the function $f\left(u\right)=2u^2-sin\left(3{sin}^{-1}\left(u^2\right)\right)$ for $u\in [0,{\displaystyle \frac{1}{2}}]$. Using variational calculus, we obtain $f\left(u\right)\le 0$; that is,

$$2u^2\le sin\left(3{sin}^{-1}\left(u^2\right)\right)\mathrm{for} \mathrm{all} u\in [0,{\displaystyle \frac{1}{2}}].$$

(2)

From (1) and (2), we obtain

$$\begin{array}{ccc}\hfill {(r-s)}^2& \le & sin\left[3{sin}^{-1}{(r-t)}^2\right]+sin\left[3{sin}^{-1}{(t-s)}^2\right]\hfill \\ & =& {\alpha }_1\left({(r-t)}^2\right)+{\alpha }_2\left({(t-s)}^2\right)\hfill \\ & =& {\alpha }_1({\rm Y}(r,t))+{\alpha }_2({\rm Y}(t,s)).\hfill \end{array}$$

(3)

The inequality (3) proves the triangle inequality. Then, $(\mathcal{S},{\rm Y})$ is a $DCM$. □

Now, the topology of the DCM is defined as follows:

Definition 3

([21]). Let $\left(\mathcal{S},{\rm Y}\right)$ be a DCM. For each sequence $\left\{{\hslash }_n\right\}\in \mathcal{S}$, we say

1. 

That $\left\{{\hslash }_n\right\}$ a Cauchy sequence if ${\displaystyle \underset{n,m\to \infty }{lim}{\rm Y}\left({\hslash }_n,{\hslash }_m\right)}$ exists and is finite;

2. 

That $\left\{{\hslash }_n\right\}$ converges to ℏ if ${\displaystyle \underset{n\to \infty }{lim}{\rm Y}\left({\hslash }_n,\hslash \right)=0}$;

3. 

That $\left(\mathcal{S},\hslash \right)$ is complete if every Cauchy sequence in $\mathcal{S}$ is convergent to some point in $\mathcal{S}$ and is noted CDCM.

Proposition 2

([21]). Let $\left(\mathcal{S},{\rm Y}\right)$ be a DCM with two continuous and non-constant control functions ${\alpha }_1,{\alpha }_2:[0,\infty )\to [0,\infty )$ such that ${\alpha }_1\left(0\right)+{\alpha }_2\left(0\right)=0$. Then, the limit of every convergent sequence is unique.

3. Main Result

This section presents some of the main results, establishing new fixed-point theorems in CDCM. By leveraging the concepts of sequential and subsequential convergence, we provide sufficient conditions for the existence and uniqueness of FPs for various classes of mappings.

Definition 4.

Let $(\mathcal{S},{\rm Y})$ be a CDCM, and $g:\mathcal{S}\to \mathcal{S}$ is a mapping.

1. 

For each sequence $\left\{{\hslash }_n\right\}$, if $\left\{g{\hslash }_n\right\}$ is convergent, then $\left\{{\hslash }_n\right\}$ also converges, and then g is called sequentially convergent.

2. 

For every sequence $\left\{{\hslash }_n\right\}$, if $\left\{g{\hslash }_n\right\}$ is convergent, then $\left\{{\hslash }_n\right\}$ has a convergent subsequence, and then g is said to be subsequentially convergent.

Theorem 1.

Let $(\mathcal{S},{\rm Y})$ be a CDCM with non-constant control functions ${\alpha }_1,{\alpha }_2:[0,\infty )\to [0,\infty )$. Let $F_1,F_2:\mathcal{S}\to \mathcal{S}$ be a mapping such that $F_1$ is continuous, subsequentially convergent, one-to-one, and

$${\rm Y}(F_1{F}_{2}s,F_1{F}_{2}t)\le q[{\rm Y}(F_{1}s,F_1{F}_{2}s)+{\rm Y}(F_{1}t,F_1{F}_{2}t)]$$

(4)

for all $s,t\in \mathcal{S}$ and $q\in (0,{\displaystyle \frac{1}{2}}).$ For ${\hslash }_0\in \mathcal{S}$, define a sequence $\left\{{\hslash }_n\right\}$ by ${\hslash }_n=F_2^n{\hslash }_0$. Suppose that the following conditions are satisfied:

1. 

${\displaystyle \underset{m,n\to \infty }{lim}\left[\sum _{i=n}^{m-2}{\alpha }_2^i\left({\alpha }_1\left(r^i{\rm Y}(F_1{\hslash }_0,F_1{\hslash }_1)\right)\right)+{\alpha }_2^{m-n-1}\left(r^{n-1}{\rm Y}(F_1{\hslash }_0,F_1{\hslash }_1)\right)\right]=0}$ where $r={\displaystyle \frac{q}{1-q}}$, and ${\alpha }_2^i\left({\alpha }_1\left(r^i{\rm Y}(F_1{\hslash }_0,F_1{\hslash }_1)\right)\right)$ and ${\alpha }_2^{m-n-1}\left(r^{n-1}{\rm Y}(F_1{\hslash }_0,F_1{\hslash }_1)\right)$ denote the composite functions.

2. 

${\alpha }_1$ and ${\alpha }_2$ are non-decreasing functions and continuous with ${\alpha }_1\left(0\right)+{\alpha }_2\left(0\right)=0$ and ${\alpha }_2\left(qs\right)<s$ for every $s\in \mathcal{S}$ and $q\in (0,1).$

3. 

${\alpha }_2$ is sub-additive, and ${\alpha }_1(ks+t)\le k{\alpha }_1\left(s\right)+{\alpha }_1\left(t\right)$ for all $s,t\in \mathcal{S}$.

Hence, $F_2$ has a unique FP.

Proof. 

Let ${\hslash }_0\in \mathcal{S}$, and define a sequence $\left\{{\hslash }_n\right\}$ in $\mathcal{S}$ inductively as follows: ${\hslash }_n=F_2{\hslash }_{n-1},n\ge 1$. Set ${{\rm Y}}_n={\rm Y}(F_1{\hslash }_n,F_1{\hslash }_{n+1})$; then we have,

$$\begin{array}{cc}\hfill {{\rm Y}}_n={\rm Y}\left(F_1{\hslash }_n,F_1{\hslash }_{n+1}\right)& ={\rm Y}\left(F_1{F}_2{\hslash }_{n-1},F_1{F}_2{\hslash }_n\right)\hfill \\ \hfill & \le q\left[{\rm Y}(F_1{\hslash }_{n-1},F_1{F}_2{\hslash }_{n-1})+{\rm Y}(F_1{\hslash }_n,F_1{F}_2{\hslash }_n)\right]\hfill \\ \hfill & =q\left[{\rm Y}(F_1{\hslash }_{n-1},F_1{F}_2{\hslash }_{n-1})+{\rm Y}(F_1{\hslash }_n,F_1{\hslash }_{n+1})\right]\hfill \\ \hfill & \le q\left[{{\rm Y}}_{n-1}+{{\rm Y}}_n)\right]\hfill \end{array}$$

which implies,

$${{\rm Y}}_n\le r{{\rm Y}}_{n-1}$$

(5)

where $r={\displaystyle \frac{q}{1-q}}<1$ as $q\in (0,{\displaystyle \frac{1}{2}})$. It follows that

$${\rm Y}\left(F_1{\hslash }_n,F_1{\hslash }_{n+1}\right)={{\rm Y}}_n\le r^n{\rm Y}\left(F_1{\hslash }_0,F_1{\hslash }_1\right).$$

