Perturbed $\mathcal{F}$-Metric Spaces: New Fixed Point Results with Applications to Caputo Fractional Initial Value Problems

Abstract

The primary objective of this study is to establish some new fixed point results in the framework of perturbed $\mathcal{F}$-metric spaces. Specifically, we develop fixed point theorems for three classes of contractions: Banach-type, rational-type, and interpolative contractions. To demonstrate the validity and applicability of the obtained results, some nontrivial examples are provided. Moreover, our findings yield various consequences that extend existing results in both $\mathcal{F}$-metric spaces and perturbed metric spaces. As an application of the main result, we examine the existence of solutions for a Caputo fractional initial value problem.

1. Introduction

Fixed point (FP) theory has become a central tool in modern analysis due to its wide applicability in nonlinear problems arising in pure and applied sciences. The celebrated Banach contraction principle, established by Stefan Banach [1], serves as a cornerstone of this theory and guarantees the existence and uniqueness of FPs for self-mappings satisfying a contractive condition in complete metric spaces (MSs). Over the years, this classical result has been extended and generalized in numerous directions to accommodate more complex structures and broader classes of mappings. In this direction, an important generalization was introduced by Kannan [2], who replaced the classical contraction condition with a weaker form involving the distances between points and their images, thereby broadening the applicability of FP results. Following these developments, Fisher [3] introduced the notion of rational contractions in FP theory, where the contraction condition is expressed through rational combinations of distances. The study of contractive mappings has evolved considerably with the introduction of interpolative techniques. In particular, Karapınar [4] developed the notion of interpolative contractions, which combine different distance expressions in a unified framework. This approach has proven to be highly effective in generalizing and unifying a wide range of existing FP results.

Metric spaces provide the fundamental framework for analyzing distance and convergence in mathematical analysis and FP theory. However, in many situations arising from applied mathematics and nonlinear analysis, the classical triangle inequality is too restrictive to capture the underlying structure of the problem. To overcome this limitation, several generalizations of MSs have been introduced in the literature. One of the most important generalizations is the notion of a b-metric space (b-MS), introduced by Bakhtin [5] in 1993, in which the triangle inequality is relaxed by a multiplicative constant $s\ge 1.$ This notion was later further developed and systematically studied by Czerwik [6], who established its fundamental properties and demonstrated its usefulness in FP theory and nonlinear analysis. Further generalizations were later proposed in the form of s-relaxed_p-MSs given by Fagin et al. [7], which incorporate both a relaxation parameter and a power-type structure in the triangle inequality. This development provides a more flexible framework that unifies several known metric-type spaces and enables the study of nonlinear problems under weaker conditions. In this context, $\mathcal{F}$-metric spaces ($\mathcal{F}$-MSs), introduced by Jleli et al. [8], have emerged as an important generalization of MSs, where the standard metric axioms are suitably relaxed while maintaining the core framework needed for FP theory. In a subsequent study, Al-Mazrooei et al. [9] investigated rational contractions in $\mathcal{F}$-MSs and proved a number of corresponding FP theorems. Later on, Hussain et al. [10] defined the notion of interpolative Ciric-type contractions in $\mathcal{F}$-MSs and established fixed disk results for such mappings. Similarly, the concept of perturbed MSs introduced by Jleli et al. [11] provides a refined extension of classical MSs by decomposing the distance function into an exact component and a perturbation term. More precisely, a mapping $\mathcal{R}$ is said to be a perturbed metric with respect to a function $\mathcal{S}$ if the difference $d=\mathcal{R}-\mathcal{S}$ defines a metric on the underlying set. This formulation preserves the intrinsic metric structure while incorporating external deviations through the perturbation mapping. Notably, every MS becomes a particular case when the perturbation vanishes. The introduction of this framework is motivated by the need to model inaccuracies, uncertainties, and disturbances that naturally arise in applications. Moreover, it offers a flexible setting for extending FP results and studying the stability of solutions under perturbations. Subsequently, Alsahlia et al. [12] investigated rational contractive mappings in the framework of perturbed MSs and derived FP theorems associated with them. Ughade et al. [13] defined the notion of interpolative contraction in perturbed MSs and obtained FP theorems for such contractions.

In addition to the theoretical developments, FP theory plays a fundamental role in the study of differential and integral equations. In particular, fractional differential equations (FDEs) have attracted considerable attention because of their ability to describe memory and hereditary phenomena arising in numerous physical, engineering, biological, and control systems. Among the various fractional operators, the Caputo fractional derivative has become one of the most widely used due to its compatibility with classical initial conditions and its effectiveness in modeling real-world processes. Consequently, the combination of FP theory and Caputo fractional calculus has emerged as an active and rapidly developing research area. In 2023, Alharbi et al. [14] studied the solution of fractional differential equations in the framework of $\mathcal{F}$-MSs by employing suitable FP techniques. In their work, they extended the theory of $\mathcal{F}$-MSs by incorporating an orthogonality structure, which enables the treatment of fractional differential equations under more generalized geometric conditions. A recent development in the study of fractional differential equations was presented in the work of Gangwar et al. [15], where they investigated the solution of implicit fractional differential equations within the framework of newly introduced perturbed spaces. In their study, the authors extended classical analytical techniques by incorporating perturbation structures into metric-type settings, which allows for greater flexibility in handling nonlinear fractional models. Recent advances have also highlighted the growing importance of the Caputo fractional operator in modern analysis beyond the study of fractional differential equations. For instance, Mateen et al. [16] established new Boole-type inequalities for the Caputo fractional operator together with computational analyses and practical applications, thereby providing sharp estimates and demonstrating the effectiveness of Caputo fractional techniques in the study of fractional integral inequalities. Furthermore, Mahajan et al. [17] developed Newton-type integral inequalities involving the Caputo fractional operator, extending classical integral inequalities to the fractional setting and enriching the theoretical foundations of fractional calculus. These recent developments clearly demonstrate the increasing significance of Caputo fractional operators in mathematical analysis and further motivate the development of new FP frameworks capable of addressing nonlinear fractional problems. Additional developments related to fractional differential equations and fractional calculus can be found in [18,19,20].

Motivated by these developments, in this paper, we introduce a new generalized structure, namely perturbed $\mathcal{F}$-MSs, which simultaneously extend both $\mathcal{F}$-MSs and perturbed MSs. This new setting provides a unified framework that captures the advantages of both approaches and allows for a broader class of mappings to be studied. Within this framework, we first establish FP results for mappings satisfying Banach contraction conditions. These results serve as a foundation for further investigations. Subsequently, we derive FP theorems for rational-type contractions, which significantly generalize the Banach-type results by incorporating nonlinear distance relationships. We then develop FP results for interpolative contractions, thereby extending and unifying several existing theorems in the literature. The validity and applicability of the obtained results are illustrated through appropriate nontrivial examples. As consequences of our main theorems, we obtain corresponding FP results in the settings of $\mathcal{F}$-MSs and perturbed MSs by considering particular cases of the proposed structure. These corollaries not only recover several known results but also improve and generalize them. As an application of our main FP results, we study the existence of solutions for a Caputo fractional initial value problem in the framework of perturbed $\mathcal{F}$-MSs. This demonstrates the effectiveness of the proposed approach and highlights its potential for addressing nonlinear problems in fractional calculus.

2. Preliminaries

This section collects the key definitions, notation, and preliminary results that will be employed throughout this paper. We start with the notion of perturbed $\mathcal{F}$-MSs, followed by a brief overview of perturbed MSs and $\mathcal{F}$-metric structures. These concepts establish the mathematical foundation required for the development of the main results in the later sections. Throughout this work, $\mathbb{R}$ denotes the set of real numbers, ${\mathbb{R}}^{+}$ represents the set of non-negative real numbers, and $\mathbb{N}$ stands for the set of natural numbers.

In 1922, Stefan Banach [1] introduced the concept of contraction mapping, which is now known as the Banach contraction. This result is fundamental in nonlinear analysis and provides a powerful tool for proving the existence and uniqueness of FPs in complete MSs. It also serves as the basis for many iterative numerical methods used in applied mathematics.

Definition 1 

([1]). Let $\left(\mathcal{W},d\right)$ be an MS and $\mathcal{T}:\mathcal{W}\to \mathcal{W}.$ Then $\mathcal{T}$ is called a Banach contraction if there exists $\phi \in [0,1)$ such that

$$d\left(\mathcal{T}\sigma ,\mathcal{T}\varkappa \right)\le \phi d\left(\sigma ,\varkappa \right),$$

for all $\sigma ,\varkappa \in \mathcal{W}.$

Later on, Fisher [3] introduced the notion of rational contraction as a generalization of Banach contraction. This extension is important because it incorporates nonlinear terms involving the mapping itself, allowing for the treatment of problems where classical linear contractive conditions are not sufficient.

Definition 2 

([3]). Let $\left(\mathcal{W},d\right)$ be an MS. A mapping $\mathcal{T}:\mathcal{W}\to \mathcal{W}$ is said to be a rational contraction if there exist ${\phi }_1,{\phi }_2\in [0,1)$ with ${\phi }_1+{\phi }_2<1$ such that

$$d(\mathcal{T}\sigma ,\mathcal{T}\varkappa )\le {\phi }_{1}d\left(\sigma ,\varkappa \right)+{\phi }_2{\displaystyle \frac{d(\sigma ,\mathcal{T}\sigma )d(\varkappa ,\mathcal{T}\varkappa )}{1+d\left(\sigma ,\varkappa \right)}},$$

for all $\sigma ,\varkappa \in \mathcal{W}$.

Karapınar [4] unified the concept of contraction with interpolation techniques to introduce the notion of interpolative contraction. This class of contractions is important because it provides more flexibility in handling nonlinear problems and has been widely applied in differential equations and optimization theory.

Definition 3 

([4]). Let $\left(\mathcal{W},d\right)$ be an MS. A mapping $\mathcal{T}:\mathcal{W}\to \mathcal{W}$ is said to be an interpolative contraction if there exist $\phi \in (0,1)$ and $\beta \in (0,1)$ such that

$$d(\mathcal{T}\sigma ,\mathcal{T}\varkappa )\le \phi {\left[d(\sigma ,\mathcal{T}\sigma )\right]}^{\beta }{\left[d(\varkappa ,\mathcal{T}\varkappa )\right]}^{1-\beta },$$

for all $\sigma ,\varkappa \in \mathcal{W}$.

The concept of a b-MS represents a significant extension of the classical metric framework. It was first introduced by Bakhtin [5] in 1993, where the standard triangle inequality is relaxed through the inclusion of a constant $s\ge 1.$ This modification allows for a broader class of distance functions while preserving the essential structure needed for analysis. Subsequently, Czerwik [6] carried out a systematic investigation of b-MSs, establishing their fundamental properties and highlighting their effectiveness in the study of FP theory and various problems in nonlinear analysis.

Definition 4 

([5]). Let $\mathcal{W}\ne \varnothing$ and $s\ge 1$. A real-valued function $d:\mathcal{W}\times \mathcal{W}\to$ $[0,+\infty )$ is said to be a b-metric if it satisfies

(b₁)

$d(\sigma ,\varkappa )=0$ if and only if $\sigma =\varkappa$.

(b₂)

$d(\sigma ,\varkappa )=d(\varkappa ,\sigma )$.

(b₃)

$d(\sigma ,\varkappa )\le s\left[d\right(\sigma ,z)+d(z,\varkappa \left)\right].$

The pair $(\mathcal{W},d)$ is then called a b-MS.

