Exploring Fixed-Point Results Using Random Sehgal Contraction in Symmetric Random Cone Metric Spaces with Applications

Abstract

This paper introduces a new concept of random Sehgal contraction in the setting of random cone metric spaces. We explore the modern advancements of traditional fixed-point theorems in a random setting, elaborating on the Sehgal–Guseman fixed-point theorem within the realm of random cone metric spaces. A significant aspect of our research is the interplay between symmetry and randomness; while symmetry provides a framework for understanding structural properties, randomness introduces complexity, which can lead to unexpected behaviors. Our research provides a deeper understanding of the classical results and incorporates a detailed example to illustrate our findings. In addition, major random fixed-point results are also established, which could be applied to nonlinear random fractional differential equations (FDEs) and integral equations as well as to random boundary value problems (BVPs) related to homogeneous random transverse bars.

1. Introduction

The Banach contraction principle (BCP), introduced in 1922 by esteemed mathematician Stefan Banach [1], stands as a cornerstone in analysis, serving as a fixed-point theorem across many fields. Specific contractive conditions apply to a mapping within a complete metric space, which can be easily assessed in various contexts. The versatility and applicability of this principle have been demonstrated by numerous generalizations over the years, providing insights into the convergence rate toward the fixed point as well as establishing the existence of fixed points. The BCP proves the existence of fixed points by a sequence of approximations. According to Bryant [2], the BCP was interpreted in a broader way in 1968. Self-mappings, even if they are not strictly contractions, could still meet the contraction criteria for the $n^{th}$ power, thereby establishing that they have fixed points.

Sehgal contractions are generalizations of the BCP. They were first proposed by Sehgal [3] in 1969. Based on Bryant’s idea, Sehgal created a contraction in which each point g in the underlying space is represented by a natural number $n\left(g\right)$. This contraction satisfies the contraction criterion and yields the fixed point for the self-mapping, being characterized by the power of the self-mapping. Fixed-point techniques for cone metric spaces (CMSs) were provided by Raja and Vaezpour [4].

In 2007, Huang and Zhang [5] presented CMSs, which enhance conventional metric spaces through the incorporation of a cone structure. They established fixed-point theorems for contractive mappings within this novel framework. These CMSs have spurred various extensions of fixed-point theorems, including those by Kannan and Chatterjea. More recently, Du [6] introduced a more generalized framework known as the topological vector space and CMS, which remains an active area of research in fixed-point theory. Subsequent research by authors such as Abbas [7], Ilic [8], Rezapour [9], and Vetro [10] further explored the existence of fixed points which meet contractive-type conditions in normal CMSs. Fixed-point results for multi-valued mappings in cone metric spaces were obtained in [11]. More related applications in this direction can be seen in [12,13,14].

Probabilistic functional analysis is a rapidly evolving field in mathematical research, fueled by its practical applications in probabilistic models. Proficiency in random operator theory becomes imperative when tackling various forms of random equations. Indeed, many mathematical models utilized in physics, engineering, and the biological sciences involve unspecified parameters or coefficients of particular significance. Therefore, it is more practical to view these equations as random operator equations. Solving these equations numerically presents considerably more difficulties compared with deterministic equations. Among others, the authors of [15,16] made significant contributions to understanding the mathematical properties of random equations.

Random nonlinear analysis constitutes a vital field of mathematics essential for investigating various types of random equations, predominantly focusing on random nonlinear operators and their characteristics. The exploration of random fixed-point theory traces back to the 1950s with the emergence of the Prague School of Probabilities [17,18,19]. By extending classical common fixed-point theorems to stochastic settings, one can derive common random fixed-point theorems. The application of random fixed-point theory proves particularly relevant in addressing numerous challenges arising in economic theory (as exemplified in, for instance, [17]) and its associated research.

The survey paper authored by Bharucha-Reid [20] ignited substantial interest among mathematicians, infusing fresh energy into the theory. Itoh [21] expanded the theorem established by Spacek and Hans to multi-valued contraction mappings (also referenced in [22,23,24,25]). Recently, there has been a surge in research focusing on applying concepts from random fixed-point theory to nonlinear random systems. This emerging area has seen significant developments, with studies exploring various aspects and implications (refer to [26,27,28,29]).

Beg [30,31] and Papageorgiou [32,33] introduced fixed-point theorems for contractive random operators by leveraging common random fixed points of pairs of compatible random operators in Polish spaces. This development was centered around utilizing correlations between these operators to establish the fixed-point results.

Scientists have been able to simulate situations with uncertainty and randomness by extending the Banach contraction principle to include randomness. Nonetheless, there are still many problems for which a random operator cannot be made to converge and stabilize. Improving the random BCP to encourage faster convergence of the sequence toward the solution is one way to tackle this problem.