(6)

For every $n,m\in \mathbb{N}$ such that $m\ge n$, using the triangle inequality, we have

$$\begin{array}{ccc}\hfill {\rm Y}({\hslash }_n,{\hslash }_m)& \le & {\alpha }_1\left({\rm Y}(F_1{\hslash }_n,F_1{\hslash }_{n+1})\right)+{\alpha }_2\left({\rm Y}(F_1{\hslash }_{n+1},F_1{\hslash }_m)\right)\hfill \\ & \le & {\alpha }_1\left({\rm Y}(F_1{\hslash }_n,F_1{\hslash }_{n+1})\right)+{\alpha }_2({\alpha }_1\left({\rm Y}(F_1{\hslash }_{n+1},F_1{\hslash }_{n+2})\right)+{\alpha }_2\left({\rm Y}(F_1{\hslash }_{n+2},F_1{\hslash }_m)\right).\hfill \end{array}$$

Since the mapping ${\alpha }_2$ is sub-additive, we get

$$\begin{array}{ccc}\hfill {\rm Y}({\hslash }_n,{\hslash }_m)& \le & {\alpha }_1\left({\rm Y}(F_1{\hslash }_n,F_1{\hslash }_{n+1})\right)+{\alpha }_2({\alpha }_1\left({\rm Y}(F_1{\hslash }_{n+1},F_1{\hslash }_{n+2})\right)+{\alpha }_2^2\left({\rm Y}(F_1{\hslash }_{n+2},F_1{\hslash }_m)\right)\hfill \\ & \le & {\alpha }_1\left({\rm Y}(F_1{\hslash }_n,F_1{\hslash }_{n+1})\right)+{\alpha }_2({\alpha }_1\left({\rm Y}(F_1{\hslash }_{n+1},F_1{\hslash }_{n+2})\right)+{\alpha }_2^2\left({\alpha }_1({\rm Y}(F_1{\hslash }_{n+2},F_1{\hslash }_{n+3})\right)\hfill \\ & +& {\alpha }_2({\rm Y}(F_1{\hslash }_{n+3},F_1{\hslash }_m)))\hfill \\ & \le & {\alpha }_1\left({\rm Y}(F_1{\hslash }_n,F_1{\hslash }_{n+1})\right)+{\alpha }_2({\alpha }_1\left({\rm Y}(F_1{\hslash }_{n+1},F_1{\hslash }_{n+2})\right)+{\alpha }_2^2\left({\alpha }_1({\rm Y}(F_1{\hslash }_{n+2},F_1{\hslash }_{n+3})\right)\hfill \\ & +& {\alpha }_2^3\left({\rm Y}(F_1{\hslash }_{n+3},F_1{\hslash }_m)\right)\hfill \\ & ⋮& \hfill \\ & \le & {\alpha }_1\left({\rm Y}(F_1{\hslash }_n,F_1{\hslash }_{n+1})\right)+{\alpha }_2({\alpha }_1\left({\rm Y}(F_1{\hslash }_{n+1},F_1{\hslash }_{n+2})\right)+{\alpha }_2^2\left({\alpha }_1({\rm Y}(F_1{\hslash }_{n+2},F_1{\hslash }_{n+3})\right)\hfill \\ & +& {\alpha }_2^3\left({\rm Y}(F_1{\hslash }_{n+3},F_1{\hslash }_{n+4})\right)+\dots +{\alpha }_2^{m-2}[{\alpha }_1\left({\rm Y}(F_1{\hslash }_{m-2},F_1{\hslash }_{m-1})\right)\hfill \\ & +& {\alpha }_2\left({\rm Y}(F_1{\hslash }_{m-1},F_1{\hslash }_m)\right)]\hfill \\ & \le & {\alpha }_1\left({\rm Y}(F_1{\hslash }_n,F_1{\hslash }_{n+1})\right)+{\alpha }_2({\alpha }_1\left({\rm Y}(F_1{\hslash }_{n+1},F_1{\hslash }_{n+2})\right)+{\alpha }_2^2\left({\alpha }_1({\rm Y}(F_1{\hslash }_{n+2},F_1{\hslash }_{n+3})\right)\hfill \\ & +& {\alpha }_2^3\left({\rm Y}(F_1{\hslash }_{n+3},F_1{\hslash }_{n+4})\right)+\dots +{\alpha }_2^{m-n-2}\left({\alpha }_1\left({\rm Y}(F_1{\hslash }_{m-2},F_1{\hslash }_{m-1})\right)\right)\hfill \\ & +& {\alpha }_2^{m-n-1}\left({\alpha }_1\left({\rm Y}(F_1{\hslash }_{m-1},F_1{\hslash }_m)\right)\right)\hfill \\ & \le & {\displaystyle \sum _{i=n}^{m-2}{\alpha }_2^{i-n}{\alpha }_1\left({\rm Y}(F_1{\hslash }_i,F_1{\hslash }_{i+1})\right)+{\alpha }_2^{m-n-1}{\alpha }_1\left({\rm Y}(F_1{\hslash }_{n-1},F_1{\hslash }_n)\right).}\hfill \end{array}$$

(7)

Using inequality (6) in (7), we get

$${\rm Y}(F_1{\hslash }_n,F_1{\hslash }_m)\le \sum _{i=n}^{m-2}{\alpha }_2^{i-n}{\alpha }_1\left(r^i{\rm Y}(F_1{\hslash }_0,F_1{\hslash }_1))\right)+{\alpha }_2^{m-n-1}\left(r^{n-1}{\rm Y}(F_1{\hslash }_0,F_1{\hslash }_1)\right).$$

(8)

Taking the limit as $n,m$ tend to infinity in (8) and using condition (1) from Theorem 1, we get

$${\displaystyle \underset{n,m\to \infty }{lim}{\rm Y}(F_1{\hslash }_n,F_1{\hslash }_m)=0.}$$

(9)

Consequently, the sequence $\left\{F_1{\hslash }_n\right\}$ is Cauchy in $\mathcal{S}$. Since the DCMS $(\mathcal{S},{\rm Y})$ is complete, it follows that $\left\{F_1{\hslash }_n\right\}$ converges to a point $s\in \mathcal{S}$; that is,

$${\displaystyle \underset{n\to \infty }{lim}{\rm Y}(F_1{\hslash }_n,s)=0.}$$

(10)