In 2003, Fagin et al. [7] proposed a further generalization of b-MSs by introducing the concept of s-relaxed_p-MSs. This framework extends the b-metric structure by incorporating an additional relaxation parameter, thereby allowing for greater flexibility in the underlying distance function. The notion is defined as follows:

Definition 5 

([7]). Let $\mathcal{W}\ne \varnothing$, and let $d:\mathcal{W}\times \mathcal{W}\to [0,+\infty )$ be a function satisfying the following conditions:

(sp₁)

$d(\sigma ,\varkappa )=0$ if and only if $\sigma =\varkappa$.

(sp₂)

$d(\sigma ,\varkappa )=d(\varkappa ,\sigma )$.

(sp₃)

There exists a constant $s\ge 1$ such that for every integer $N\ge 2$ and for every finite sequence ${\left(u_i\right)}_{i=1}^N\subset \mathcal{W}$ with $u_1=\sigma$ and $u_N=\varkappa$, the inequality

$$d(\sigma ,\varkappa )>0\Rightarrow d(\sigma ,\varkappa )\le s\left({\displaystyle \sum _{i=1}^{N-1}}d(u_i,u_{i+1})\right)$$

holds. In this case, the pair $(\mathcal{W},d)$ is said to be an s-relaxed_p-MS.

Jleli et al. [8] proposed a notable generalization of all the above metric structures by introducing the concept of an $\mathcal{F}$-MS. To describe this framework, let $\mathcal{F}$ denote the family of functions $\xi :(0,+\infty )\to \mathbb{R}$ satisfying the following conditions:

(${\mathcal{F}}_1$)

$\xi$ is non-decreasing, i.e., for $0<s<t,$ one has $\xi \left(s\right)\le \xi \left(t\right);$

(${\mathcal{F}}_2$)

For any sequence $\left\{t_n\right\}\subseteq {\mathbb{R}}^{+}$,

$$\underset{n\to +\infty }{lim}{t}_n=0\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}\underset{n\to +\infty }{lim}\xi \left(t_n\right)=-\infty .$$

Definition 6 

([8]). Let $\mathcal{W}\ne \varnothing$, and let $d_{\mathcal{F}}:\mathcal{W}\times \mathcal{W}\to [0,+\infty )$ be a function. Assume that there exists a pair $(\xi ,\chi )\in \mathcal{F}\times [0,+\infty )$ such that

(fm₁)

$d_{\mathcal{F}}(\sigma ,\varkappa )=0$ if and only if $\sigma =\varkappa$.

(fm₂)

$d_{\mathcal{F}}(\sigma ,\varkappa )=d_{\mathcal{F}}(\varkappa ,\sigma )$.

(fm₃)

For every $\sigma ,\varkappa \in \mathcal{W}$, for each integer $N\ge 2$, and for every finite sequence ${\left(u_i\right)}_{i=1}^N\subset \mathcal{W}$ with $u_1=\sigma$ and $u_N=\varkappa ,$ the following implication holds:

$$d_{\mathcal{F}}(\sigma ,\varkappa )>0\Rightarrow \xi \left(d_{\mathcal{F}}(\sigma ,\varkappa )\right)\le \xi \left({\displaystyle \sum _{i=1}^{N-1}}{d}_{\mathcal{F}}(u_i,u_{i+1})\right)+\chi .$$

Then the function $d_{\mathcal{F}}$ is called an $\mathcal{F}$-metric on $\mathcal{W}$, and the pair $(\mathcal{W},d_{\mathcal{F}})$ is said to be an $\mathcal{F}$-MS.

On the other hand, Jleli et al. [11] introduced the concept of a perturbed MS as a natural refinement of the classical notion of an MS. The main idea behind this approach is to view the distance between two points as consisting of two parts: an exact component that preserves the usual metric behavior and an additional term that represents a perturbation. This perspective allows one to retain the fundamental structure of an MS while incorporating possible deviations arising from external influences. In particular, the classical MS appears as a special case when the perturbation term is absent. The motivation for introducing this framework lies in its ability to model uncertainties, errors, and disturbances that commonly occur in real-world applications.

Definition 7 

([11]). Let $\mathcal{R},\mathcal{S}:\mathcal{W}\times \mathcal{W}\to [0,+\infty )$ be two functions. The function $\mathcal{R}$ is said to be a perturbed metric on $\mathcal{W}$ with respect to $\mathcal{S}$ if the mapping

$$(\mathcal{R}-\mathcal{S}):\mathcal{W}\times \mathcal{W}\to {\mathbb{R}}^{+}$$

is a metric on $\mathcal{W}$. That is, for all $\sigma ,\varkappa ,z\in \mathcal{W}$, the following conditions hold:

(P₁)

$(\mathcal{R}-\mathcal{S})(\sigma ,\varkappa )=0$ if and only if $\sigma =\varkappa$;

(P₂)

$(\mathcal{R}-\mathcal{S})(\sigma ,\varkappa )=(\mathcal{R}-\mathcal{S})(\varkappa ,\sigma )$;

(P₃)

$(\mathcal{R}-\mathcal{S})(\sigma ,\varkappa )\le (\mathcal{R}-\mathcal{S})(\sigma ,z)+(\mathcal{R}-\mathcal{S})(z,\varkappa )$.

In this setting, $\mathcal{S}$ is referred to as the perturbation mapping, $d=\mathcal{R}-\mathcal{S}$ is called the exact metric, and the triple ($\mathcal{W},\mathcal{R},\mathcal{S}$) is known as a perturbed MS.

Example 1. 

Let $\mathcal{W}=\mathbb{R}$, and the function $\mathcal{R}:\mathcal{W}\times \mathcal{W}\to [0,+\infty )$ is defined as

$$\mathcal{R}(\sigma ,\varkappa )=|\sigma -\varkappa |+{\sigma }^2{\varkappa }^4,$$

for all $\sigma ,\varkappa \in \mathcal{W}.$ Define $\mathcal{S}:\mathcal{W}\times \mathcal{W}\to [0,+\infty )$ by

$$\mathcal{S}(\sigma ,\varkappa )={\sigma }^2{\varkappa }^4,$$

for all $\sigma ,\varkappa \in \mathcal{W}.$ Then the induced function $d:\mathcal{W}\times \mathcal{W}\to [0,+\infty )$ defined by

$$d(\sigma ,\varkappa )=|\sigma -\varkappa |,$$

for all $\sigma ,\varkappa \in \mathcal{W},$ is a metric on $\mathcal{W}.$ Hence, $\mathcal{R}$ is a perturbed metric on $\mathcal{W}$ with respect to $\mathcal{S}$.

It is important to note that $\mathcal{R}$ itself is not a metric. Indeed,

$$\mathcal{R}(1,1)=1\ne 0,$$

which violates the identity of indiscernibles.

In this spirit, we introduce the concept of a perturbed $\mathcal{F}$-MS, which naturally extends both $\mathcal{F}$-MSs and perturbed MSs.

Definition 8. 

Let $\mathcal{W}\ne \varnothing$, and let $\mathcal{R}:\mathcal{W}\times \mathcal{W}\to [0,+\infty )$ be a continuous function, while $\mathcal{S}:\mathcal{W}\times \mathcal{W}\to [0,+\infty )$ is a given auxiliary function. The function $\mathcal{R}$ is said to be a perturbed $\mathcal{F}$-metric on $\mathcal{W}$ with respect to $\mathcal{S}$ if the mapping

$$d_{\mathcal{F}}=\mathcal{R}-\mathcal{S}:\mathcal{W}\times \mathcal{W}\to {\mathbb{R}}^{+}$$

defines an $\mathcal{F}$-metric on $\mathcal{W}.$ That is, there exist a function $\xi \in \mathcal{F}$ and a constant $\chi \ge 0$ such that the following axioms hold for all $\sigma ,\varkappa \in \mathcal{W}$.

($P{\mathcal{F}}_1$) $(\mathcal{R}-\mathcal{S})(\sigma ,\varkappa )=0$ if and only if $\sigma =\varkappa$.

($P{\mathcal{F}}_2$) $(\mathcal{R}-\mathcal{S})(\sigma ,\varkappa )=(\mathcal{R}-\mathcal{S})(\varkappa ,\sigma )$.

($P{\mathcal{F}}_3$) For every integer $N\ge 2$, and for every finite sequence ${\left(u_i\right)}_{i=1}^N\subset \mathcal{W}$ with $u_1=\sigma$ and $u_N=\varkappa ,$ the following implication holds: If

$$\mathcal{R}(\sigma ,\varkappa )-\mathcal{S}(\sigma ,\varkappa )>0,$$

then

$$\xi \left(\mathcal{R}(\sigma ,\varkappa )-\mathcal{S}(\sigma ,\varkappa )\right)\le \xi \left({\displaystyle \sum _{i=1}^{N-1}}\left(\mathcal{R}\left(u_i,u_{i+1}\right)-\mathcal{S}\left(u_i,u_{i+1}\right)\right)\right)+\chi .$$

The pair $(\mathcal{W},d_{\mathcal{F}})$ is called an $\mathcal{F}$-MS. Then, the triple $\left(\mathcal{W},\mathcal{R},\mathcal{S}\right)$ is referred to as a perturbed $\mathcal{F}$-MS.

Example 2. 

Let $\mathcal{W}=\mathbb{R}$, and define the mappings

$$\mathcal{R}(\sigma ,\varkappa )=\left|\sigma -\varkappa \right|+1,$$

and

$$\mathcal{S}(\sigma ,\varkappa )=1,$$

for all $\sigma ,\varkappa \in \mathcal{W}.$ Then the induced function

$$d_{\mathcal{F}}(\sigma ,\varkappa )=\mathcal{R}(\sigma ,\varkappa )-\mathcal{S}(\sigma ,\varkappa )$$

reduces to

$$d_{\mathcal{F}}(\sigma ,\varkappa )=\left|\sigma -\varkappa \right|.$$

It is straightforward to verify that $d_{\mathcal{F}}$ defines an $\mathcal{F}$-metric on $\mathcal{W}$ with respect to the function $\xi \left(t\right)=ln\left(t\right),$ for $t>0,$ and with $\chi =0.$ Hence $\left(\mathcal{W},\mathcal{R},\mathcal{S}\right)$ is a perturbed $\mathcal{F}$-MS.

Example 3. 