Rezapour and Hamlbarani [9] made a significant advance in fixed-point theory for cone metric spaces in 2008 by doing away with the need for normalcy. The idea of a random cone metric space (RCMS) was first presented by Mehta et al. [34], who proved that a random fixed point exists in weak contraction scenarios.

Building on Ri’s research, Aydi et al. [35] extensively analyzed Ri’s findings, specifically focusing on exploring random solutions for the nonlinear Caputo-type FDE. Meanwhile, during the same timeframe, Baleanu et al. [36] directed their attention toward confirming the existence of a solution for the FDE. Moreover, their research extended to exploration of the dynamic behavior of a bead in random motion along a wire [37].

The motivation for this research arises from the need to expand fixed-point theory by exploring random Sehgal contractions in RCMSs. This study builds on the foundational Sehgal–Guseman fixed-point theorem ([38], Theorem 1) by extending it to encompass random operators and contractions. It provides a modern perspective which extends beyond traditional approaches like Sehgal and Banach contractions in metric spaces. The exploration of random Sehgal contractions strengthens the theoretical basis of fixed-point theorems. Additionally, it reveals practical applications for addressing nonlinear random FDEs and random BVPs. A detailed example highlights the accelerated convergence properties of random Sehgal contractions, emphasizing their effectiveness in solving complex mathematical problems involving uncertainty and randomness. By establishing significant random fixed-point results and showcasing the utility of random Sehgal contractions in nonlinear systems, this research advances the field. It offers fresh insights and solutions which extend beyond conventional contraction principles in metric spaces.

The strength of the current work on random Sehgal contractions in RCMSs lies in its ability to handle uncertainty and randomness more effectively than traditional Sehgal and Banach contractions in metric spaces. By extending the contraction principles to random operators, this approach demonstrates accelerated convergence and unique solution capabilities, particularly in solving nonlinear random FDEs and random BVPs. The adaptability of random Sehgal contractions to stochastic environments enhances their performance, making them a powerful tool in modern fixed-point theory for addressing complex mathematical problems with varying degrees of uncertainty.

This paper extends the Sehgal–Guseman-type theorems [38] to a random operator which satisfies a random contraction condition for a particular natural number $n\in \mathbb{N}$. It is within this framework that we establish the existence of a random fixed point for a given random operator. An alternative approach to identifying random fixed points is presented here by introducing the concept of random Sehgal contraction. Although the random operator may not adhere to a random BCP, there exists a corresponding $n\left(g\right)$ which fulfills the contraction condition. By examining a nontrivial case, we illustrate the effectiveness of random Sehgal contraction, which reveals accelerated convergence. In this investigation, we establish significant results regarding random fixed points, employing them to address nonlinear FDEs involving random fractions. Additionally, we utilize these findings to tackle random BVPs linked with homogeneous, random transverse bars.

2. Some Basic Results and Definitions

In this section, we present definitions, notation, and preliminary results concerning random variables. These foundational elements will be employed consistently throughout our investigation.

Definition 1

([34]). Consider a topological vector space E. A subset $\mathcal{P}\subset E$ is said to be a cone if it fulfills the following criteria:

(C1) 

$\mathcal{P}$ is a nonempty closed set with $\mathcal{P}\ne \left\{0\right\}$;

(C2) 

For any $p,q\in \mathcal{P}$ and any non-negative scalars $a,b\in \mathbb{R}$, $ap+bq\in \mathcal{P}$;

(C3) 

If $p\in \mathcal{P}$ and $-p\in \mathcal{P}$, then $p=0$.

For a given cone $\mathcal{P}$ in E, one defines a partial ordering ${\le }_{\mathcal{P}}$ on E as follows:

$p{\le }_{\mathcal{P}}q$ if and only if $q-p\in \mathcal{P}$, and $p{\ll }_{\mathcal{P}}q$ if and only if $q-p\in \mathrm{Int}\left(\mathcal{P}\right)$, where $\mathrm{Int}\left(\mathcal{P}\right)$ denotes the interior of $\mathcal{P}$.

Definition 2

([5]). Let $\mathcal{P}$ be a cone in a topological vector space E. Consider a nonempty set Q and a mapping $d:Q\times Q\to E$ satisfying the following conditions:

(1) 

For any $x,y\in Q$, $d(x,y){\ge }_{\mathcal{P}}0$, and $d(x,y)=0$ if and only if $x=y$;

(2) 

For any $x,y\in Q$, $d(x,y)=d(y,x)$;

(3) 

For any $x,y,z\in Q$, $d(x,y){\le }_{\mathcal{P}}d(x,z)+d(z,y)$.

The pair $(Q,d)$ is called a cone metric space (in relation to a cone $\mathcal{P}$ in E).