Since $F_1$ is subsequentially convergent, then the sequence $\left\{{\hslash }_n\right\}$ has a convergent subsequence denoted by $\left\{{\hslash }_{n_k}\right\}$ such that

$${\displaystyle \underset{k\to \infty }{lim}{\hslash }_{n_k}=u.}$$

(11)

From the continuity of $F_1$, it follows that

$${\displaystyle \underset{k\to \infty }{lim}{F}_1{\hslash }_{n_k}=F_{1}u.}$$

(12)

From (10) and (12), we conclude that $F_{1}u=s$. Then, by using the triangle inequality and (4), we get

$$\begin{array}{ccc}\hfill {\rm Y}(F_1{F}_{2}u,F_{1}u)& \le & {\alpha }_1\left({\rm Y}(F_1{F}_{2}u,F_1{F}_2^{n_k}{\hslash }_0)\right)+{\alpha }_2\left({\rm Y}(F_1{F}_2^{n_k}{\hslash }_0,F_{1}u)\right)\hfill \\ & \le & {\alpha }_1\left(q\left[{\rm Y}(F_{1}u,F_1{F}_{2}u)+{\rm Y}(F_1{F}_2^{n_{k-1}}{\hslash }_0,F_1{F}_2^{n_k}{\hslash }_0)\right]\right)+{\alpha }_2\left({\rm Y}(F_1{F}_2^{n_k}{\hslash }_0,F_{1}u)\right).\hfill \end{array}$$

Using the property of ${\alpha }_1$, we get

$$\begin{array}{ccc}\hfill {\rm Y}(F_1{F}_{2}u,F_{1}u)& \le & q{\alpha }_1\left({\rm Y}(F_{1}u,F_1{F}_{2}u)\right)+{\alpha }_1\left({\rm Y}(F_1{F}_2^{n_{k-1}}{\hslash }_0,F_1{F}_2^{n_k}{\hslash }_0)\right)+{\alpha }_2\left({\rm Y}(F_1{F}_2^{n_k}{\hslash }_0,F_{1}u)\right)\hfill \\ & \le & q{\rm Y}(F_{1}u,F_1{F}_{2}u)+{\alpha }_1\left({\rm Y}(F_1{F}_2^{n_{k-1}}{\hslash }_0,F_1{F}_2^{n_k}{\hslash }_0)\right)+{\alpha }_2\left({\rm Y}(F_1{F}_2^{n_k}{\hslash }_0,F_{1}u)\right)\hfill \\ & \le & {\displaystyle \frac{1}{1-q}}{\alpha }_1\left({\rm Y}(F_1{F}_2^{n_{k-1}}{\hslash }_0,F_1{F}_2^{n_k}{\hslash }_0)\right)+{\displaystyle \frac{1}{1-q}}{\alpha }_2\left({\rm Y}(F_1{F}_2^{n_k}{\hslash }_0,F_{1}u)\right).\hfill \end{array}$$

Since $F_2^{n_k}{\hslash }_0={\hslash }_{n_k}$, we obtain

$${\rm Y}(F_1{F}_{2}u,F_{1}u)\le {\displaystyle \frac{1}{1-q}}{\alpha }_1\left({\rm Y}(F_1{\hslash }_{n_{k-1}},F_1{\hslash }_{n_k})\right)+{\displaystyle \frac{1}{1-q}}{\alpha }_2\left({\rm Y}(F_1{\hslash }_{n_k},F_{1}u)\right).$$

(13)

Let k tend to infinity in (13) and using the continuity of ${\alpha }_1$ and ${\alpha }_2$, we obtain

$$\begin{array}{ccc}\hfill {\rm Y}(F_1{F}_{2}u,F_{1}u)& \le & {\displaystyle {\displaystyle \frac{1}{1-q}}{\alpha }_1\left({\rm Y}(\underset{k\to \infty }{lim}{F}_1{\hslash }_{n_{k-1}},\underset{k\to \infty }{lim}{F}_1{\hslash }_{n_k})\right)}\hfill \\ & +& {\displaystyle {\displaystyle \frac{1}{1-q}}{\alpha }_2\left({\rm Y}(\underset{k\to \infty }{lim}{F}_1{\hslash }_{n_k},F_{1}u)\right)}\hfill \\ & \le & {\displaystyle \frac{1}{1-q}}{\alpha }_1\left({\rm Y}(F_{1}u,F_{1}u)\right)+{\displaystyle \frac{1}{1-q}}{\alpha }_2\left({\rm Y}(F_{1}u,F_{1}u)\right)\hfill \\ & \le & {\displaystyle \frac{1}{1-q}}\left[{\alpha }_1\left(0\right)+{\alpha }_2\left(0\right)\right].\hfill \end{array}$$

Since ${\alpha }_1\left(0\right)+{\alpha }_2\left(0\right)=0$, then ${\displaystyle \underset{k\to \infty }{lim}{\rm Y}(F_1{F}_{2}u,F_{1}u)=0}$. Then, $F_1{F}_{2}u=F_{1}u$. Since $F_1$ is one-to-one mapping, $F_{2}u=u$; that is, u is an FP of $F_2$.

Finally, let us prove the uniqueness of the FP. Let $u_1,u_2$ be two FPs of $F_2$, then $F_2{u}_1=u_1$, and $F_2{u}_2=u_2$.

$$\begin{array}{ccc}\hfill {\rm Y}(F_1{u}_1,F_1{u}_2)={\rm Y}(F_1{F}_2{u}_1,F_1{F}_2{u}_2)& \le & k\left[{\rm Y}(F_1{u}_1,F_1{F}_2{u}_1)+{\rm Y}(F_1{u}_2,F_1{F}_2{u}_2)\right]\hfill \\ & =& k\left[{\rm Y}(F_1{u}_1,F_1{u}_1)+{\rm Y}(F_1{u}_2,F_1{u}_2)\right]=0.\hfill \end{array}$$

Therefore, $F_1{u}_1=F_1{u}_2$. Since $F_2$ is one-to-one mapping, $u_1=u_2$. □

Next, we consider an important special case of the general FP results established in Theorem 1, namely Kannan-type contractions, whose definition is given as follows.

Definition 5.

Let $(\mathcal{S},{\rm Y})$ be a DCM. A self-mapping $T:\mathcal{S}\to \mathcal{S}$ is said to be a Kannan-type contraction if there exists $q\in (0,{\displaystyle \frac{1}{2}})$ such that

$${\rm Y}(Ts,Tt)\le q[{\rm Y}(s,Ts)+{\rm Y}(t,Tt\left)\right]for alls,t\in \mathcal{S}.$$

(14)

It is easy to see that (14) is a special case of (4) introduced in Theorem 1. Indeed, it is sufficient to consider the mapping F as an identity mapping. Therefore, we can state the following FP result, namely, under the same conditions as in Theorem 1, the Kannan-type contraction on a CDCM has a unique FP.