Let $\mathcal{W}=\mathcal{C}\left([0,\mathsf{\Lambda }],\mathbb{R}\right)$ be the Banach space of all real-valued continuous functions on $[0,\mathsf{\Lambda }],$ endowed with the supremum norm

$${∥\sigma ∥}_{\infty }=\underset{t\in [0,\mathsf{\Lambda }]}{sup}\left|\sigma \left(t\right)\right|.$$

Define $\mathcal{R},\mathcal{S}:\mathcal{W}\times \mathcal{W}\to {\mathbb{R}}^{+}$ by

$$\mathcal{R}\left(\sigma ,\varkappa \right)=\underset{t\in [0,\mathsf{\Lambda }]}{sup}\left|\sigma \left(t\right)-\varkappa \left(t\right)\right|,$$

and

$$\mathcal{S}\left(\sigma ,\varkappa \right)=\lambda \underset{t\in [0,\mathsf{\Lambda }]}{sup}\left|\sigma \left(t\right)-\varkappa \left(t\right)\right|,$$

for all $\sigma ,\varkappa \in \mathcal{W},$ where $\lambda \in [0,1).$ Then

$$d_{\mathcal{F}}\left(\sigma ,\varkappa \right)=\left(1-\lambda \right)\underset{t\in [0,\mathsf{\Lambda }]}{sup}\left|\sigma \left(t\right)-\varkappa \left(t\right)\right|.$$

Since $0<\left(1-\lambda \right)\le 1,$ it follows directly that

$$d_{\mathcal{F}}\left(\sigma ,\varkappa \right)=\left(1-\lambda \right)\underset{t\in [0,\mathsf{\Lambda }]}{sup}\left|\sigma \left(t\right)-\varkappa \left(t\right)\right|\ge 0,$$

and

$$d_{\mathcal{F}}\left(\sigma ,\varkappa \right)=0\mathit{if}\mathit{and}\mathit{only}\sigma =\varkappa .$$

Also,

$$\begin{array}{c}\hfill d_{\mathcal{F}}\left(\sigma ,\varkappa \right)=\left(1-\lambda \right)\underset{t\in [0,\mathsf{\Lambda }]}{sup}\left|\sigma \left(t\right)-\varkappa \left(t\right)\right|\\ \hfill =\left(1-\lambda \right)\underset{t\in [0,\mathsf{\Lambda }]}{sup}\left|\varkappa \left(t\right)-\sigma \left(t\right)\right|\\ \hfill =d_{\mathcal{F}}\left(\varkappa ,\sigma \right).\end{array}$$

For the generalized triangle-type condition, let ${\left(u_i\right)}_{i=1}^N\subset \mathcal{W}$ with $u_1=\sigma$ and $u_N=\varkappa .$ Then, by the triangle inequality of the absolute value,

$$\left|\sigma \left(t\right)-\varkappa \left(t\right)\right|\le {\displaystyle \sum _{i=1}^{N-1}}\left|u_i\left(t\right)-u_{i+1}\left(t\right)\right|,$$

for all $t\in [0,\mathsf{\Lambda }].$ Taking the supremum over $t\in [0,\mathsf{\Lambda }],$ we obtain

$$\underset{t\in [0,\mathsf{\Lambda }]}{sup}\left|\sigma \left(t\right)-\varkappa \left(t\right)\right|\le {\displaystyle \sum _{i=1}^{N-1}}\underset{t\in [0,\mathsf{\Lambda }]}{sup}\left|u_i\left(t\right)-u_{i+1}\left(t\right)\right|.$$

Multiplying by $(1-\lambda )$ gives

$$d_{\mathcal{F}}\left(\sigma ,\varkappa \right)\le {\displaystyle \sum _{i=1}^{N-1}}{d}_{\mathcal{F}}(u_i,u_{i+1}).$$

Since $\xi \left(t\right)=lnt$ is non-decreasing on $(0,+\infty )$, it follows that

$$\xi \left(d_{\mathcal{F}}\left(\sigma ,\varkappa \right)\right)\le \xi \left({\displaystyle \sum _{i=1}^{N-1}}{d}_{\mathcal{F}}(u_i,u_{i+1})\right)+\chi ,$$

with $\chi =ln1=0.$ Hence, $(\mathcal{W},\mathcal{R},\mathcal{S})$ is a perturbed $\mathcal{F}$-MS.

Definition 9. 

Let $\left(\mathcal{W},\mathcal{R},\mathcal{S}\right)$ be a perturbed $\mathcal{F}$-MS, and let $\left\{{\sigma }_n\right\}$ be a sequence in $\mathcal{W}.$ Then

(i) The sequence $\left\{{\sigma }_n\right\}$ converges to $\sigma \in \mathcal{W}$ if

$$\underset{n\to +\infty }{lim}(\mathcal{R}-\mathcal{S})({\sigma }_n,\sigma )=0.$$

(ii) The sequence $\left\{{\sigma }_n\right\}$ is a Cauchy sequence if

$$\underset{n,m\to +\infty }{lim}(\mathcal{R}-\mathcal{S})({\sigma }_n,{\sigma }_m)=0.$$

(iii) The space $\left(\mathcal{W},\mathcal{R},\mathcal{S}\right)$ is complete if every Cauchy sequence in $\mathcal{W}$ converges with respect to an $\mathcal{F}$-metric $d_{\mathcal{F}}=\mathcal{R}-\mathcal{S}.$

3. Main Results

3.1. Fixed Point Theorem for Banach Contraction

In this subsection, we present our leading FP results in the setting of perturbed $\mathcal{F}$-MSs. Building on the framework introduced in the previous section, we establish a Banach contraction theorem for perturbed mappings. This result extends the classical contraction principle to the newly defined structure $\left(\mathcal{W},\mathcal{R},\mathcal{S}\right),$ providing sufficient conditions for the existence and uniqueness of FPs under a contractive perturbation.

Theorem 1. 

Let $(\mathcal{W},\mathcal{R},\mathcal{S})$ be a complete perturbed $\mathcal{F}$-MS, and let $\mathcal{T}:$ $\mathcal{W}$ $\to \mathcal{W}$. Assume that there exists a constant $\phi \in [0,1)$ such that

$$\mathcal{R}\left(\mathcal{T}\sigma ,\mathcal{T}\varkappa \right)\le \phi \mathcal{R}\left(\sigma ,\varkappa \right),$$

(1)

for all $\sigma ,\varkappa \in \mathcal{W}.$ Then $\mathcal{T}$ admits a unique FP.

Proof. 

Let ${\sigma }_0\in \mathcal{W}$ be arbitrary. Define a sequence {${\sigma }_n$} by

$${\sigma }_{n+1}=\mathcal{T}{\sigma }_n,$$

for all $n\in \mathbb{N}.$ By (1) with $\left(\sigma ,\varkappa \right)=\left({\sigma }_0,{\sigma }_1\right),$ we have

$$\begin{array}{c}\hfill \mathcal{R}\left({\sigma }_1,{\sigma }_2\right)=\mathcal{R}\left(\mathcal{T}{\sigma }_0,\mathcal{T}{\sigma }_1\right)\\ \hfill \le \phi \mathcal{R}\left({\sigma }_0,{\sigma }_1\right),\end{array}$$

which yields

$$\mathcal{R}\left({\sigma }_1,{\sigma }_2\right)\le \phi \mathcal{R}\left({\sigma }_0,{\sigma }_1\right).$$

(2)

Similarly, by (1) with $\left(\sigma ,\varkappa \right)=\left({\sigma }_1,{\sigma }_2\right),$ we have

$$\begin{array}{c}\hfill \mathcal{R}\left({\sigma }_2,{\sigma }_3\right)=\mathcal{R}\left(\mathcal{T}{\sigma }_1,\mathcal{T}{\sigma }_2\right)\\ \hfill \le \phi \mathcal{R}\left({\sigma }_1,{\sigma }_2\right),\end{array}$$

which yields

$$\mathcal{R}\left({\sigma }_2,{\sigma }_3\right)\le \phi \mathcal{R}\left({\sigma }_1,{\sigma }_2\right),$$

which implies by (2) that

$$\mathcal{R}\left({\sigma }_2,{\sigma }_3\right)\le {\phi }^2\mathcal{R}\left({\sigma }_0,{\sigma }_1\right).$$

(3)

By continuing this process and applying mathematical induction, we obtain

$$\mathcal{R}\left({\sigma }_n,{\sigma }_{n+1}\right)\le {\phi }^n\mathcal{R}\left({\sigma }_0,{\sigma }_1\right),$$

(4)

for all $n\in \mathbb{N}.$ Since $d_{\mathcal{F}}(\sigma ,\varkappa )=\mathcal{R}(\sigma ,\varkappa )-\mathcal{S}(\sigma ,\varkappa )\le \mathcal{R}(\sigma ,\varkappa )$$\left(\mathrm{because}\mathcal{S}(\sigma ,\varkappa )\ge 0\right),$ it follows that

$$d_{\mathcal{F}}\left({\sigma }_n,{\sigma }_{n+1}\right)\le \mathcal{R}\left({\sigma }_n,{\sigma }_{n+1}\right)\le {\phi }^n\mathcal{R}\left({\sigma }_0,{\sigma }_1\right),$$

for all $n\in \mathbb{N}.$ For $m>n$, we have

$$\begin{array}{c}\hfill {\displaystyle \sum _{i=n}^{m-1}}{d}_{\mathcal{F}}\left({\sigma }_i,{\sigma }_{i+1}\right)\le {\displaystyle \sum _{i=n}^{m-1}}{\phi }^i\mathcal{R}\left({\sigma }_0,{\sigma }_1\right)\le {\displaystyle \sum _{i=n}^{\infty }}{\phi }^i\mathcal{R}\left({\sigma }_0,{\sigma }_1\right)\\ \hfill ={\displaystyle \frac{{\phi }^n}{1-\phi }}\mathcal{R}\left({\sigma }_0,{\sigma }_1\right).\end{array}$$

Since $\phi \in [0,1),$ the right-hand side tends to zero as $n\to +\infty .$ Using properties (${\mathcal{F}}_1$) and (${\mathcal{F}}_2$), there exists $n\left(ϵ\right)$ such that

$$\xi \left({\displaystyle \sum _{i=n}^{m-1}}{d}_{\mathcal{F}}\left({\sigma }_i,{\sigma }_{i+1}\right)\right)<\xi \left(ϵ\right)-\chi ,$$

(5)

for $m>n\ge n\left(ϵ\right).$ Using ($P{\mathcal{F}}_3$) and (5), we obtain $d_{\mathcal{F}}\left({\sigma }_n,{\sigma }_m\right)>0,$ $m>n\ge n\left(ϵ\right)$, which implies

$$\xi \left(d_{\mathcal{F}}\left({\sigma }_n,{\sigma }_m\right)\right)\le \xi \left({\displaystyle \sum _{i=n}^{m-1}}{d}_{\mathcal{F}}\left({\sigma }_i,{\sigma }_{i+1}\right)\right)+\chi <\xi \left(ϵ\right),$$

which yields by (${\mathcal{F}}_1$) that

$$d_{\mathcal{F}}\left({\sigma }_n,{\sigma }_m\right)<ϵ,$$

$m>n\ge n\left(ϵ\right).$ This shows that $\left\{{\sigma }_n\right\}$ is Cauchy sequence in $\mathcal{F}$-MS ($\mathcal{W},d_{\mathcal{F}}$); that is, $\left\{{\sigma }_n\right\}$ is a perturbed Cauchy sequence in the perturbed $\mathcal{F}$-MS ($\mathcal{W},\mathcal{R},\mathcal{S}$). By the completeness of the perturbed $\mathcal{F}$-MS $(\mathcal{W},\mathcal{R},\mathcal{S})$, we deduce that there exists ${\sigma }^{*}\in \mathcal{W}$ such that

$$\underset{n\to +\infty }{lim}{d}_{\mathcal{F}}\left({\sigma }_n,{\sigma }^{*}\right)=0.$$

(6)

We now show that ${\sigma }^{*}$ is an FP of $\mathcal{T}$. By (1), we have

$$\mathcal{R}\left({\sigma }_{n+1},\mathcal{T}{\sigma }^{*}\right)=\mathcal{R}\left(\mathcal{T}{\sigma }_n,\mathcal{T}{\sigma }^{*}\right)\le \phi \mathcal{R}\left({\sigma }_n,{\sigma }^{*}\right).$$

Taking the limit as $n\to +\infty$ and using the continuity of the function $\mathcal{R},$ we get

$$\mathcal{R}\left({\sigma }^{*},\mathcal{T}{\sigma }^{*}\right)\le 0,$$

so

$$\mathcal{R}\left({\sigma }^{*},\mathcal{T}{\sigma }^{*}\right)=0.$$

Since $d_{\mathcal{F}}=\mathcal{R}-\mathcal{S}$ is an $\mathcal{F}$-metric on $\mathcal{W},$ we have

$$d_{\mathcal{F}}\left({\sigma }^{*},\mathcal{T}{\sigma }^{*}\right)=\mathcal{R}\left({\sigma }^{*},\mathcal{T}{\sigma }^{*}\right)-\mathcal{S}\left({\sigma }^{*},\mathcal{T}{\sigma }^{*}\right)\le \mathcal{R}\left({\sigma }^{*},\mathcal{T}{\sigma }^{*}\right)=0.$$

Thus ${\sigma }^{*}$ is an FP of $\mathcal{T}$. Let ${\sigma }^{*}$ and ${\sigma }^{/}$ be two FPs of $\mathcal{T}$. Then, by (1), we have

$$\mathcal{R}\left({\sigma }^{*},{\sigma }^{/}\right)=\mathcal{R}\left(\mathcal{T}{\sigma }^{*},\mathcal{T}{\sigma }^{/}\right)\le \phi \mathcal{R}\left({\sigma }^{*},{\sigma }^{/}\right).$$

Since $0\le \phi <1,$ this implies $\mathcal{R}\left({\sigma }^{*},{\sigma }^{/}\right)=0.$ Thus

$$d_{\mathcal{F}}(\sigma ,\varkappa )=\mathcal{R}(\sigma ,\varkappa )-\mathcal{S}(\sigma ,\varkappa )\le \mathcal{R}(\sigma ,\varkappa ).$$

Consequently,

$$d_{\mathcal{F}}\left({\sigma }^{*},{\sigma }^{/}\right)=0,$$

which implies ${\sigma }^{*}={\sigma }^{/}.$ Hence, the FP is unique.    □

Example 4. 