Throughout this paper, we suppose that the cone $\mathcal{P}\subset E$ is such that $\mathrm{Int}\left(\mathcal{P}\right)\ne \varnothing$ and use the notation ≤ instead of ${\le }_{\mathcal{P}}$ and ≪ instead of ${\ll }_{\mathcal{P}}$.

Let $(Q,d)$ be a cone metric space. A sequence $\left\{p_n\right\}$ in Q has the following properties:

(1)

It is convergent to $p_0\in Q$ if for every $\epsilon \in E$ such that $0\ll \epsilon$, there is $n_0\in \mathbb{N}$ such that $d(p_n,p_0)\ll \epsilon$ for every $n\ge n_0$;

(2)

There is a Cauchy sequence if for every $\epsilon \in E$ with $0\ll \epsilon$, there is $n_0\in \mathbb{N}$ such that $d(p_n,p_m)\ll \epsilon$ for every $n,m\ge n_0$.

(3)

$(Q,d)$ is complete if every Cauchy sequence in Q converges.

The Bryant fixed-point theorem, initially introduced by Bryant in 1968 and referenced in [2], serves as a notable extension of the BCP.

Theorem 1

([2]). Let f be a self-mapping of a complete metric space $(M,d)$. If the iterated mapping $f^n$ demonstrates contraction-like behavior for a specific $n\in \mathbb{N}$, then the set $\mathrm{Fix}\left(f\right)$ contains exactly one element.

It is worth noting that although $f^n$ is continuous, this does not automatically ensure the continuity of f itself.

In 1969, Sehgal ([3], Theorem) made a significant contribution by unveiling a novel result regarding the contractive iteration of every point within complete metric spaces. This discovery extended the principles of Banach’s contraction.

Theorem 2

([3]). Let $(M,d)$ be a complete metric space, $f:M\to M$ be a continuous self-mapping of M, and $k\in [0,1)$. If for each $x\in M$ there exists $n\left(x\right)\in \mathbb{N}$ such that

$$\begin{array}{c}\hfill d(f^{n\left(x\right)}\left(x\right),f^{n\left(x\right)}\left(y\right))\le kd(x,y)\end{array}$$

for all $y\in M$, then the set $\mathrm{Fix}\left(f\right)$ contains exactly one element.

Random Variable

In this subsection, $(\mathcal{M},\mathsf{\Sigma })$ will be a measurable space, where $\mathsf{\Sigma }$ is a $\sigma$-algebra on $\mathcal{M}$, M will be a nonempty subset of a cone metric space $(Q,d)$ (related to a cone $\mathcal{P}$), and $2^M$ will denote the set of all nonempty closed subsets of M.

Definition 3

([5]). A mapping $F:\mathcal{M}\to 2^M$ is measurable if for each open subset U of M, the inverse image $F^{\leftarrow }\left(U\right)\in \mathsf{\Sigma }$, where $F^{\leftarrow }\left(U\right)=\{\mathfrak{m}\in \mathcal{M}:F\left(\mathfrak{m}\right)\cap U\ne \varnothing \}$.

Definition 4

([34]). A measurable mapping $\xi :\mathcal{M}\to M$ is called a measurable selector of a measurable mapping $F:\mathcal{M}\to 2^M$ if for each $\mathfrak{m}\in \mathcal{M}$, $\xi \left(\mathfrak{m}\right)$ belongs to $F\left(\mathfrak{m}\right)$.

Definition 5

([34]). A mapping $f:\mathcal{M}\times M\to Q$ is a random operator (continuous random operator) if for any fixed $q\in M$, the mapping $f(.,q):\mathcal{M}\to Q$ is measurable (continuous).

Definition 6

([34]). A measurable mapping $\xi :\mathcal{M}\to M$ is said to be a random fixed point of a random operator $f:\mathcal{M}\times M\to Q$ if $f(\mathfrak{m},\xi (\mathfrak{m}\left)\right)=\xi \left(\mathfrak{m}\right)$ holds for every $\mathfrak{m}\in \mathcal{M}$.

The concepts of symmetry and random fixed points have been explored through various mathematical frameworks, particularly in symmetric spaces. Symmetric spaces provide a structured environment where fixed-point theorems can be applied under certain contraction conditions and offer insights into the distribution of fixed points in random permutations. These studies reveal the intricate relationship between symmetry and fixed points, highlighting both theoretical and practical implications.