Corollary 1.

Let $(\mathcal{S},{\rm Y})$ be a CDCM with non-constant control functions ${\alpha }_1,{\alpha }_2:[0,\infty )\to [0,\infty )$. Let $T:\mathcal{S}\to \mathcal{S}$ be a continuous mapping and

$${\rm Y}(Ts,Tt)\le q[{\rm Y}(s,Ts)+{\rm Y}(t,Tt\left)\right]$$

(15)

for all $s,t\in \mathcal{S}$ and $k\in (0,{\displaystyle \frac{1}{2}}).$ For ${\hslash }_0\in \mathcal{S}$, define a sequence $\left\{{\hslash }_n\right\}$ by ${\hslash }_n=T^n{\hslash }_0$. Suppose that the following conditions are satisfied:

1. 

${\displaystyle \underset{m,n\to \infty }{lim}\left[\sum _{i=n}^{m-2}{\alpha }_2^i\left({\alpha }_1\left(r^i{\rm Y}({\hslash }_0,{\hslash }_1)\right)\right)+{\alpha }_2^{m-n-1}\left(r^{n-1}{\rm Y}({\hslash }_0,{\hslash }_1)\right)\right]=0}$, where $r={\displaystyle \frac{q}{1-q}}$, and ${\alpha }_2^i\left({\alpha }_1\left(r^i{\rm Y}({\hslash }_0,{\hslash }_1)\right)\right)$ and ${\alpha }_2^{n-1}\left(r^{n-1}{\rm Y}(F{\hslash }_0,F{\hslash }_1)\right)$ denote the composite functions.

2. 

${\alpha }_1$ and ${\alpha }_2$ are continuous and non-decreasing functions with ${\alpha }_1\left(0\right)={\alpha }_2\left(0\right)=0$ and ${\alpha }_2\left(ks\right)<s$ for every $s\in \mathcal{S}$ and $k\in (0,1).$

3. 

${\alpha }_2$ is sub-additive and ${\alpha }_1(ks+t)\le k{\alpha }_1\left(s\right)+{\alpha }_1\left(t\right)$ for all $s,t\in \mathcal{S}$.

Therefore, T has a unique FP.

To conclude this section, we introduce a new class of mappings, known as $\gamma$-admissible mappings, which will be instrumental in formulating the final FP result.

Definition 6.

Let $\mathcal{S}$ be a nonempty set, $T:\mathcal{S}\to \mathcal{S}$ and $\gamma ,\lambda :\mathcal{S}\times \mathcal{S}\to [0,\infty )$. T is γ-admissible with respect to λ if

$$\forall x_1,x_2\in \mathcal{S},\gamma (x_1,x_2)\ge \lambda (x_1,x_2)⟹\gamma (T{x}_1,T{x}_2)\ge \lambda (T{x}_1,T{x}_2).$$

(16)

Notation:

We consider a new family $\Psi :[0,\infty )\to [0,\infty )$ such that

• $\psi$ is an upper semicontinuous mapping from right;
• For all $s\in (0,\infty )$, $\psi \left(s\right)<s$;
• $\psi \left(0\right)=0$.

Using the notion of $\gamma$ admissibility together with the family of functions $\Psi$, we are now able to formulate a general FP result. This theorem extends classical contraction principles to mappings that respect the $\gamma$-admissible structure, providing conditions for the existence and uniqueness of FPs in CDCM.

Theorem 2.

Let $(\mathcal{S},{\rm Y})$ be a CDCM with non-constant control functions ${\alpha }_1,{\alpha }_2$. Let $\psi \in \Psi$. Suppose that the continuous mapping $T:\mathcal{S}\to \mathcal{S}$ satisfies the following hypothesis:

(i) 

T is γ-admissible with respect to λ;

(ii) 

If $x,y\in \mathcal{S}$ and $\gamma (s,t)\ge \lambda (s,t)$, then ${\rm Y}(Ts,Tt)\le \psi ({\rm Y}(s,t\left)\right)$;

(iii) 

There exists $s_0\in \mathcal{S}$ such that $\gamma (s_0,T{s}_0)\ge \lambda (s_0,T{s}_0)$;

(iv) 

The functions ${\alpha }_1$ and ${\alpha }_2$ are continuous.

Therefore, T has a unique FP.

Proof. 

Since $s_0\in \mathcal{S}$, there exists $s_1$ such that $s_1=T{s}_0$. Then, we can build the sequence $\left\{s_n\right\}$ in $\mathcal{S}$ such that

$$s_{n+1}=T{s}_n,\mathrm{for} \mathrm{all}n\in \mathbb{N}.$$

(17)

Assume that $s_n\ne s_{n+1}$ for all $n\in \mathbb{N}$; otherwise, $s_n=s_{n+1}$. Since $\gamma (s_0,s_1)=\gamma (s_0,T{s}_0)\ge \lambda (s_0,T{s}_0)$ and T is $\gamma$-admissible with respect to $\lambda$, we obtain that

$$\gamma (s_1,s_2)=\gamma (T{s}_0,T{s}_1)\ge \lambda (T{s}_0,T{s}_1)=\lambda (s_1,s_2).$$

(18)

By continuing the process as above, we have

$$\gamma (s_n,s_{n+1})\ge \lambda (s_n,s_{n+1})\mathrm{for} \mathrm{all}n\in \mathbb{N}.$$

(19)

By applying (ii), we get

$${\rm Y}(s_n,s_{n+1})={\rm Y}(T{s}_{n-1},T{s}_n)\le \psi ({\rm Y}(s_{n-1},s_n))\mathrm{for} \mathrm{all}n\in \mathbb{N}.$$

(20)

Then, $\{{\rm Y}(s_n,s_{n+1})\}$ is a nonincreasing sequence. It follows that there exists $q\ge 0$ such that

$${\displaystyle \underset{n\to \infty }{lim}{\rm Y}(s_n,s_{n+1})=q.}$$

(21)

We claim that $q=0$. Suppose that $q>0$. Since $\psi$ is upper semicontinuous from the right using (20), we have

$$\begin{array}{ccc}\hfill q& =& {\displaystyle \underset{n\to \infty }{lim\; sup}{\rm Y}(s_n,s_{n+1})}\hfill \\ & \le & {\displaystyle \underset{n\to \infty }{lim\; sup}\psi ({\rm Y}(s_{n-1},s_n))\le \psi \left(q\right)<q,}\hfill \end{array}$$

which is a contradiction. Then,

$${\displaystyle \underset{n\to \infty }{lim}{\rm Y}(s_n,s_{n+1})=0.}$$

(22)