Let $\mathcal{W}=\mathbb{R}.$ Define $\mathcal{R},\mathcal{S}:\mathcal{W}\times \mathcal{W}\to [0,+\infty )$ by

$$\mathcal{S}\left(\sigma ,\varkappa \right)=\left|\sigma -\varkappa \right|,$$

and

$$\mathcal{R}\left(\sigma ,\varkappa \right)=2\left|\sigma -\varkappa \right|.$$

Then

$$d_{\mathcal{F}}\left(\sigma ,\varkappa \right)=\mathcal{R}\left(\sigma ,\varkappa \right)-\mathcal{S}\left(\sigma ,\varkappa \right)=\left|\sigma -\varkappa \right|$$

is an $\mathcal{F}$-metric by choosing $\xi \left(t\right)=lnt,$ for $t>0$ and $\chi =0.$ It is trivial to show that ξ satisfies conditions (${\mathcal{F}}_1$)–(${\mathcal{F}}_3$). Thus $\left(\mathcal{W},\mathcal{R},\mathcal{S}\right)$ is a perturbed $\mathcal{F}$-MS. Moreover, $(\mathcal{W},d_{\mathcal{F}})=(\mathbb{R},\left|·\right|)$ is complete (every Cauchy sequence of real numbers converges). So $\left(\mathcal{W},\mathcal{R},\mathcal{S}\right)$ is a complete perturbed $\mathcal{F}$-MS. Define $\mathcal{T}:$ $\mathcal{W}$ $\to \mathcal{W}$ by

$$\mathcal{T}\sigma ={\displaystyle \frac{\sigma }{2}},\mathit{for}\mathit{all}\sigma \in \mathcal{W}.$$

Now for any $\sigma ,\varkappa \in \mathcal{W},$ we have

$$\mathcal{R}\left(\mathcal{T}\sigma ,\mathcal{T}\varkappa \right)=\left|{\displaystyle \frac{\sigma }{2}}-{\displaystyle \frac{\varkappa }{2}}\right|={\displaystyle \frac{1}{2}}\left|\sigma -\varkappa \right|=\phi \mathcal{R}\left(\sigma ,\varkappa \right).$$

Therefore contraction condition (1) holds with $\phi ={\displaystyle \frac{1}{2}}$. Thus all the hypotheses of Theorem 1 are satisfied, and $\mathcal{T}$ has a unique FP $0\in \mathcal{W}.$

3.2. Fixed Point Result for Rational-Type Contraction

In this subsection, we establish our next FP result under a rational-type contractive condition in the framework of perturbed $\mathcal{F}$-MSs. This theorem extends the previous contraction principles by incorporating a rational expression involving the distances between points and their images, leading to a more general form of contraction. Under this setting, we prove the existence and uniqueness of an FP for the considered self-mapping.

Theorem 2. 

Let $\left(\mathcal{W},\mathcal{R},\mathcal{S}\right)$ be a complete perturbed $\mathcal{F}$-MS, and let $\mathcal{T}:$ $\mathcal{W}$ $\to \mathcal{W}$ be a self-mapping. Suppose that there exists a constant $\phi \in [0,1)$ such that

$$\mathcal{R}\left(\mathcal{T}\sigma ,\mathcal{T}\varkappa \right)\le \phi \mathcal{M}\left(\sigma ,\varkappa \right),$$

(7)

where

$$\mathcal{M}\left(\sigma ,\varkappa \right)=max\left\{\mathcal{R}\left(\sigma ,\varkappa \right),{\displaystyle \frac{\mathcal{R}\left(\sigma ,\mathcal{T}\sigma \right)\mathcal{R}\left(\varkappa ,\mathcal{T}\varkappa \right)}{1+\mathcal{R}\left(\sigma ,\varkappa \right)}}\right\},$$

(8)

for all $\sigma ,\varkappa \in \mathcal{W}$; then the mapping $\mathcal{T}$ admits a unique FP ${\sigma }^{*}$ in $\mathcal{W}$.

Proof. 

Let ${\sigma }_0\in \mathcal{W}$ be arbitrary. Define a sequence {${\sigma }_n$} by

$${\sigma }_{n+1}=\mathcal{T}{\sigma }_n,$$

for all $n\in \mathbb{N}.$ By (7) with $\left(\sigma ,\varkappa \right)=\left({\sigma }_0,{\sigma }_1\right),$ we have

$$\mathcal{R}\left({\sigma }_1,{\sigma }_2\right)=\mathcal{R}\left(\mathcal{T}{\sigma }_0,\mathcal{T}{\sigma }_1\right)\le \phi \mathcal{M}\left({\sigma }_0,{\sigma }_1\right),$$

where

$$\begin{array}{c}\hfill \mathcal{M}\left({\sigma }_0,{\sigma }_1\right)=max\left\{\mathcal{R}\left({\sigma }_0,{\sigma }_1\right),{\displaystyle \frac{\mathcal{R}\left({\sigma }_0,\mathcal{T}{\sigma }_0\right)\mathcal{R}\left({\sigma }_1,\mathcal{T}{\sigma }_1\right)}{1+\mathcal{R}\left({\sigma }_0,{\sigma }_1\right)}}\right\}\\ \hfill =max\left\{\mathcal{R}\left({\sigma }_0,{\sigma }_1\right),{\displaystyle \frac{\mathcal{R}\left({\sigma }_0,{\sigma }_1\right)\mathcal{R}\left({\sigma }_1,{\sigma }_2\right)}{1+\mathcal{R}\left({\sigma }_0,{\sigma }_1\right)}}\right\}.\end{array}$$

If $\mathcal{M}\left({\sigma }_0,{\sigma }_1\right)=\mathcal{R}\left({\sigma }_0,{\sigma }_1\right),$ then we get

$$\mathcal{R}\left({\sigma }_1,{\sigma }_2\right)=\mathcal{R}\left(\mathcal{T}{\sigma }_0,\mathcal{T}{\sigma }_1\right)\le \phi \mathcal{R}\left({\sigma }_0,{\sigma }_1\right).$$

If $\mathcal{M}\left({\sigma }_0,{\sigma }_1\right)={\displaystyle \frac{\mathcal{R}\left({\sigma }_0,{\sigma }_1\right)\mathcal{R}\left({\sigma }_1,{\sigma }_2\right)}{1+\mathcal{R}\left({\sigma }_0,{\sigma }_1\right)}},$ then we get

$$\begin{array}{c}\hfill \mathcal{R}\left({\sigma }_1,{\sigma }_2\right)=\mathcal{R}\left(\mathcal{T}{\sigma }_0,\mathcal{T}{\sigma }_1\right)\le \phi {\displaystyle \frac{\mathcal{R}\left({\sigma }_0,{\sigma }_1\right)\mathcal{R}\left({\sigma }_1,{\sigma }_2\right)}{1+\mathcal{R}\left({\sigma }_0,{\sigma }_1\right)}}\\ \hfill \le \phi \mathcal{R}\left({\sigma }_1,{\sigma }_2\right),\end{array}$$

which is a contradiction to the fact that $\phi <1.$ Hence

$$\mathcal{R}\left({\sigma }_1,{\sigma }_2\right)=\mathcal{R}\left(\mathcal{T}{\sigma }_0,\mathcal{T}{\sigma }_1\right)\le \phi \mathcal{R}\left({\sigma }_0,{\sigma }_1\right).$$

Similarly, by (7) with $\left(\sigma ,\varkappa \right)=\left({\sigma }_1,{\sigma }_2\right),$ we have

$$\mathcal{R}\left({\sigma }_2,{\sigma }_3\right)=\mathcal{R}\left(\mathcal{T}{\sigma }_1,\mathcal{T}{\sigma }_2\right)\le \phi \mathcal{M}\left({\sigma }_1,{\sigma }_2\right),$$

where

$$\begin{array}{c}\hfill \mathcal{M}\left({\sigma }_1,{\sigma }_2\right)=max\left\{\mathcal{R}\left({\sigma }_1,{\sigma }_2\right),{\displaystyle \frac{\mathcal{R}\left({\sigma }_1,\mathcal{T}{\sigma }_1\right)\mathcal{R}\left({\sigma }_2,\mathcal{T}{\sigma }_2\right)}{1+\mathcal{R}\left({\sigma }_1,{\sigma }_2\right)}}\right\}\\ \hfill =max\left\{\mathcal{R}\left({\sigma }_1,{\sigma }_2\right),{\displaystyle \frac{\mathcal{R}\left({\sigma }_1,{\sigma }_2\right)\mathcal{R}\left({\sigma }_2,{\sigma }_3\right)}{1+\mathcal{R}\left({\sigma }_1,{\sigma }_2\right)}}\right\}.\end{array}$$

If $\mathcal{M}\left({\sigma }_1,{\sigma }_2\right)=\mathcal{R}\left({\sigma }_1,{\sigma }_2\right),$ then we get

$$\mathcal{R}\left({\sigma }_2,{\sigma }_3\right)=\mathcal{R}\left(\mathcal{T}{\sigma }_1,\mathcal{T}{\sigma }_2\right)\le \phi \mathcal{R}\left({\sigma }_1,{\sigma }_2\right).$$

If $\mathcal{M}\left({\sigma }_1,{\sigma }_2\right)={\displaystyle \frac{\mathcal{R}\left({\sigma }_1,{\sigma }_2\right)\mathcal{R}\left({\sigma }_2,{\sigma }_3\right)}{1+\mathcal{R}\left({\sigma }_1,{\sigma }_2\right)}},$ then we get

$$\begin{array}{c}\hfill \mathcal{R}\left({\sigma }_2,{\sigma }_3\right)=\mathcal{R}\left(\mathcal{T}{\sigma }_1,\mathcal{T}{\sigma }_2\right)\le \phi {\displaystyle \frac{\mathcal{R}\left({\sigma }_1,{\sigma }_2\right)\mathcal{R}\left({\sigma }_2,{\sigma }_3\right)}{1+\mathcal{R}\left({\sigma }_1,{\sigma }_2\right)}}\\ \hfill \le \phi \mathcal{R}\left({\sigma }_2,{\sigma }_3\right),\end{array}$$

which is a contradiction to the fact that $\phi <1.$ Hence

$$\mathcal{R}\left({\sigma }_2,{\sigma }_3\right)=\mathcal{R}\left(\mathcal{T}{\sigma }_1,\mathcal{T}{\sigma }_2\right)\le \phi \mathcal{R}\left({\sigma }_1,{\sigma }_2\right).$$