Definition 7

([34]). Consider a mapping $\rho :\mathcal{M}\times M\to \mathcal{P}$ which fulfills the following criteria for a selector $\mathfrak{m}\in \mathcal{M}$:

(1) 

$\rho \left(p\right(\mathfrak{m}),q(\mathfrak{m}\left)\right)$ is nonnegative, and it equals zero if and only if $p\left(\mathfrak{m}\right)=q\left(\mathfrak{m}\right)$ for all $p\left(\mathfrak{m}\right),q\left(\mathfrak{m}\right)\in \mathcal{M}\times M$;

(2) 

$\rho \left(p\right(\mathfrak{m}),q(\mathfrak{m}\left)\right)=\rho \left(q\right(\mathfrak{m}),p(\mathfrak{m}\left)\right)$ for all $p\left(\mathfrak{m}\right),q\left(\mathfrak{m}\right)\in M$, $\mathfrak{m}\in \mathcal{M}$, and $p\left(\mathfrak{m}\right),q\left(\mathfrak{m}\right)\in \mathcal{M}\times M$;

(3) 

$\rho \left(p\right(\mathfrak{m}),q(\mathfrak{m}\left)\right)\le \rho \left(p\right(\mathfrak{m}),u(\mathfrak{m}\left)\right)+\rho \left(u\right(\mathfrak{m}),q(\mathfrak{m}\left)\right)$ for all $p\left(\mathfrak{m}\right),q\left(\mathfrak{m}\right),u\left(\mathfrak{m}\right)\in M$, and $\mathfrak{m}\in \mathcal{M}$ as a selector.

(4) 

The function $\rho \left(p\right(\mathfrak{m}),q(\mathfrak{m}\left)\right)$ is non-increasing and left-continuous for any $p\left(\mathfrak{m}\right),q\left(\mathfrak{m}\right)\in M$, and $\mathfrak{m}\in \mathcal{M}$.

The mapping ρ is called a random cone metric, and the pair $(M,\rho )$ is called a random cone metric space (shortly RCMS).

3. Main Results

This section presents the main results, followed by the necessary definitions and concepts.

Definition 8.

Consider an RCMS $(Q,d)$ defined with respect to a cone $\mathcal{P}$. Let M be a nonempty, separable closed subset of Q, and define a continuous random operator $f:\mathcal{M}\times Q⟶Q$ as a random Sehgal contraction under the condition that for each $\mathfrak{m}\in \mathcal{M}$, there exists $k\left(\mathfrak{m}\right)\in [0,1)$ such that

$$\begin{array}{c}\hfill d(f^{n\left(x\right(\mathfrak{m}\left)\right)}\left(x\left(\mathfrak{m}\right)\right),f^{n\left(x\right(\mathfrak{m}\left)\right)}\left(y\left(\mathfrak{m}\right)\right))\le k\left(\mathfrak{m}\right)d(x\left(\mathfrak{m}\right),y\left(\mathfrak{m}\right));x\left(\mathfrak{m}\right),y\left(\mathfrak{m}\right)\in Q.\end{array}$$

(1)

Lemma 1.

Consider a complete RCMS denoted by $(M,\rho )$ associated with a cone $\mathcal{P}$, where $f:\mathcal{M}\times M⟶Q$ is a continuous random operator, and $\beta \in [0,1)$. For each $x\left(\mathfrak{m}\right)\in Q$, there exists $n\left(x\right(\mathfrak{m})\in \mathbb{N}$ such that

$$\begin{array}{c}\hfill d(f^{n\left(x\right(\mathfrak{m}\left)\right)}\left(x\left(\mathfrak{m}\right)\right),f^{n\left(x\right(\mathfrak{m}\left)\right)}y\left(\mathfrak{m}\right))\le k\left(\mathfrak{m}\right)d(x\left(\mathfrak{m}\right),y\left(\mathfrak{m}\right));\forall y\left(\mathfrak{m}\right)\in Q.\end{array}$$

Then,

$$\begin{array}{c}\hfill r\left(x\left(\mathfrak{m}\right)\right):=\underset{n}{sup}d(f^n\left(x\left(\mathfrak{m}\right)\right),x\left(\mathfrak{m}\right));\forall x\left(\mathfrak{m}\right)\in Q\end{array}$$

is finite.

Proof. 

Let $p\left(\mathfrak{m}\right)\in Q$, and let

$$\begin{array}{c}\hfill \ell \left(p\left(\mathfrak{m}\right)\right)=max\{\rho (f^k\left(p\left(\mathfrak{m}\right)\right),p\left(\mathfrak{m}\right)):k=1,2,\dots ,n\left(p\left(\mathfrak{m}\right)\right)\}.\end{array}$$