Let $i,j\in \mathcal{S}$ be such that $s_i\ne s_j$ for all $i\ne j$. Suppose without loss of generality that $i<j$. Using the triangle inequality, we obtain

$$\begin{array}{ccc}\hfill {\rm Y}(s_i,s_j)& \le & {\alpha }_1({\rm Y}(s_i,s_{i+1}))+{\alpha }_2({\rm Y}(s_{i+1},s_j))\hfill \\ & \le & {\alpha }_1({\rm Y}(s_i,s_{i+1}))+{\alpha }_2({\alpha }_1({\rm Y}(s_{i+1},s_{i+2}))+{\alpha }_2({\rm Y}(s_{i+2},s_j)))\hfill \\ & ⋮& \\ & \le & {\displaystyle \sum _{k=n}^{j-2}{\alpha }_2^{k-i}{\alpha }_1({\rm Y}(s_k,s_{k+1}))+{\alpha }_2^{j-i-1}{\alpha }_1({\rm Y}(s_{j-1},s_j)).}\hfill \end{array}$$

Let $i,j$ tend to infinity, we obtain, using (22), ${\displaystyle \underset{i,j\to \infty }{lim}{\rm Y}(s_i,s_j)=0.}$ Then, $\left\{s_n\right\}$ is a Cauchy sequence, and $\left\{s_n\right\}$ converges to some $s\in \mathcal{S}$. Then, using the continuity of T, we obtain

$${\displaystyle s=\underset{n\to \infty }{lim}{s}_{n+1}=\underset{n\to \infty }{lim}T{s}_n=Ts.}$$

Therefore, s is an FP of T. □

In the following section, we present applications of these general FP results. In particular, Section 4 is devoted to the case of graphic contractions, where the abstract theory developed here is applied in a graph-theoretic framework.

4. Fixed-Point Results for Graphic Contractions

This section provides an application of the main results established in Section 3 to the setting of graphic contractions where a DCMS is enriched with a graph structure. This connection illustrates the applicability of our main results beyond the purely theoretical setting and provides a concrete demonstration of their usefulness.

Before delving into FP results in graph theory within DCMs, it is important to recall some fundamental concepts from graph theory. These notions form the building blocks to understanding the more complex structures and results discussed later.

Notation: a graph G is defined as a pair of sets: $G=(V,E)$, where

• V is the set of vertices (or nodes).
• E is the set of edges, where each edge is a pair of vertices.

Definition 7

([22]). Let G be a graph. A path between two vertices u and v in G of length q ($q\in \mathbb{N}\cup \left\{0\right\}$) is a sequence ${\left(s_i\right)}_{i=0}^q$ of $q+1$ distinct vertices such that $s_0=u$, $s_n=v$ and $(s_i,s_{i+1})\in E\left(G\right)$, for $i=1,2,...,q$.

Definition 8

([22]). Consider a vertex u in a graph G. The subgraph $G_u$, which consists of all the vertices and edges that are part of some path in G starting at u, is referred to as the component of G that contains u. The equivalence class ${\left[u\right]}_G$ on the vertex set $V\left(G\right)$, defined by the relation R (where $uRv$ if there is a path from u to v), satisfies the property that the set of vertices in $G_u$, denoted $V\left(G_u\right)$, is equal to ${\left[u\right]}_G$.

Let $(\mathcal{S},{\rm Y})$ be a CDCM. Graph G can be transformed into a weighted graph by assigning each edge a weight that corresponds to the distance between its vertices, given by the double-composed metric shown in Figure 1.

Notation: we denote by ${\mathcal{S}}^f=\{s\in \mathcal{S}/(s,fs)\in E\left(G\right)or(fs,s)\in E\left(G\right)\}$.

Definition 9.

Let $(\mathcal{S},{\rm Y})$ be a CDCM endowed with a graph G. The mapping $f:\mathcal{S}\to \mathcal{S}$ is referred to as a $G_{\psi }$ contraction if it satisfies the following conditions:

• $$for alls,t\in \mathcal{S},(s,t)\in E\left(G\right)⟹(fs,ft)\in E\left(G\right),$$

  (23)
• there exists a function $\psi :{\mathbb{R}}^{+}⟶{\mathbb{R}}^{+}$ such that

  $${\rm Y}(fs,f^{2}s)\le \psi ({\rm Y}(s,fs))for all s\in {\mathcal{S}}^f,$$

  (24)

  where ψ is a nondecreasing function, and ${\displaystyle \underset{n\to \infty }{lim}{\psi }^n\left(z\right)=0}$ for all $z>0$.

Definition 10.

A mapping $f:\mathcal{S}⟶\mathcal{S}$ is said to be orbitally G continuous if, for all $u,v\in \mathcal{S}$ and any positive sequence ${\left\{s_n\right\}}_{n\in \mathbb{N}}$,

$$f^{s_n}u⟶v,(f^{s_n}u,f^{s_{n+1}}u)\in E\left(G\right)⟹f\left(f^{s_n}u\right)⟶fv as n\to \infty .$$

Theorem 3.

Let $(\mathcal{S},{\rm Y})$ be a CDCM with non-constant control functions ${\alpha }_1,{\alpha }_2:[0,\infty )\to [0,\infty )$ endowed with a graph G. Let $f:\mathcal{S}\to \mathcal{S}$ be a $G_{\psi }$ contraction that is orbitally G continuous. We assume the following property $\left(P\right)$: for any ${\left\{s_n\right\}}_{n\in \mathbb{N}}$ in $\mathcal{S}$, if $s_n⟶s$ and $(s_n,s_{n+1})\in E\left(G\right)$, then there is a subsequence ${\left\{s_{k_n}\right\}}_{n\in \mathbb{N}}$ with $(s_{k_n},s)\in E\left(G\right)$.

Additionally, suppose that the following conditions are satisfied:

• $${\displaystyle \underset{m,n\to \infty }{lim}\left[\sum _{i=n}^{m-2}{\alpha }_2^i\left({\alpha }_1\left({\psi }^i({\rm Y}(s,fs))\right)\right)+{\alpha }_2^{m-n-1}\left({\alpha }_1\left({\psi }^{n-1}({\rm Y}(s,fs))\right)\right)\right]=0,}$$

  (25)

  where ${\alpha }_2^i\left({\alpha }_1\left({\psi }^i({\rm Y}(s,fs))\right)\right)$ and ${\alpha }_2^{n-1}\left({\alpha }_1\left({\psi }^{n-1}({\rm Y}(s,fs))\right)\right)$ denote the composite functions.
• ${\alpha }_1$ and ${\alpha }_2$ are non-decreasing functions and continuous with ${\alpha }_1\left(0\right)$, ${\alpha }_2\left(0\right)$, which exist and are finite, and ${\alpha }_2\left(ks\right)<s$, for every $s\in \mathcal{S}$ and $k\in (0,1).$
• ${\alpha }_2$ is sub-additive and ${\alpha }_1(ks+t)\le k{\alpha }_1\left(s\right)+{\alpha }_1\left(t\right)$ for all $s,t\in \mathcal{S}$.

Then, the restriction $f_{{|\left[s\right]}_{\tilde{G}}}$ has a unique FP.

Proof. 