By continuing this process and applying mathematical induction, we obtain

$$\mathcal{R}\left({\sigma }_n,{\sigma }_{n+1}\right)\le {\phi }^n\mathcal{R}\left({\sigma }_0,{\sigma }_1\right),$$

for all $n\in \mathbb{N}.$ Following the same arguments employed in the proof of Theorem 1, it can be established that $\left\{{\sigma }_n\right\}$ is a Cauchy sequence in the $\mathcal{F}$-MS ($\mathcal{W},d_{\mathcal{F}}$). Consequently, $\left\{{\sigma }_n\right\}$ is also a perturbed Cauchy sequence in the perturbed $\mathcal{F}$-MS ($\mathcal{W},\mathcal{R},\mathcal{S}$). Since $(\mathcal{W},\mathcal{R},\mathcal{S})$ is complete, there exists an element ${\sigma }^{*}\in \mathcal{W}$ such that

$$\underset{n\to +\infty }{lim}{d}_{\mathcal{F}}\left({\sigma }_n,{\sigma }^{*}\right)=0.$$

(9)

We now show that ${\sigma }^{*}$ is an FP of $\mathcal{T}$. By (7), we have

$$\begin{array}{c}\hfill \mathcal{R}\left({\sigma }_{n+1},\mathcal{T}{\sigma }^{*}\right)=\mathcal{R}\left(\mathcal{T}{\sigma }_n,\mathcal{T}{\sigma }^{*}\right)\le \phi max\left\{\mathcal{R}\left({\sigma }_n,{\sigma }^{*}\right),{\displaystyle \frac{\mathcal{R}\left({\sigma }_n,\mathcal{T}{\sigma }_n\right)\mathcal{R}\left({\sigma }^{*},\mathcal{T}{\sigma }^{*}\right)}{1+\mathcal{R}\left({\sigma }_n,{\sigma }^{*}\right)}}\right\}\\ \hfill =\phi max\left\{\mathcal{R}\left({\sigma }_n,{\sigma }^{*}\right),{\displaystyle \frac{\mathcal{R}\left({\sigma }_n,{\sigma }_{n+1}\right)\mathcal{R}\left({\sigma }^{*},\mathcal{T}{\sigma }^{*}\right)}{1+\mathcal{R}\left({\sigma }_n,{\sigma }^{*}\right)}}\right\}.\end{array}$$

Taking the limit as $n\to +\infty$ and using the continuity of the function $\mathcal{R},$ we get

$$\mathcal{R}\left({\sigma }^{*},\mathcal{T}{\sigma }^{*}\right)\le 0,$$

so

$$\mathcal{R}\left({\sigma }^{*},\mathcal{T}{\sigma }^{*}\right)=0.$$

Since $d_{\mathcal{F}}=\mathcal{R}-\mathcal{S}$ is a metric on $\mathcal{W},$ we have

$$d_{\mathcal{F}}\left({\sigma }^{*},\mathcal{T}{\sigma }^{*}\right)=\mathcal{R}\left({\sigma }^{*},\mathcal{T}{\sigma }^{*}\right)-\mathcal{S}\left({\sigma }^{*},\mathcal{T}{\sigma }^{*}\right)\le \mathcal{R}\left({\sigma }^{*},\mathcal{T}{\sigma }^{*}\right)=0.$$

Thus ${\sigma }^{*}$ is an FP of $\mathcal{T}$. Let ${\sigma }^{*}$ and ${\sigma }^{/}$ be two FPs of $\mathcal{T}$. Then, by (7), we have

$$\mathcal{R}\left({\sigma }^{*},{\sigma }^{/}\right)=\mathcal{R}\left(\mathcal{T}{\sigma }^{*},\mathcal{T}{\sigma }^{/}\right)\le \phi \mathcal{M}\left({\sigma }^{*},{\sigma }^{/}\right),$$

(10)

where

$$\mathcal{M}\left({\sigma }^{*},{\sigma }^{/}\right)=max\left\{\mathcal{R}\left({\sigma }^{*},{\sigma }^{/}\right),{\displaystyle \frac{\mathcal{R}\left({\sigma }^{*},\mathcal{T}{\sigma }^{*}\right)\mathcal{R}\left({\sigma }^{/},\mathcal{T}{\sigma }^{/}\right)}{1+\mathcal{R}\left({\sigma }^{*},{\sigma }^{/}\right)}}\right\}.$$

If

$$\mathcal{M}\left({\sigma }^{*},{\sigma }^{/}\right)=\mathcal{R}\left({\sigma }^{*},{\sigma }^{/}\right).$$

Then from (10), we get

$$\mathcal{R}\left({\sigma }^{*},{\sigma }^{/}\right)\le \phi \mathcal{R}\left({\sigma }^{*},{\sigma }^{/}\right),$$

which implies

$$\left(1-\phi \right)\mathcal{R}\left({\sigma }^{*},{\sigma }^{/}\right)\le 0.$$

Since $0\le \phi <1,$$\left(1-\phi \right)\ne 0.$ Thus we conclude that

$$\mathcal{R}\left({\sigma }^{*},{\sigma }^{/}\right)=0.$$

Moreover,

$$d_{\mathcal{F}}(\sigma ,\varkappa )=\mathcal{R}(\sigma ,\varkappa )-\mathcal{S}(\sigma ,\varkappa )\le \mathcal{R}(\sigma ,\varkappa ).$$

Consequently,

$$d_{\mathcal{F}}\left({\sigma }^{*},{\sigma }^{/}\right)=0,$$

which implies ${\sigma }^{*}={\sigma }^{/}.$ Hence, the FP is unique. Moreover, if

$$\mathcal{M}\left({\sigma }^{*},{\sigma }^{/}\right)={\displaystyle \frac{\mathcal{R}\left({\sigma }^{*},\mathcal{T}{\sigma }^{*}\right)\mathcal{R}\left({\sigma }^{/},\mathcal{T}{\sigma }^{/}\right)}{1+\mathcal{R}\left({\sigma }^{*},{\sigma }^{/}\right)}},$$

then from (10), we get

$$\mathcal{R}\left({\sigma }^{*},{\sigma }^{/}\right)\le \phi {\displaystyle \frac{\mathcal{R}\left({\sigma }^{*},\mathcal{T}{\sigma }^{*}\right)\mathcal{R}\left({\sigma }^{/},\mathcal{T}{\sigma }^{/}\right)}{1+\mathcal{R}\left({\sigma }^{*},{\sigma }^{/}\right)}}=0,$$

which implies

$$\mathcal{R}\left({\sigma }^{*},{\sigma }^{/}\right)=0.$$

Moreover,

$$d_{\mathcal{F}}(\sigma ,\varkappa )=\mathcal{R}(\sigma ,\varkappa )-\mathcal{S}(\sigma ,\varkappa )\le \mathcal{R}(\sigma ,\varkappa ).$$

Consequently,

$$d_{\mathcal{F}}\left({\sigma }^{*},{\sigma }^{/}\right)=0,$$

which implies ${\sigma }^{*}={\sigma }^{/}.$ Hence, the FP is unique.    □

Example 5. 

Let $\mathcal{W}=[0,1].$ Define $\mathcal{R},\mathcal{S}:\mathcal{W}\times \mathcal{W}\to [0,+\infty )$ by

$$\mathcal{S}\left(\sigma ,\varkappa \right)={\sigma }^2+{\varkappa }^2,$$

and

$$\mathcal{R}\left(\sigma ,\varkappa \right)=\left|\sigma -\varkappa \right|+{\sigma }^2+{\varkappa }^2.$$

Clearly, $\mathcal{R}$ is continuous. Then

$$d_{\mathcal{F}}\left(\sigma ,\varkappa \right)=\mathcal{R}\left(\sigma ,\varkappa \right)-\mathcal{S}\left(\sigma ,\varkappa \right)=\left|\sigma -\varkappa \right|$$

is an $\mathcal{F}$-metric by choosing $\xi \left(t\right)=lnt,$ for $t>0$ and $\chi =ln3.$ It is trivial to show that ξ satisfies conditions (${\mathcal{F}}_1$)–(${\mathcal{F}}_2$). Consequently, $\left(\mathcal{W},\mathcal{R},\mathcal{S}\right)$ is a complete perturbed $\mathcal{F}$-MS. Define $\mathcal{T}:$ $\mathcal{W}$ $\to \mathcal{W}$ by

$$\mathcal{T}\sigma ={\displaystyle \frac{\sigma }{3}},\mathit{for}\mathit{all}\sigma \in \mathcal{W}.$$

Now for all $\sigma ,\varkappa \in \mathcal{W},$ we have

$$\begin{array}{c}\hfill \mathcal{R}\left(\mathcal{T}\sigma ,\mathcal{T}\varkappa \right)=\left|{\displaystyle \frac{\sigma }{3}}-{\displaystyle \frac{\varkappa }{3}}\right|+{\left({\displaystyle \frac{\sigma }{3}}\right)}^2+{\left({\displaystyle \frac{\varkappa }{3}}\right)}^2\\ \hfill ={\displaystyle \frac{1}{3}}\left|\sigma -\varkappa \right|+{\displaystyle \frac{{\sigma }^2+{\varkappa }^2}{9}}.\end{array}$$

Since

$$\begin{array}{c}\hfill \mathcal{M}\left(\sigma ,\varkappa \right)=max\left\{\mathcal{R}\left(\sigma ,\varkappa \right),{\displaystyle \frac{\mathcal{R}\left(\sigma ,\mathcal{T}\sigma \right)\mathcal{R}\left(\varkappa ,\mathcal{T}\varkappa \right)}{1+\mathcal{R}\left(\sigma ,\varkappa \right)}}\right\}\\ \hfill \ge \mathcal{R}\left(\sigma ,\varkappa \right)=\left|\sigma -\varkappa \right|+{\sigma }^2+{\varkappa }^2,\end{array}$$

we have

$$\begin{array}{c}\hfill \mathcal{R}\left(\mathcal{T}\sigma ,\mathcal{T}\varkappa \right)={\displaystyle \frac{1}{3}}\left|\sigma -\varkappa \right|+{\displaystyle \frac{{\sigma }^2+{\varkappa }^2}{9}}\\ \hfill \le {\displaystyle \frac{1}{3}}\left(\left|\sigma -\varkappa \right|+{\sigma }^2+{\varkappa }^2\right)=\phi \mathcal{R}\left(\sigma ,\varkappa \right)\le \phi \mathcal{M}\left(\sigma ,\varkappa \right),\end{array}$$

where $\phi ={\displaystyle \frac{1}{3}}.$ Thus all the conditions of Theorem are satisfied, and 0 is a unique FP of $\mathcal{T}$.

3.3. Fixed Point Theorem for Interpolative Contraction

Now we present an FP theorem for interpolative contractions in the framework of perturbed $\mathcal{F}$-MSs. The result obtained in this subsection is independent of the previously established fixed point theorems, while it simultaneously extends Kannan’s FP theorem to the setting of perturbed $\mathcal{F}$-MSs. This is achieved by introducing an interpolative structure between different distance terms, which provides a more flexible framework for controlling the behavior of the mapping.