For every positive integer n, there is an integer $s\ge 0$ such that s times the value of n in terms of $p\left(\mathfrak{m}\right)$ is less than n and is less than or equal to $(s+1)$ times the value of n in terms of $p\left(\mathfrak{m}\right)$:

$$\begin{array}{cc}\hfill \rho (f^n\left(p\left(\mathfrak{m}\right)\right),p\left(\left(\mathfrak{m}\right)\right))& \le \rho (f^{n\left(p\right(\mathfrak{m}\left)\right)}·f^{n-n\left(p\right(\mathfrak{m}\left)\right)}\left(p\left(\mathfrak{m}\right)\right),f^{n\left(p\right(\mathfrak{m}\left)\right)}\left(p\left(\mathfrak{m}\right)\right))+\rho (f^{n\left(p\right(\mathfrak{m}\left)\right)}\left(p\left(\mathfrak{m}\right)\right),p\left(\mathfrak{m}\right))\hfill \\ \hfill & \le k\rho (f^{n-n\left(p\right(\mathfrak{m}\left)\right)}\left(p\left(\mathfrak{m}\right)\right),p\left(\mathfrak{m}\right))+\ell \left(p\left(\left(\mathfrak{m}\right)\right)\right)\hfill \\ \hfill & \le \ell \left(p\left(\mathfrak{m}\right)\right)+k\ell \left(p\left(\mathfrak{m}\right)\right)+k^2\ell \left(p\left(\mathfrak{m}\right)\right)+\dots +k^s\ell \left(p\left(\mathfrak{m}\right)\right)\hfill \\ \hfill & \le \ell \left(p\right(\mathfrak{m}\left)\right)/(1-k)\forall n\ge 0.\hfill \end{array}$$

Thus, $r\left(p\left(\mathfrak{m}\right)\right)={\sup }_n\rho (f^n\left(p\left(\mathfrak{m}\right)\right),p\left(\mathfrak{m}\right))$ is finite. □

Now, we will introduce Guseman’s random fixed-point theorem by relaxing the requirement of continuity.

Theorem 3.

Let $(Q,\rho )$ be a complete separable RCMS associated with a cone $\mathcal{P}$. Let $f:\mathcal{M}\times Q\to Q$ represent a random operator for some $\lambda \in [0,1)$. Suppose that for all $p\left(\mathfrak{m}\right)\in Q$, there exists $n\left(p\right(\mathfrak{m}\left)\right)\in \mathbb{N}$ such that

$$\begin{array}{c}\hfill \rho \left(f^{n\left(p\right(\mathfrak{m}\left)\right)}p\left(\mathfrak{m}\right),f^{n\left(p\right(\mathfrak{m}\left)\right)}q\left(\mathfrak{m}\right)\right)\le \lambda \rho (p\left(\mathfrak{m}\right),q\left(\mathfrak{m}\right)),\forall q\left(\mathfrak{m}\right)\in Q.\end{array}$$

(2)

The function f possesses a unique random fixed point $u\left(\mathfrak{m}\right)$, and as the number of iterations tends toward infinity, applying f repeatedly to an initial value $p\left(\mathfrak{m}\right)$ converges to $u\left(\mathfrak{m}\right)$ for all $p\left(\mathfrak{m}\right)\in Q$.

Proof. 

Given $p\left(\mathfrak{m}\right)\in Q$, we define the sequence $\left\{p_n\left(\mathfrak{m}\right)\right\}$ as follows:

$$\begin{array}{cc}\hfill & p_0\left(\mathfrak{m}\right)=p\left(\mathfrak{m}\right),\hfill \\ \hfill & p_1\left(\mathfrak{m}\right)=f^{k\left(p_0\left(\mathfrak{m}\right)\right)}{p}_0\left(\mathfrak{m}\right),\hfill \\ \hfill & p_2\left(\mathfrak{m}\right)=f^{k\left(p_1\left(\mathfrak{m}\right)\right)}{p}_1\left(\mathfrak{m}\right),\hfill \\ \hfill & ⋮\hfill \\ \hfill & p_{n+1}\left(\mathfrak{m}\right)=f^{k\left(p_n\left(\mathfrak{m}\right)\right)}{p}_n\left(\mathfrak{m}\right),\hfill \\ \hfill & ⋮\hfill \end{array}$$

We establish that the sequence $\left\{p_n\left(\mathfrak{m}\right)\right\}$ converges as a Cauchy sequence. Referring to Equation (2), this can be expressed as

$$\begin{array}{cc}\hfill \rho \left(p_{n+1}\left(\mathfrak{m}\right),p_n\left(\mathfrak{m}\right)\right)& =\rho \left(f^{k\left(p_n\left(\mathfrak{m}\right)\right)}{p}_n\left(\mathfrak{m}\right)\right)\hfill \\ \hfill & =\rho \left(f^{k\left(p_{n-1}\left(\mathfrak{m}\right)\right)}{p}_{n-1}\left(\mathfrak{m}\right),f^{k\left(p_{n-1}\left(\mathfrak{m}\right)\right)}{f}^{k\left(p_n\left(\mathfrak{m}\right)\right)}{p}_{n-1}\left(\mathfrak{m}\right)\right)\hfill \\ \hfill & \le \lambda \rho \left(p_{n-1}\left(\mathfrak{m}\right),f^{k\left(p_n\left(\mathfrak{m}\right)\right)}{p}_{n-1}\left(\mathfrak{m}\right)\right).\hfill \\ \hfill & ⋮\hfill \\ \hfill & \le {\lambda }^n\rho \left(p_0\left(\mathfrak{m}\right),f^{k\left(p_n\left(\mathfrak{m}\right)\right)}{p}_0\left(\mathfrak{m}\right)\right).\hfill \end{array}$$