Let $s\in {\mathcal{S}}^f$, then $(s,fs)\in E\left(G\right)$ or $(fs,s)\in E\left(G\right)$. Let us suppose w.l.o.g that $(s,fs)\in E\left(G\right)$. We derive through induction that

$$(f^{n}s,f^{n+1}s)\in E\left(G\right)\mathrm{for} \mathrm{all}n\in \mathbb{N}.$$

(26)

Thus, we have

$$\begin{array}{ccc}\hfill {\rm Y}(f^{n}s,f^{n+1}s)& \le & \psi ({\rm Y}(f^{n-1}s,f^{n}s))\hfill \\ & \le & {\psi }^2({\rm Y}(f^{n-2}s,f^{n-1}s))\hfill \\ & \le & ⋮\hfill \\ & \le & {\psi }^n({\rm Y}(s,fs)).\hfill \end{array}$$

(27)

Let $n,m\in \mathbb{N}$; using the triangle inequality, we get

$$\begin{array}{ccc}\hfill {\rm Y}(f^{n}s,f^{n+m}s)& \le & {\alpha }_1({\rm Y}(f^{n}s,f^{n+1}s))+{\alpha }_2({\rm Y}(f^{n+1}s,f^{n+m}s))\hfill \\ & \le & {\alpha }_1({\rm Y}(f^{n}s,f^{n+1}s))+{\alpha }_2\left[{\alpha }_1({\rm Y}(f^{n+1}s,f^{n+2}s))+{\alpha }_2({\rm Y}(f^{n+2}s,f^{n+m}s))\right]\hfill \\ & \le & {\alpha }_1({\rm Y}(f^{n}s,f^{n+1}s))+{\alpha }_2\left({\alpha }_1({\rm Y}(f^{n+1}s,f^{n+2}s))\right)+{\alpha }_2^2({\rm Y}(f^{n+2}s,f^{n+m}s))\hfill \\ & \le & {\alpha }_1({\rm Y}(f^{n}s,f^{n+1}s))+{\alpha }_2\left({\alpha }_1({\rm Y}(f^{n+1}s,f^{n+2}s))\right)+{\alpha }_2^2[{\alpha }_1({\rm Y}(f^{n+2}s,f^{n+3}s)\hfill \\ & +& {\alpha }_2({\rm Y}(f^{n+3}s,f^{n+m}s))]\hfill \\ & \le & {\alpha }_1({\rm Y}(f^{n}s,f^{n+1}s))+{\alpha }_2\left({\alpha }_1({\rm Y}(f^{n+1}s,f^{n+2}s))\right)+{\alpha }_2^2({\alpha }_1({\rm Y}(f^{n+2}s,f^{n+3}s))\hfill \\ & +& {\alpha }_2^3({\rm Y}(f^{n+3}s,f^{n+m}s)))\hfill \\ & \le & ⋮\hfill \\ & \le & {\alpha }_1({\rm Y}(f^{n}s,f^{n+1}s))+{\alpha }_2\left({\alpha }_1({\rm Y}(f^{n+1}s,f^{n+2}s))\right)+{\alpha }_2^2\left({\alpha }_1({\rm Y}(f^{n+2}s,f^{n+3}s))\right)\hfill \\ & +& \dots +{\alpha }_2^{m-n-2}\left({\alpha }_1({\rm Y}(f^{m-2}s,f^{m-1}s))\right)+{\alpha }_2^{m-n-1}\left({\alpha }_1({\rm Y}(f^{m-1}s,f^{m}s))\right)\hfill \\ & \le & {\displaystyle \sum _{i=n}^{m-2}{\alpha }_2^{i-n}\left({\alpha }_1({\rm Y}(f^{i}s,f^{i+1}s)\right)+{\alpha }_2^{m-n-1}\left({\alpha }_1({\rm Y}(f^{m-1}s,f^{m}s))\right).}\hfill \end{array}$$

Using (27), we obtain

$${\displaystyle {\rm Y}(f^{n}s,f^{n+m}s)\le \sum _{i=n}^{m-2}{\alpha }_2^{i-n}\left({\alpha }_1\left({\psi }^i({\rm Y}(s,fs))\right)\right)+{\alpha }_2^{m-n-1}\left({\alpha }_1\left({\psi }^{m-1}({\rm Y}(s,fs))\right)\right).}$$

(28)

Now, from (25) and (28), it follows that

$${\displaystyle \underset{m,n\to \infty }{lim}{\rm Y}(f^{n}s,f^{n+m}s)=0.}$$

(29)

Hence, the sequence ${\left\{f^{n}s\right\}}_n$ is a Cauchy sequence. Exploiting the completeness of $(\mathcal{S},{\rm Y})$, there exists $u_1\in \mathcal{S}$ so that

$${\displaystyle \underset{n\to \infty }{lim}{f}^{n}s=u_1.}$$

(30)

We claim that u is a, FP of $f_{{|\left[s\right]}_{\tilde{G}}}$.

Since $s\in {\mathcal{S}}^f$, $f^{n}s\in {\mathcal{S}}^f$, for every $n\in \mathbb{N}$, then $(s,fs)\in E\left(G\right)$. From property $\left(P\right)$ a subsequence ${\left\{f^{k_n}s\right\}}_n$ of ${\left\{f^{n}s\right\}}_n$ exists such that $(f^{k_n}s,u_1)\in E\left(G\right)$ for all $n\in \mathbb{N}$.

On the other hand, a path in G can be created by the points $s,fs,...,f^{k_1}s,u_1$, which allows us to affirm that $u_1\in {\left[s\right]}_{\tilde{G}}$. With respect to the orbitally G continuity of f, we get

$$f\left(f^{k_n}s\right)⟶f{u}_1 as n\to \infty .$$

(31)

Then, using (30) and (31), we deduce that $u_1$ is an FP of $f_{{|\left[s\right]}_{\tilde{G}}}$.

Assume that there exist $u_1,u_2$ such that $f{u}_1=u_1$ and $f{u}_2=u_2$. We have

$$\begin{array}{ccc}\hfill {\rm Y}(u_1,u_2)& \le & {\alpha }_1({\rm Y}(u_1,f^n{u}_1))+{\alpha }_2({\rm Y}(f^n{u}_1,u_2))\hfill \\ & =& {\alpha }_1({\rm Y}(u_1,u_1))+{\alpha }_2({\rm Y}(f^n{u}_1,u_2))\hfill \\ & =& {\alpha }_1\left(0\right)+{\alpha }_2({\rm Y}(u_1,f^n{u}_2))\hfill \\ & \le & {\displaystyle {\alpha }_1\left(0\right)+\underset{n\to \infty }{lim}\left({\alpha }_2({\rm Y}(u_1,f^n{u}_2))\right)}\hfill \\ & \le & {\displaystyle {\alpha }_1\left(0\right)+{\alpha }_2(\underset{n\to \infty }{lim}{\rm Y}(u_1,f^n{u}_2)).}\hfill \end{array}$$

(32)

Using (32) and (30), we obtain

$${\rm Y}(u_1,u_2)\le {\alpha }_1\left(0\right)+{\alpha }_2\left(0\right)=0.$$

Then, ${\rm Y}(u_1,u_2)=0$ and $u_1=u_2$, hence the uniqueness of the FP. □

Definition 11.