Theorem 3. 

Let $\left(\mathcal{W},\mathcal{R},\mathcal{S}\right)$ be a complete perturbed $\mathcal{F}$-MS, and let $\mathcal{T}:\mathcal{W}\to \mathcal{W}$ be a self-mapping. Consider the existence of $\phi \in (0,1)$ and $\beta \in (0,1)$ such that

$$\mathcal{R}(\mathcal{T}\sigma ,\mathcal{T}\varkappa )\le \phi {\left[\mathcal{R}(\sigma ,\mathcal{T}\sigma )\right]}^{\beta }{\left[\mathcal{R}(\varkappa ,\mathcal{T}\varkappa )\right]}^{1-\beta },$$

(11)

for all $\sigma ,\varkappa \in \mathcal{W}$ with $\sigma ,\varkappa \notin Fix\left(\mathcal{T}\right)$; then $\mathcal{T}$ has an FP in $\mathcal{W}.$

Proof. 

Let ${\sigma }_0\in \mathcal{W}$ be arbitrary, and define the Picard sequence {${\sigma }_n$} by

$${\sigma }_{n+1}=\mathcal{T}{\sigma }_n,$$

for all $n\in \mathbb{N}.$ If for some $n,$${\sigma }_{n+1}={\sigma }_n,$ then ${\sigma }_n$ is an FP, and the proof is complete. Hence, assume that ${\sigma }_{n+1}\ne {\sigma }_n,$ for all $n\in \mathbb{N}.$ Using the contractive condition with $\sigma ={\sigma }_{n-1}$ and $\varkappa ={\sigma }_n,$ we obtain

$$\begin{array}{c}\hfill \mathcal{R}({\sigma }_n,{\sigma }_{n+1})=\mathcal{R}(\mathcal{T}{\sigma }_{n-1},\mathcal{T}{\sigma }_n)\\ \hfill \le \phi {\left[\mathcal{R}({\sigma }_{n-1},\mathcal{T}{\sigma }_{n-1})\right]}^{\beta }{\left[\mathcal{R}({\sigma }_n,\mathcal{T}{\sigma }_n)\right]}^{1-\beta }\\ \hfill =\phi {\left[\mathcal{R}({\sigma }_{n-1},{\sigma }_n)\right]}^{\beta }{\left[\mathcal{R}({\sigma }_n,{\sigma }_{n+1})\right]}^{1-\beta }.\end{array}$$

If $\mathcal{R}({\sigma }_n,{\sigma }_{n+1})>0,$ dividing both sides by ${\left[\mathcal{R}({\sigma }_n,{\sigma }_{n+1})\right]}^{1-\beta }$ yields

$${\left[\mathcal{R}({\sigma }_n,{\sigma }_{n+1})\right]}^{\beta }\le \phi {\left[\mathcal{R}({\sigma }_{n-1},{\sigma }_n)\right]}^{\beta }.$$

Taking the power ${\displaystyle \frac{1}{\beta }}$ on both sides, we obtain

$$\mathcal{R}({\sigma }_n,{\sigma }_{n+1})\le {\phi }^{{\displaystyle \frac{1}{\beta }}}\mathcal{R}({\sigma }_{n-1},{\sigma }_n).$$

Let $q={\phi }^{{\displaystyle \frac{1}{\beta }}}\in \left(0,1\right).$ By induction,

$$\mathcal{R}({\sigma }_n,{\sigma }_{n+1})\le q^n\mathcal{R}({\sigma }_{n-1},{\sigma }_n),$$

for all $n\in \mathbb{N}.$ Since $d_{\mathcal{F}}(\sigma ,\varkappa )=\mathcal{R}(\sigma ,\varkappa )-\mathcal{S}(\sigma ,\varkappa )\le \mathcal{R}(\sigma ,\varkappa )$$\left(\mathrm{because}\mathcal{S}(\sigma ,\varkappa )\ge 0\right),$ it follows that

$$d_{\mathcal{F}}\left({\sigma }_n,{\sigma }_{n+1}\right)\le \mathcal{R}\left({\sigma }_n,{\sigma }_{n+1}\right)\le q^n\mathcal{R}\left({\sigma }_0,{\sigma }_1\right),$$

for all $n\in \mathbb{N}.$ Continuing the same way as in the proof of Theorem 1, we can prove that $\left\{{\sigma }_n\right\}$ is a Cauchy sequence in $\mathcal{F}$-MS $(\mathcal{W},d_{\mathcal{F}})$, that is, $\left\{{\sigma }_n\right\}$ is a perturbed Cauchy sequence in the perturbed $\mathcal{F}$-MS ($\mathcal{W},\mathcal{R},\mathcal{S}$). By the completeness of the perturbed $\mathcal{F}$-MS $(\mathcal{W},\mathcal{R},\mathcal{S})$, we deduce that there exists ${\sigma }^{*}\in \mathcal{W}$ such that

$$\underset{n\to +\infty }{lim}{d}_{\mathcal{F}}\left({\sigma }_n,{\sigma }^{*}\right)=0.$$

We now show that ${\sigma }^{*}$ is an FP of $\mathcal{T}$. By (11), we have

$$\begin{array}{c}\hfill \mathcal{R}\left({\sigma }_{n+1},\mathcal{T}{\sigma }^{*}\right)=\mathcal{R}\left(\mathcal{T}{\sigma }_n,\mathcal{T}{\sigma }^{*}\right)\le \phi {\left[\mathcal{R}({\sigma }_n,\mathcal{T}{\sigma }_n)\right]}^{\beta }{\left[\mathcal{R}({\sigma }^{*},\mathcal{T}{\sigma }^{*})\right]}^{1-\beta }\\ \hfill =\phi {\left[\mathcal{R}({\sigma }_n,{\sigma }_{n+1})\right]}^{\beta }{\left[\mathcal{R}({\sigma }^{*},\mathcal{T}{\sigma }^{*})\right]}^{1-\beta }.\end{array}$$

Taking the limit as $n\to +\infty$ and using the continuity of the function $\mathcal{R},$ we get

$$\mathcal{R}\left({\sigma }^{*},\mathcal{T}{\sigma }^{*}\right)\le 0,$$

so

$$\mathcal{R}\left({\sigma }^{*},\mathcal{T}{\sigma }^{*}\right)=0.$$

Since $d_{\mathcal{F}}=\mathcal{R}-\mathcal{S}$ is a metric on $\mathcal{W},$ we have

$$d_{\mathcal{F}}\left({\sigma }^{*},\mathcal{T}{\sigma }^{*}\right)=\mathcal{R}\left({\sigma }^{*},\mathcal{T}{\sigma }^{*}\right)-\mathcal{S}\left({\sigma }^{*},\mathcal{T}{\sigma }^{*}\right)\le \mathcal{R}\left({\sigma }^{*},\mathcal{T}{\sigma }^{*}\right)=0.$$

Thus ${\sigma }^{*}$ is an FP of $\mathcal{T}$.    □

Example 6. 

Let $\mathcal{W}=\{0,1,2,3\}.$ Define $\mathcal{R},\mathcal{S}:\mathcal{W}\times \mathcal{W}\to [0,+\infty )$ by

$$\mathcal{S}\left(\sigma ,\varkappa \right)=1,$$

and

$$\mathcal{R}\left(\sigma ,\varkappa \right)=\left|\sigma -\varkappa \right|+1.$$

Then

$$d_{\mathcal{F}}\left(\sigma ,\varkappa \right)=\mathcal{R}\left(\sigma ,\varkappa \right)-\mathcal{S}\left(\sigma ,\varkappa \right)=\left|\sigma -\varkappa \right|,$$

is an $\mathcal{F}$-metric by choosing $\xi \left(t\right)=lnt,$ for $t>0$ and $\chi =ln3.$ It is trivial to show that ξ satisfies conditions (${\mathcal{F}}_1$)–(${\mathcal{F}}_2$). Consequently, $\left(\mathcal{W},\mathcal{R},\mathcal{S}\right)$ is a perturbed $\mathcal{F}$-MS. Since $\mathcal{W}$ is a finite set, any perturbed $\mathcal{F}$-MS defined on it is automatically complete. Define $\mathcal{T}:$ $\mathcal{W}$ $\to \mathcal{W}$ by

$$\mathcal{T}\sigma =\left\{\begin{array}{c}1,\mathit{if}\sigma \in \{0,1,3\},\\ 2,\mathit{if}\sigma =2,\end{array}\right.$$

for all $\sigma ,\varkappa \in \mathcal{W}.$ Here $Fix\left(\mathcal{T}\right)=\{1,2\}.$ Consider

$$\mathcal{W}\setminus Fix\left(\mathcal{T}\right)=\{0,3\}.$$

Now

$$\mathcal{T}0=1\mathit{and}\mathcal{T}3=1,$$

and

$$\mathcal{R}\left(\mathcal{T}0,\mathcal{T}3\right)=\mathcal{R}(1,1)=\left|1-1\right|+1=1.$$

Next,

$$\mathcal{R}\left(0,\mathcal{T}0\right)=\mathcal{R}(0,1)=\left|0-1\right|+1=2$$

$$\mathcal{R}\left(3,\mathcal{T}3\right)=\mathcal{R}(3,1)=\left|3-1\right|+1=3.$$

Choose, $\beta ={\displaystyle \frac{1}{2}}.$ Then

$$\begin{array}{c}\hfill \mathcal{R}\left(\mathcal{T}0,\mathcal{T}3\right)=1<1.225\approx {\displaystyle \frac{1}{2}}\left(2.45\right)\approx {\displaystyle \frac{1}{2}}\left(\sqrt{6}\right)\approx {\displaystyle \frac{1}{2}}\left(2^{{\displaystyle \frac{1}{2}}}·3^{{\displaystyle \frac{1}{2}}}\right)\\ \hfill =\phi {\left[\mathcal{R}\left(0,\mathcal{T}0\right)\right]}^{\beta }{\left[\mathcal{R}\left(3,\mathcal{T}3\right)\right]}^{1-\beta },\end{array}$$

where $\phi ={\displaystyle \frac{1}{2}}\in (0,1).$ Hence all the conditions of Theorem 2 are satisfied, and $\mathcal{T}$ has two FPs that are 1 and 2.

4. Fixed Point Theorems in Generalized Metric Structures

4.1. Fixed Point Results in $\mathcal{F}$-Metric Spaces

In this subsection, we establish FP results in $\mathcal{F}$-MSs, which arise as a particular case of the newly introduced perturbed $\mathcal{F}$-MSs. We first derive a key result that extends and refines the main theorem of Jleli et al. [8], serving as a foundational step for our subsequent developments in this framework.

Corollary 1 

([8]). Let $\left(\mathcal{W},d_{\mathcal{F}}\right)$ be a complete $\mathcal{F}$-MS, and let $\mathcal{T}:\mathcal{W}\to \mathcal{W}$ be a self-mapping. Consider the existence of $\phi \in (0,1)$ such that

$$d_{\mathcal{F}}(\mathcal{T}\sigma ,\mathcal{T}\varkappa )\le \phi d_{\mathcal{F}}(\sigma ,\varkappa ),$$

(12)

for all $\sigma ,\varkappa \in \mathcal{W}$; then $\mathcal{T}$ has a unique FP in $\mathcal{W}.$

Proof. 

Let us put $\mathcal{R}=d_{\mathcal{F}}$ and $\mathcal{S}=0$. Then the triple $(\mathcal{W},\mathcal{R},\mathcal{S})$ reduces to an $\mathcal{F}$-MS, and hence it is trivially a perturbed $\mathcal{F}$-MS. Therefore, all the hypotheses of Theorem 1 are satisfied, and the result follows directly.    □

Corollary 2. 