By applying the triangle inequality, we can establish the following relationship for $m>n$:

$$\begin{array}{cc}\hfill \rho \left(p_n\left(\mathfrak{m}\right),p_m\left(\mathfrak{m}\right)\right)& \le \rho \left(p_n\left(\mathfrak{m}\right),p_{n+1}\left(\mathfrak{m}\right)\right)+\rho \left(p_{n+1}\left(\mathfrak{m}\right),p_{n+2}\left(\mathfrak{m}\right)\right)+\cdots +\rho \left(p_{m-1}\left(\mathfrak{m}\right),p_m\left(\mathfrak{m}\right)\right)\hfill \\ \hfill & \le {\lambda }^n\rho \left(p_0\left(\mathfrak{m}\right),f^{k\left(p_n\left(\mathfrak{m}\right)\right)}{p}_0\left(\mathfrak{m}\right)\right)+{\lambda }^{n+1}\rho \left(p_0\left(\mathfrak{m}\right),f^{k\left(p_{n+1}\right)}{p}_0\right)+\cdots +{\lambda }^{m-1}\hfill \\ \hfill & \rho \left(p_0\left(\mathfrak{m}\right),f^{k\left(p_{m-1}\left(\mathfrak{m}\right)\right)}{p}_0\left(\mathfrak{m}\right)\right)\hfill \\ \hfill & ={\lambda }^n\rho \left(p_0\left(\mathfrak{m}\right),f^{k\left(p_n\left(\mathfrak{m}\right)\right)}{p}_0\left(\mathfrak{m}\right)\right)+\hfill \\ \hfill & \lambda \rho \left(p_0\left(\mathfrak{m}\right),f^{k\left(p_{n+1}\left(\mathfrak{m}\right)\right)}{p}_0\left(\mathfrak{m}\right)\right)+\cdots +{\lambda }^{m-n-1}\rho \left(p_0\left(\mathfrak{m}\right),f^{k\left(p_{m-1}\left(\mathfrak{m}\right)\right)}{p}_0\left(\mathfrak{m}\right)\right)\hfill \\ \hfill & \Rightarrow \rho \left(p_n\left(\mathfrak{m}\right),p_m\left(\mathfrak{m}\right)\right)⟶0,\mathrm{as}n⟶+\infty .\hfill \end{array}$$

Based on Lemma 1, it follows that for $m>n$, we have

$$\begin{array}{cc}\hfill \parallel \rho & \left(p_0\left(\mathfrak{m}\right),f^{k\left(p_n\left(\mathfrak{m}\right)\right)}{p}_0\left(\mathfrak{m}\right)\right)+\lambda \rho \left(p_0\left(\mathfrak{m}\right),f^{k\left(p_{n+1}\left(\mathfrak{m}\right)\right)}{p}_0\left(\mathfrak{m}\right)\right)+\cdots +\hfill \\ & {\lambda }^{m-n-1}\rho \left(p_0\left(\mathfrak{m}\right),f^{k\left(p_{m-1}\left(\mathfrak{m}\right)\right)}{p}_0\left(\mathfrak{m}\right)\right)\parallel \hfill \\ \hfill & \le ∥\rho \left(p_0\left(\mathfrak{m}\right),f^{k\left(p_n\right)}{p}_0\left(\mathfrak{m}\right)\right)∥+∥\lambda \rho \left(p_0\left(\mathfrak{m}\right),f^{k\left(p_{n+1}\left(\mathfrak{m}\right)\right)}{p}_0\left(\mathfrak{m}\right)\right)∥+\cdots \hfill \\ \hfill & +∥{\lambda }^{m-n-1}\rho \left(p_0\left(\mathfrak{m}\right),f^{k\left(p_{m-1}\left(\mathfrak{m}\right)\right)}{p}_0\left(\mathfrak{m}\right)\right)∥\hfill \\ \hfill & \le (1+\lambda +\cdots +\lambda )·r\left(p_0\left(\mathfrak{m}\right)\right)\le {\displaystyle \frac{1}{1-\lambda }}r\left(p_0\left(\mathfrak{m}\right)\right)<+\infty .\hfill \end{array}$$

This implies that the distance between $p_n\left(\mathfrak{m}\right)$ and $p_m\left(\mathfrak{m}\right)$ tends toward zero as n approaches infinity, and we establish that $\left\{p_n\left(\mathfrak{m}\right)\right\}$ forms a Cauchy sequence. Through the completeness of $(Q,\rho )$, we conclude that $p_n\left(\mathfrak{m}\right)$ converges to $u\left(\mathfrak{m}\right)$ for some $u\left(\mathfrak{m}\right)\in Q$. Our next goal is to show that $fu\left(\mathfrak{m}\right)=u\left(\mathfrak{m}\right)$.