Let $(\mathcal{S},{\rm Y})$ be a CDCM endowed with a graph G. The mapping $f:\mathcal{S}\to \mathcal{S}$ is said to be a G contraction if the following hypothesis are fulfilled:

• $$for all s,t\in \mathcal{S},(s,t)\in E\left(G\right)⟹(fs,ft)\in E\left(G\right),$$

  (33)
• $${\rm Y}(fs,ft)\le k{\rm Y}(s,t)for all s,t\in {\mathcal{S}}^f and k\in (0,1).$$

  (34)

Theorem 4.

Let $(\mathcal{S},{\rm Y})$ be a CDCM with non-constant control functions ${\alpha }_1,{\alpha }_2:[0,\infty )\to [0,\infty )$ endowed with a graph G. Let $f:\mathcal{S}\to \mathcal{S}$ be a G contraction that is orbitally G continuous. Suppose that the following property $\left(P_r\right)$ holds: for any ${\left\{s_n\right\}}_{n\in \mathbb{N}}$ in $\mathcal{S}$, if $s_n⟶s$ and $(s_n,s_{n+1})\in E\left(G\right)$, then there is a subsequence ${\left\{s_{k_n}\right\}}_{n\in \mathbb{N}}$ with $(s_{k_n},s)\in E\left(G\right)$.

Furthermore, suppose that the following conditions are met:

•  

  $${\displaystyle \underset{m,n\to \infty }{lim}\left[\sum _{i=n}^{m-2}{\alpha }_2^i\left({\alpha }_1\left(k^i{\rm Y}(s,fs)\right)\right)+{\alpha }_2^{m-n-1}\left({\alpha }_1(k^{n-1}{\rm Y}(s,fs))\right)\right]=0,}$$

  (35)

  where ${\alpha }_2^i\left({\alpha }_1(k^i{\rm Y}(s,fs))\right)$ and ${\alpha }_2^{n-1}\left({\alpha }_1(k^{n-1}{\rm Y}(s,fs))\right)$ denote the composite functions.
• ${\alpha }_2$ is sub-additive.

Then, the restriction $f_{{|\left[s\right]}_{\tilde{G}}}$ has a unique FP.

Proof. 

Let $s\in {\mathcal{S}}^f$; then, $(s,fs)\in E\left(G\right)$ or $(fs,s)\in E\left(G\right)$. We suppose without loss of generality that $(s,fs)\in E\left(G\right)$. We obtain by induction

$$(f^{n}s,f^{n+1}s)\in E\left(G\right)\mathrm{for} \mathrm{all}n\in \mathbb{N}.$$

(36)

Thus, we have

$$\begin{array}{ccc}\hfill {\rm Y}(f^{n}s,f^{n+1}s)& \le & k{\rm Y}(f^{n-1}s,f^{n}s)\hfill \\ & \le & k^2{\rm Y}(f^{n-2}s,f^{n-1}s)\hfill \\ & \le & ⋮\hfill \\ & \le & k^n{\rm Y}(s,fs).\hfill \end{array}$$

(37)

Let $n,m\in \mathbb{N}$. By applying the triangle inequality and the sub-additivity of the function ${\alpha }_2$, and following a similar approach to the proof of Theorem 3, we obtain

$${\displaystyle {\rm Y}(f^{n}s,f^{n+m}s)\le \sum _{i=n}^{m-2}{\alpha }_2^{i-n}\left({\alpha }_1({\rm Y}(f^{i}s,f^{i+1}s)\right)+{\alpha }_2^{m-n-1}\left({\alpha }_1({\rm Y}(f^{m-1}s,f^{m}s))\right).}$$

Using (37), we obtain

$${\displaystyle {\rm Y}(f^{n}s,f^{n+m}s)\le \sum _{i=n}^{m-2}{\alpha }_2^{i-n}\left({\alpha }_1(k^i{\rm Y}(s,fs))\right)+{\alpha }_2^{m-n-1}\left({\alpha }_1(k^{m-1}{\rm Y}(s,fs))\right).}$$

(38)

Now, from (35) and (38), we infer that

$${\displaystyle \underset{n,m\to \infty }{lim}{\rm Y}(f^{n}s,f^{n+m}s)=0.}$$

(39)

Therefore, the sequence ${\left\{f^{n}s\right\}}_n$ is a Cauchy sequence. Given the completeness $(\mathcal{S},{\rm Y})$, there exists $u\in \mathcal{S}$ such that

$${\displaystyle \underset{n\to \infty }{lim}{f}^{n}s=u.}$$

(40)

Let us prove that u is an FP of the restriction of f to ${\left[s\right]}_{\tilde{G}}$.

Since $s\in {\mathcal{S}}^f$, $f^{n}s\in {\mathcal{S}}^f$, for every $n\in \mathbb{N}$, then $(s,fs)\in E\left(G\right)$. From property $\left(P_r\right)$ there exists a subsequence ${\left\{f^{k_n}s\right\}}_n$ of ${\left\{f^{n}s\right\}}_n$ such that $(f^{k_n}s,u)\in E\left(G\right)$ for all $n\in \mathbb{N}$.

On the other hand, a path G can be constructed by utilizing the points $s,fs,...,f^{k_1}s,u$, which allows us to affirm that $u\in {\left[s\right]}_{\tilde{G}}$. Given the orbitally G continuity of f, we get

$$f\left(f^{k_n}s\right)⟶fu as n\to \infty .$$

(41)

Hence, using (40) and (41), we deduce that u is a FP of $f_{{|\left[s\right]}_{\tilde{G}}}$.

Assume that there exist $u_1,u_2$ such that $f{u}_1=u_1$ and $f{u}_2=u_2$. We have

$${\rm Y}(u_1,u_2)={\rm Y}(f{u}_1,f{u}_2)\le k{\rm Y}(u_1,u_2).$$

Then, ${\rm Y}(u_1,u_2)=0$, and $u_1=u_2$, hence the uniqueness of the FP. □

To illustrate the applicability of Theorem 4, we now present a concrete example within the setting of a DCMS.

Example 3.

Let $\mathcal{S}=[0,1]$. Define the functions ${\alpha }_1,{\alpha }_2:[0,\infty )\to [0,\infty )$ by ${\alpha }_1\left(a\right)={\alpha }_2\left(a\right)=\sqrt{a}/2$, for all $a\ge 0$.

We define a function ${\rm Y}:{\mathcal{S}}^2\to [0,\infty )$ by ${\rm Y}(a_1,a_2)={\displaystyle \frac{1}{4}}{|a_1-a_2|}^2$. Consider $f:\mathcal{S}\to \mathcal{S}$ such that $fa={\displaystyle \frac{a}{2}}$. Then, f has a unique FP.