Let $\left(\mathcal{W},d_{\mathcal{F}}\right)$ be a complete $\mathcal{F}$-MS, and let $\mathcal{T}:$ $\mathcal{W}$ $\to \mathcal{W}$ be a self-mapping. Suppose that there exists a constant $\phi \in [0,1)$ such that

$$d_{\mathcal{F}}\left(\mathcal{T}\sigma ,\mathcal{T}\varkappa \right)\le \phi \mathcal{M}\left(\sigma ,\varkappa \right),$$

where

$$\mathcal{M}\left(\sigma ,\varkappa \right)=max\left\{d_{\mathcal{F}}\left(\sigma ,\varkappa \right),{\displaystyle \frac{d_{\mathcal{F}}\left(\sigma ,\mathcal{T}\sigma \right)d_{\mathcal{F}}\left(\varkappa ,\mathcal{T}\varkappa \right)}{1+d_{\mathcal{F}}\left(\sigma ,\varkappa \right)}}\right\},$$

for all $\sigma ,\varkappa \in \mathcal{W}$, then the mapping $\mathcal{T}$ admits a unique FP ${\sigma }^{*}$ in $\mathcal{W}$.

Proof. 

Take $\mathcal{R}=d_{\mathcal{F}}$ and $\mathcal{S}=0$ in Theorem 2.    □

Corollary 3. 

Let $\left(\mathcal{W},d_{\mathcal{F}}\right)$ be a complete $\mathcal{F}$-MS and let $\mathcal{T}:\mathcal{W}\to \mathcal{W}$ be a self mapping. Suppose the existence of $\phi \in (0,1)$ and $\beta \in (0,1)$ such that

$$d_{\mathcal{F}}(\mathcal{T}\sigma ,\mathcal{T}\varkappa )\le \phi {\left[d_{\mathcal{F}}(\sigma ,\mathcal{T}\sigma )\right]}^{\beta }{\left[d_{\mathcal{F}}(\varkappa ,\mathcal{T}\varkappa )\right]}^{1-\beta },$$

for all $\sigma ,\varkappa \in \mathcal{W}$ with $\sigma ,\varkappa \notin Fix\left(\mathcal{T}\right)$; then $\mathcal{T}$ has an FP in $\mathcal{W}.$

Proof. 

Take $\mathcal{R}=d_{\mathcal{F}}$ and $\mathcal{S}=0$ in Theorem 3.    □

4.2. Fixed Point Results in Perturbed Metric Spaces

This subsection is devoted to establishing FP results in the setting of perturbed MSs, which arise as a special case of the more general framework of perturbed $\mathcal{F}$-MSs. In this direction, we derive a principal result in the spirit of Jleli et al. [11], thereby showing that their theorem can be recovered as a natural consequence of our approach.

Corollary 4 

([11]). Let $\left(\mathcal{W},\mathcal{R},\mathcal{S}\right)$ be a complete perturbed MS, and let $\mathcal{T}:$ $\mathcal{W}$ $\to \mathcal{W}$ be a self-mapping. Assume that there exists a constant $\phi \in [0,1)$ such that

$$\mathcal{R}\left(\mathcal{T}\sigma ,\mathcal{T}\varkappa \right)\le \phi \mathcal{R}\left(\sigma ,\varkappa \right),$$

for all $\sigma ,\varkappa \in \mathcal{W}.$ Then $\mathcal{T}$ admits a unique FP.

Proof. 

Take $\xi \left(t\right)=lnt,$ for $t>0$ and $\chi =0$ in Theorem 1.    □

Corollary 5. 

Let $\left(\mathcal{W},\mathcal{R},\mathcal{S}\right)$ be a complete perturbed MS, and let $\mathcal{T}:$$\mathcal{W}$$\to \mathcal{W}$ be a self-mapping. Suppose that there exists a constant $\phi \in [0,1)$ such that

$$\mathcal{R}\left(\mathcal{T}\sigma ,\mathcal{T}\varkappa \right)\le \phi \mathcal{M}\left(\sigma ,\varkappa \right),$$

where

$$\mathcal{M}\left(\sigma ,\varkappa \right)=max\left\{\mathcal{R}\left(\sigma ,\varkappa \right),{\displaystyle \frac{\mathcal{R}\left(\sigma ,\mathcal{T}\sigma \right)\mathcal{R}\left(\varkappa ,\mathcal{T}\varkappa \right)}{1+\mathcal{R}\left(\sigma ,\varkappa \right)}}\right\},$$

for all $\sigma ,\varkappa \in \mathcal{W}$; then the mapping $\mathcal{T}$ admits a unique FP ${\sigma }^{*}$ in $\mathcal{W}$.

Proof. 

Take $\xi \left(t\right)=lnt,$ for $t>0$ and $\chi =0$ in Theorem 2.    □

Consequently, our result reduces to the principal result established by Ughade et al. [13] in the setting of perturbed MSs.

Corollary 6 

([13]). Let $\left(\mathcal{W},\mathcal{R},\mathcal{S}\right)$ be a complete perturbed MS, and let $\mathcal{T}:\mathcal{W}\to \mathcal{W}$. Consider the existence of $\phi \in (0,1)$ and $\beta \in (0,1)$ such that

$$\mathcal{R}(\mathcal{T}\sigma ,\mathcal{T}\varkappa )\le \phi {\left[\mathcal{R}(\sigma ,\mathcal{T}\sigma )\right]}^{\beta }{\left[\mathcal{R}(\varkappa ,\mathcal{T}\varkappa )\right]}^{1-\beta },$$

for all $\sigma ,\varkappa \in \mathcal{W}$ with $\sigma ,\varkappa \notin Fix\left(\mathcal{T}\right)$; then $\mathcal{T}$ has an FP in $\mathcal{W}.$

Proof. 

Take $\xi \left(t\right)=lnt,$ for $t>0$ and $\chi =0$ in Theorem 3.    □

5. Existence of Solutions for Caputo Fractional Initial Value Problems via Fixed Point Theory in Perturbed $\mathcal{F}$-Metric Spaces

Fractional differential equations, especially those defined through the Caputo derivative, are essential in describing systems that exhibit memory and hereditary behavior in fields such as physics, biology, medicine, engineering, economics, and materials science. The intrinsic nonlocal character of fractional operators, however, makes the determination of exact solutions highly difficult and, in many cases, unattainable. To address this challenge, FP theory serves as an effective and structured approach to establish solutions of such equations.

By transforming a Caputo fractional initial value problem into an associated Volterra-type integral equation, one can utilize FP theorems within appropriate functional settings, including perturbed $\mathcal{F}$-MSs. This reformulation enables the establishment of a unique continuous solution without relying on explicit expressions. Moreover, it offers a solid analytical framework for studying nonlinear and nonlocal phenomena that naturally arise in fractional-order systems.

We examine the following Caputo fractional initial value problem involving the Caputo derivative

$$\begin{array}{c}\hfill {}^{C}D_{\alpha }\sigma \left(t\right)=f(t,\sigma \left(t\right)),t\in \left[0,\mathsf{\Lambda }\right]\\ \hfill \sigma \left(0\right)={\sigma }_0,\end{array}$$

(13)

where $0<\alpha <1.$ This problem can be equivalently rewritten as the Volterra-type integral equation

$$\sigma \left(t\right)={\sigma }_0+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\displaystyle \underset{0}{\overset{t}{\int }}}{\left(t-s\right)}^{\alpha -1}f\left(s,\sigma \left(s\right)\right)ds,$$

where

• $\sigma :[0,\mathsf{\Lambda }]\to \mathbb{R}$ represents the unknown continuous function to be determined.
• $f:[0,\mathsf{\Lambda }]\to \mathbb{R}$ is a prescribed continuous function describing the system input or nonlinear term.
• $\mathsf{\Gamma }\left(\alpha \right)$ denotes the Gamma function.

Let $\mathcal{W}=\mathcal{C}\left([0,\mathsf{\Lambda }],\mathbb{R}\right)$ denote the Banach space of all real-valued continuous functions defined on the interval $[0,\mathsf{\Lambda }],$ equipped with the supremum norm

$${∥\sigma ∥}_{\infty }=\underset{t\in [0,\mathsf{\Lambda }]}{sup}\left|\sigma \left(t\right)\right|.$$

Define the mappings $\mathcal{R},\mathcal{S}:\mathcal{W}\times \mathcal{W}\to \mathbb{R}$ by

$$\mathcal{R}\left(\sigma ,\varkappa \right)=\underset{t\in [0,\mathsf{\Lambda }]}{sup}\left|\sigma \left(t\right)-\varkappa \left(t\right)\right|,$$

and

$$\mathcal{S}\left(\sigma ,\varkappa \right)=\lambda \underset{t\in [0,\mathsf{\Lambda }]}{sup}\left|\sigma \left(t\right)-\varkappa \left(t\right)\right|,$$

for all $\sigma ,\varkappa \in \mathcal{W},$ where $\lambda \in [0,1).$ Consequently, the associated perturbed $\mathcal{F}$-metric is given by

$$d_{\mathcal{F}}\left(\sigma ,\varkappa \right)=\left(1-\lambda \right)\underset{t\in [0,\mathsf{\Lambda }]}{sup}\left|\sigma \left(t\right)-\varkappa \left(t\right)\right|.$$

With these definitions, the triple $(\mathcal{W},\mathcal{R},\mathcal{S})$ forms a complete perturbed $\mathcal{F}$-MS. Moreover, this structure is generated by the function $\xi \left(t\right)=lnt,$ for $t>0$, with the constant $\chi =ln3$.

Theorem 4. 

Let $(\mathcal{W},\mathcal{R},\mathcal{S})$ be a complete perturbed $\mathcal{F}$-MS associated with the $\mathcal{F}$-metric $d_{\mathcal{F}}$ given above, and a continuous mapping $f:[0,\mathsf{\Lambda }]\times \mathbb{R}\to \mathbb{R}$ fulfills

$$\left|f\left(s,\sigma \right)-f\left(s,\varkappa \right)\right|\le \eta \left|\sigma -\varkappa \right|,$$

for all $s\in$$[0,\mathsf{\Lambda }]$ and $\sigma ,\varkappa \in \mathbb{R},$ where $\eta >0$.

Define $\mathcal{T}:$$\mathcal{W}$$\to \mathcal{W}$ by

$$\left(\mathcal{T}\sigma \right)\left(t\right)={\sigma }_0+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\displaystyle \underset{0}{\overset{t}{\int }}}{\left(t-s\right)}^{\alpha -1}f\left(s,\sigma \left(s\right)\right)ds,t\in [0,\mathsf{\Lambda }].$$

Assume that the constant

$$\phi =\left({\displaystyle \frac{\eta {\mathsf{\Lambda }}^{\alpha }}{\mathsf{\Gamma }\left(\alpha \right)}}\right)$$

satisfies $\phi <1.$ Then $\mathcal{T}$ has a unique FP ${\sigma }^{*}\in \mathcal{W}.$ Moreover, this FP represents the unique continuous solution of the corresponding fractional initial value problem (13).

Proof. 