For any given $u\left(\mathfrak{m}\right)$, there exists a natural number $k\left(u\right(\mathfrak{m}\left)\right)$ such that

$$\begin{array}{c}\hfill \rho \left(f^{k\left(u\right(\mathfrak{m}\left)\right)}u\left(\mathfrak{m}\right),f^{k\left(u\right(\mathfrak{m}\left)\right)}{p}_n\left(\mathfrak{m}\right)\right)\le \lambda \rho \left(u\left(\mathfrak{m}\right),p_n\left(\mathfrak{m}\right)\right)\to 0,\mathrm{as}n\to +\infty .\end{array}$$

Furthermore, $f^{k\left(u\right(\mathfrak{m}\left)\right)}{p}_n\left(\mathfrak{m}\right)\to f^{k\left(u\right(\mathfrak{m}\left)\right)}u\left(\mathfrak{m}\right)$ as $n\to +\infty$.

Therefore, according to [5], It follows that

$$\begin{array}{c}\hfill \rho \left(f^{k\left(u\right(\mathfrak{m}\left)\right)}{p}_n\left(\mathfrak{m}\right),p_n\left(\mathfrak{m}\right)\right)\to \rho \left(f^{k\left(u\right(\mathfrak{m}\left)\right)}u\left(\mathfrak{m}\right),u\left(\mathfrak{m}\right)\right).\end{array}$$

Subsequently, from Equation (2), we can derive

$$\begin{array}{cc}\hfill \rho \left(f^{k\left(u\right)}{p}_n\left(\mathfrak{m}\right),p_n\left(\mathfrak{m}\right)\right)& =\rho \left(f^{k\left(p_{n-1}\left(\mathfrak{m}\right)\right)}{f}^{k\left(u\right)}{p}_{n-1}\left(\mathfrak{m}\right),f^{k\left(p_{n-1}\left(\mathfrak{m}\right)\right)}{p}_{n-1}\left(\mathfrak{m}\right)\right)\hfill \\ \hfill & \le \lambda \rho \left(f^{k\left(u\right)}{p}_{n-1}\left(\mathfrak{m}\right),p_{n-1}\left(\mathfrak{m}\right)\right).\hfill \end{array}$$

By iterating this reasoning n times, we arrive at the following expression:

$$\begin{array}{c}\hfill \rho \left(f^{k\left(u\right(\mathfrak{m}\left)\right)}{p}_n\left(\mathfrak{m}\right),p_n\left(\mathfrak{m}\right)\right)\le {\lambda }^n\rho \left(f^{k\left(u\right(\mathfrak{m}\left)\right)}{p}_0\left(\mathfrak{m}\right),p_0\left(\mathfrak{m}\right)\right).\end{array}$$

Since $∥\rho \left(f^{k\left(u\right(\mathfrak{m}\left)\right)}{p}_0\left(\mathfrak{m}\right),p_0\left(\mathfrak{m}\right)\right)∥\le r\left(p_0\left(\mathfrak{m}\right)\right)$, this suggests that ${\lambda }^n\rho (f^{k\left(u\right(\mathfrak{m}\left)\right)}{p}_0\left(\mathfrak{m}\right),$$p_0\left(\mathfrak{m}\right))$ converges to zero in the RCMS as n tends toward infinity. Thus, $\rho (f^{k\left(u\right(\mathfrak{m}\left)\right)}{p}_n\left(\mathfrak{m}\right),$$p_n\left(\mathfrak{m}\right))\to 0$. Therefore, $\rho \left(f^{k\left(u\right(\mathfrak{m}\left)\right)}u\left(\mathfrak{m}\right),u\left(\mathfrak{m}\right)\right)=0$; that is, $f^{k\left(u\right(\mathfrak{m}\left)\right)}u\left(\mathfrak{m}\right)=u\left(\mathfrak{m}\right)$. Following Equation (2), it can be concluded that $u\left(\mathfrak{m}\right)$ is the unique random fixed point of $f^{k\left(u\right(\mathfrak{m}\left)\right)}$. However, for $fu\left(\mathfrak{m}\right)$, we have the following:

$$\begin{array}{c}\hfill f^{k\left(u\right(\mathfrak{m}\left)\right)}fu\left(\mathfrak{m}\right)=f{f}^{k\left(u\right(\mathfrak{m}\left)\right)}u\left(\mathfrak{m}\right)=fu\left(\mathfrak{m}\right),\end{array}$$

In other words, $fu\left(\mathfrak{m}\right)$ is also the random fixed point of $f^{k\left(u\right(\mathfrak{m}\left)\right)}$. Therefore, $fu\left(\mathfrak{m}\right)=u\left(\mathfrak{m}\right)$.