Beyond demonstrating the existence and uniqueness result, this example highlights how the abstract framework can be applied in a concrete setting. Moreover, when interpreted in the context of the graph structure introduced in Section 4, the mapping f can be viewed as a $G_{\psi }$ contraction on a simple graph, thereby connecting the theoretical results with practical applications. This illustrates not only the theoretical significance of the theorem but also its potential use in modeling problems where metric and graph structures interact.

Proof. 

Let us begin by proving that $(\mathcal{S},{\rm Y})$ is a DCM. Indeed, it is easy to see that for all $a_1,a_2\in \mathcal{S}$, ${\rm Y}(a_1,a_2)=0⟺a_1=a_2$ and ${\rm Y}(a_1,a_2)={\rm Y}(a_2,a_1)$. Let $a_1,a_2,a_3\in [0,1]$, then we have

$$\begin{array}{ccc}\hfill {\rm Y}(a_1,a_2)={\displaystyle \frac{1}{4}}{|a_1-a_2|}^2& =& {\displaystyle \frac{1}{4}}{|a_1-a_3+a_3-a_2|}^2\hfill \\ & \le & {\displaystyle \frac{1}{4}}\left(2|a_1-a_3{|}^2+2{|a_3-a_2|}^2\right).\hfill \end{array}$$

Since $a_1^2\le a_1$ for all $a_1\in [0,1]$, we get

$$\begin{array}{ccc}\hfill {\rm Y}(a_1,a_2)={\displaystyle \frac{1}{4}}{|a_1-a_2|}^2& \le & {\displaystyle \frac{1}{2}}\sqrt{|a_1-a_3{|}^2}+{\displaystyle \frac{1}{2}}\sqrt{|a_3-a_2{|}^2}\hfill \\ & =& {\alpha }_1\left(\right|a_1-a_3{|}^2)+{\alpha }_2\left(\right|a_3-a_2{|}^2)\hfill \\ & =& {\alpha }_1({\rm Y}(a_1,a_3))+{\alpha }_2({\rm Y}(a_3,a_2)).\hfill \end{array}$$

Then, $(\mathcal{S},{\rm Y})$ is a DCM.

Now, let us verify the hypothesis of Theorem 4.

At first, note that ${\rm Y}(f{a}_1,f{a}_2)={\rm Y}({\displaystyle \frac{a_1}{2}},{\displaystyle \frac{a_2}{2}})={\displaystyle \frac{1}{4}}|{\displaystyle \frac{a_1}{2}}-{\displaystyle \frac{a_2}{2}}{|}^2={\displaystyle \frac{1}{16}}|a_1-a_2{|}^2\le ={\displaystyle \frac{1}{4}}{|a_1-a_2|}^2$. Then, (34) is satisfied.

Consider a graph G consisting of a set of vertices $V\left(G\right):=[0,1]$ and $E\left(G\right)={[0,1]}^2$. Then, f satisfies the preserving edge condition (33). Hence, $f:\mathcal{S}\to \mathcal{S}$ is a G contraction. Moreover, f is orbitally G continuous since f is continuous and satisfies the property $\left(P_r\right)$.

On the other hand, ${\alpha }_2\left(a_1\right)={\displaystyle \frac{{\sqrt{a}}_1}{2}}$ is a sub-additive function. Now, let us check hypothesis (35). We have

$$\begin{array}{ccc}\hfill {\displaystyle \underset{p,q\to \infty }{lim}\left[\sum _{i=q}^{p-2}{\alpha }_2^i\left({\alpha }_1\left(k^i{\rm Y}(a_1,f{a}_1)\right)\right)+{\alpha }_2^{p-q-1}\left({\alpha }_1(k^{q-1}{\rm Y}(a_1,f{a}_1))\right)\right]}& =& \\ \hfill {\displaystyle \underset{p,q\to \infty }{lim}\left[\sum _{i=q}^{p-2}{\left({\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{1}{2}}\sqrt{{\left({\displaystyle \frac{1}{2}}\right)}^i{\rm Y}(a_1,f{a}_1)}}\right)}^i+{\left({\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{1}{2}}\sqrt{{\left({\displaystyle \frac{1}{2}}\right)}^{q-1}{\rm Y}(a_1,f{a}_1)}}\right)}^{p-q-1}\right]}& =& \\ \hfill {\displaystyle \underset{p,q\to \infty }{lim}\left[\sum _{i=q}^{p-2}{\left({\displaystyle \frac{1}{2}}\right)}^i{\left({\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{1}{2}}\right)}^i{\displaystyle \frac{a_1^2}{4}}{)}^{\frac{1}{2}}\right)}^{\frac{i}{2}}+{\left({\displaystyle \frac{1}{2}}\right)}^{p-q-1}{\left({\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{1}{2}}\right)}^{q-1}{\displaystyle \frac{a_1^2}{4}}{)}^{\frac{1}{2}}\right)}^{\frac{p-q-1}{2}}\right]}& =& \\ \hfill {\displaystyle \underset{p,q\to \infty }{lim}\left[\sum _{i=q}^{p-2}{\left({\displaystyle \frac{1}{2}}\right)}^i{\left({\displaystyle \frac{1}{2}}\right)}^{\frac{i}{2}}{\left({\displaystyle \frac{1}{2}}\right)}^{\frac{i^2}{4}}{\left({\displaystyle \frac{a_1^2}{4}}\right)}^{\frac{i}{4}}+{\left({\displaystyle \frac{1}{2}}\right)}^{p-q-1}{\left({\displaystyle \frac{1}{2}}\right)}^{\frac{p-q-1}{2}}{\left({\displaystyle \frac{1}{2}}\right)}^{\frac{q-1}{2}\frac{p-q-1}{2}}{\left({\displaystyle \frac{a_1^2}{4}}\right)}^{\frac{p-q-1}{4}}\right]}& =& 0.\hfill \end{array}$$

Hence, all the hypotheses of Theorem 4 are satisfied, which implies that f has a unique FP. □

5. Conclusions

In this paper, we explored double-composed metric spaces, a concept where the control functions are composed with the metric in the triangle inequality. Firstly, we established and proved some new FP theorems. Next, we investigated double-composed metric spaces endowed with a graph structure, a concept that blends the properties of metric spaces with the combinatorial and topological features of graphs. We presented two FP theorems and an example to illustrate the usefulness of our results. The results presented in this paper also have potential applications in a variety of domains. In particular, FP techniques may serve as powerful tools in network analysis, where they can be employed to study stability, connectivity, and equilibrium states of complex networks. Likewise, in distributed systems, such results can be utilized to analyze convergence behavior and synchronization in large-scale interconnected frameworks.

Future research on this metric could explore many contractions under different assumptions. Additionally, the investigation of graph theory in the context of metric spaces may lead to new insights into the geometry of spaces with non-trivial graph structures.