Let $\sigma ,\varkappa \in \mathcal{W}$. Then, for each $t\in [0,\mathsf{\Lambda }],$ we have

$$\left(\mathcal{T}\sigma \right)\left(t\right)-\left(\mathcal{T}\varkappa \right)\left(t\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\displaystyle \underset{0}{\overset{t}{\int }}}{\left(t-s\right)}^{\alpha -1}\left(f\left(s,\sigma \left(s\right)\right)-f\left(s,\varkappa \left(s\right)\right)\right)ds.$$

Taking absolute values, it follows that

$$\begin{array}{c}\hfill \left|\left(\mathcal{T}\sigma \right)\left(t\right)-\left(\mathcal{T}\varkappa \right)\left(t\right)\right|\le {\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\displaystyle \underset{0}{\overset{t}{\int }}}{\left(t-s\right)}^{\alpha -1}\left|f\left(s,\sigma \left(s\right)\right)-f\left(s,\varkappa \left(s\right)\right)\right|ds\\ \hfill \le {\displaystyle \frac{\eta }{\mathsf{\Gamma }\left(\alpha \right)}}{\displaystyle \underset{0}{\overset{t}{\int }}}{\left(t-s\right)}^{\alpha -1}\left|\sigma \left(s\right)-\varkappa \left(s\right)\right|ds.\end{array}$$

where the Lipschitz condition on f was used. Since

$$\left|\sigma \left(s\right)-\varkappa \left(s\right)\right|\le {∥\sigma -\varkappa ∥}_{\infty },\mathrm{for}\mathrm{all}s\in [0,T],$$

we obtain

$$\left|\left(\mathcal{T}\sigma \right)\left(t\right)-\left(\mathcal{T}\varkappa \right)\left(t\right)\right|\le {\displaystyle \frac{\eta }{\mathsf{\Gamma }\left(\alpha \right)}}{∥\sigma -\varkappa ∥}_{\infty }{\displaystyle \underset{0}{\overset{t}{\int }}}{\left(t-s\right)}^{\alpha -1}ds.$$

Evaluating the integral

$${\displaystyle \underset{0}{\overset{t}{\int }}}{\left(t-s\right)}^{\alpha -1}ds={\displaystyle \frac{t^{\alpha }}{\alpha }}\le {\displaystyle \frac{{\mathsf{\Lambda }}^{\alpha }}{\alpha }},$$

we arrive at

$$\left|\left(\mathcal{T}\sigma \right)\left(t\right)-\left(\mathcal{T}\varkappa \right)\left(t\right)\right|\le {\displaystyle \frac{\eta {\mathsf{\Lambda }}^{\alpha }}{\alpha \mathsf{\Gamma }\left(\alpha \right)}}{∥\sigma -\varkappa ∥}_{\infty }.$$

Taking the supremum over $t\in [0,\mathsf{\Lambda }],$ it follows that

$${∥\mathcal{T}\sigma -\mathcal{T}\varkappa ∥}_{\infty }\le {\displaystyle \frac{\eta {\mathsf{\Lambda }}^{\alpha }}{\alpha \mathsf{\Gamma }\left(\alpha \right)}}{∥\sigma -\varkappa ∥}_{\infty }.$$

Now

$$\begin{array}{c}\hfill \mathcal{R}\left(\mathcal{T}\sigma ,\mathcal{T}\varkappa \right)={∥\mathcal{T}\sigma -\mathcal{T}\varkappa ∥}_{\infty }\le \left({\displaystyle \frac{\eta {\mathsf{\Lambda }}^{\alpha }}{\alpha \mathsf{\Gamma }\left(\alpha \right)}}\right){∥\sigma -\varkappa ∥}_{\infty }\\ \hfill =\left({\displaystyle \frac{\eta {\mathsf{\Lambda }}^{\alpha }}{\alpha \mathsf{\Gamma }\left(\alpha \right)}}\right)\mathcal{R}\left(\sigma ,\varkappa \right).\end{array}$$

Moreover, $\phi =\left({\displaystyle \frac{\eta {\mathsf{\Lambda }}^{\alpha }}{\alpha \mathsf{\Gamma }\left(\alpha \right)}}\right)<1.$ Then,

$$\mathcal{R}\left(\mathcal{T}\sigma ,\mathcal{T}\varkappa \right)\le \phi \mathcal{R}\left(\sigma ,\varkappa \right).$$

Hence, all the conditions required by Theorem 1 are satisfied for the operator $\mathcal{T}$. Consequently, there exists a unique element ${\sigma }^{*}\in \mathcal{W}$ such that $\mathcal{T}{\sigma }^{*}={\sigma }^{*}$. Moreover, this FP ${\sigma }^{*}$ coincides with the unique continuous solution of the Caputo fractional initial value problem

$${}^{C}D^{\alpha }\sigma \left(t\right)=f(t,\sigma \left(t\right)),t\in \left[0,\mathsf{\Lambda }\right],\sigma \left(0\right)={\sigma }_0.$$

    □

Example 7. 

Consider the following fractional initial value problem involving the Caputo derivative

$$\left\{\begin{array}{c}{}^{C}D^{0.8}\sigma \left(t\right)={\displaystyle \frac{t}{4}}\sigma \left(t\right)+cost,t\in \left[0,2\right]\\ \sigma \left(0\right)=1.\end{array}\right.$$

(14)

Here

• $\alpha =0.8$ and $\mathsf{\Lambda }=2$.
• $f(s,\sigma \left(s\right))={\displaystyle \frac{s}{2}}\sigma +coss.$
• ${\sigma }_0=1.$
• Lipschitz constant $\eta ={\displaystyle \frac{\mathsf{\Lambda }}{4}}={\displaystyle \frac{1}{2}}.$
• $\mathcal{W}=C\left([0,2],\mathbb{R}\right).$

Define $\mathcal{R},\mathcal{S}$ as in the theorem

$$\mathcal{R}\left(\sigma ,\varkappa \right)=\underset{t\in [0,1]}{sup}\left|\sigma \left(t\right)-\varkappa \left(t\right)\right|,$$

and

$$\mathcal{S}\left(\sigma ,\varkappa \right)=\lambda \underset{t\in [0,1]}{sup}\left|\sigma \left(t\right)-\varkappa \left(t\right)\right|,$$

for all $\sigma ,\varkappa \in \mathcal{W},$ where $\lambda \in [0,1).$ Then ($\mathcal{W},\mathcal{R},\mathcal{S}$) is a complete perturbed $\mathcal{F}$-MS with $\xi \left(t\right)=lnt,$ for $t>0$ and $\chi =ln3.$ Now

$$={\displaystyle \frac{\eta {\mathsf{\Lambda }}^{\alpha }}{\alpha \mathsf{\Gamma }\left(\alpha \right)}}=\left({\displaystyle \frac{\frac{1}{2}\times {\left(2\right)}^{0.8}}{0.8\mathsf{\Gamma }\left(0.8\right)}}\right)=\left({\displaystyle \frac{0.8706}{0.9314}}\right)\approx 0.935<1.$$

Consequently, all the required assumptions are fulfilled, and hence ${\sigma }^{*}\in \mathcal{C}\left([0,1],\mathbb{R}\right)$ is the unique continuous solution of the Caputo fractional initial value problem (14).

Example 8. 

Consider the following fractional initial value problem involving the Caputo derivative

$$\left\{\begin{array}{c}{}^{C}D^{0.6}\sigma \left(t\right)={\displaystyle \frac{1}{3}}\sigma \left(t\right)+e^{-t},t\in \left[0,1\right]\\ \sigma \left(0\right)=2.\end{array}\right.$$

(15)

Here

• $\alpha =0.6$ and $\mathsf{\Lambda }=1$.
• $f(s,\sigma )={\displaystyle \frac{1}{3}}\sigma +e^{-s}.$
• $\sigma \left(0\right)=2.$
• $\eta ={\displaystyle \frac{1}{3}}.$
• $\mathcal{W}=C\left([0,1],\mathbb{R}\right).$

Define $\mathcal{R},\mathcal{S}$ as in the theorem

$$\mathcal{R}\left(\sigma ,\varkappa \right)=\underset{t\in [0,1]}{sup}\left|\sigma \left(t\right)-\varkappa \left(t\right)\right|,$$

and

$$\mathcal{S}\left(\sigma ,\varkappa \right)=\lambda \underset{t\in [0,1]}{sup}\left|\sigma \left(t\right)-\varkappa \left(t\right)\right|,$$

for all $\sigma ,\varkappa \in \mathcal{W},$ where $\lambda \in [0,1).$ Then ($\mathcal{W},\mathcal{R},\mathcal{S}$) is a complete perturbed $\mathcal{F}$-MS with $\xi \left(t\right)=lnt,$ for $t>0$ and $\chi =ln3.$ Now compute

$$\phi ={\displaystyle \frac{\eta {\mathsf{\Lambda }}^{\alpha }}{\alpha \mathsf{\Gamma }\left(\alpha \right)}}=\left({\displaystyle \frac{\frac{1}{3}\times {\left(1\right)}^{0.6}}{0.6\mathsf{\Gamma }\left(0.6\right)}}\right)=\left({\displaystyle \frac{0.33}{0.8935}}\right)\approx 0.373<1.$$

It follows that ${\sigma }^{*}\in \mathcal{C}\left([0,1],\mathbb{R}\right)$ gives the unique continuous solution of the Caputo fractional initial value problem (15).

6. Conclusions

This study introduces a new generalization of metric structures, namely perturbed $\mathcal{F}$-MSs, which extend the frameworks of both $\mathcal{F}$-MSs and perturbed MSs. Within this setting, some FP results are established under different types of contractive conditions, including Banach-type, rational-type, and interpolative contractions. These results not only broaden the scope of classical FP theory but also provide a unified approach for handling nonlinear mappings that may not satisfy standard contraction assumptions.

To support the theoretical findings, a number of carefully constructed and nontrivial examples are presented, illustrating the applicability and effectiveness of the proposed results. In addition, various corollaries and consequences are derived, demonstrating how the obtained theorems generalize and extend several known results available in the existing literature on $\mathcal{F}$-MSs and perturbed MSs. Furthermore, the practical relevance of the developed theory is highlighted through its application to a Caputo fractional initial value problem. By employing the established FP results, the existence of solutions to the considered problem is ensured, thereby confirming the usefulness of the proposed framework in addressing problems arising in fractional differential equations.

7. Future Work

The framework developed in this study opens several promising directions for further investigation. One natural extension is the introduction and analysis of more refined structures, such as orthogonal perturbed $\mathcal{F}$-MSs and perturbed extended $\mathcal{F}$-MSs. These generalized settings are expected to provide a richer geometric structure and may lead to stronger and more flexible FP results.

Another potential direction is the study of different classes of contractive conditions in the perturbed $\mathcal{F}$-metric setting. In particular, it would be of interest to establish FP results for cyclic contractions, graphic contractions, and recently introduced polynomial-type contractions. Such generalizations could significantly broaden the applicability of the theory and allow it to capture a wider class of nonlinear operators.

Furthermore, the extension of the present results to multivalued mappings constitutes an important line of research. Investigating the existence of FPs and best proximity points for set-valued operators in perturbed $\mathcal{F}$-MSs may yield new insights, especially in optimization and equilibrium problems. Similarly, the incorporation of fuzzy mappings into this framework could provide useful tools for handling uncertainty and imprecision in applied problems.

In addition, the applicability of perturbed $\mathcal{F}$-MSs can be explored in greater depth by studying various classes of differential and integral equations. Future work may focus on the solvability of nonlinear ordinary differential equations, fractional differential equations, and Volterra–Fredholm integral equations using the proposed approach. These investigations would further demonstrate the effectiveness of the developed theory in modeling real-world phenomena.

Overall, the extension of perturbed $\mathcal{F}$-MSs to more generalized settings, along with the development of new contractive conditions and applications, offers a wide scope for future research and may lead to significant advancements in FP theory and its applications.