We now show that ${lim}_{n\to +\infty }{f}^{n}p\left(\mathfrak{m}\right)=u\left(\mathfrak{m}\right)$. When n is large enough, this can be expressed as $n=q^{\prime }·k\left(u\left(\mathfrak{m}\right)\right)+q$, where $q^{\prime }>0$ and $0\le q$:

$$\begin{array}{cc}\hfill \rho \left(u\left(\mathfrak{m}\right),f^{n}p\left(\mathfrak{m}\right)\right)& =\rho \left(f^{k\left(u\right(\mathfrak{m}\left)\right)}u\left(\mathfrak{m}\right),f^{k\left(u\right(\mathfrak{m}\left)\right)}{f}^{k\left(u\right(\mathfrak{m})+q)}p\left(\mathfrak{m}\right)\right)\hfill \\ \hfill & \le \lambda \rho \left(u\left(\mathfrak{m}\right),f^{k\left(u\right(\mathfrak{m})+q)}p\left(\mathfrak{m}\right)\right)\hfill \\ \hfill & \le \cdots \le {\lambda }^{q^{\prime }}\rho \left(u\left(\mathfrak{m}\right),f^{q}p\left(\mathfrak{m}\right)\right)\hfill \\ \hfill & \le {\lambda }^{q^{\prime }}\left(\rho (u\left(\mathfrak{m}\right),p\left(\mathfrak{m}\right))+\rho \left(p\left(\mathfrak{m}\right),f^{q}p\left(\mathfrak{m}\right)\right)\right)\to 0,\mathrm{as}{q}^{\prime }\to +\infty .\hfill \end{array}$$

This demonstrates that

$$\begin{array}{cc}\hfill & ∥\rho (u,p\left(\mathfrak{m}\right))+\rho \left(p,f^{q}p\left(\mathfrak{m}\right)\right)∥\hfill \\ \hfill & \le k·\left(\parallel \rho (u\left(\mathfrak{m}\right),p\left(\mathfrak{m}\right))\parallel +∥\rho \left(p\left(\mathfrak{m}\right),f^{q}p\left(\mathfrak{m}\right)\right)∥\right)\hfill \\ \hfill & <+\infty .\hfill \end{array}$$

Since both n and $q^{\prime }$ tend toward infinity, and $0\le \lambda <1$, we can conclude that

$$\underset{n\to +\infty }{lim}{f}^{n}p\left(\mathfrak{m}\right)=u\left(\mathfrak{m}\right).$$

□

The following remark will help in proving the next theorem.

Remark 1.

(1) 

$u\left(\mathfrak{m}\right)\ll w$ if $u\left(\mathfrak{m}\right)\le v\left(\mathfrak{m}\right)$ and $v\left(\mathfrak{m}\right)\ll w\left(\mathfrak{m}\right)$;

(2) 

$u\left(\mathfrak{m}\right)=0$ if $0\le u\left(\mathfrak{m}\right)\ll \mathfrak{m};\forall \mathfrak{m}\in$ Int($\mathcal{P}$);

(3) 

$a_n\ll \mathfrak{m}$ if $\mathfrak{m}\in \mathrm{Int}(\mathcal{P}$ for all $n_0\in \mathbb{N}$ such that $\forall n_0<n$. Then, it holds that $0\le a_n$ and $a_n\to 0$.

Theorem 4.

For a complete, separable RCMS $(Q,\rho )$, consider $f:\mathcal{M}\times Q\to Q$ to be a continuous random operator. There exists $n\left(p\right(\mathfrak{m}\left)\right)\in \mathbb{N}$ such that for every $p\left(\mathfrak{m}\right)\in Q$ and some $\lambda \in [0,1)$, the inequality

$$\begin{array}{c}\hfill \rho \left(f^{n\left(p\right(\mathfrak{m}\left)\right)}p\left(\mathfrak{m}\right),f^{n\left(p\right(\mathfrak{m}\left)\right)}q\left(\mathfrak{m}\right)\right)\le \lambda \rho (p\left(\mathfrak{m}\right),q\left(\mathfrak{m}\right)),\end{array}$$

(3)

holds for each $q\left(\mathfrak{m}\right)\in Q$. If $O_f(p\left(\mathfrak{m}\right);+\infty )$ forms a bounded random orbit around a random point $p\left(\mathfrak{m}\right)\in Q$, then f possesses a unique random fixed point $u\left(\mathfrak{m}\right)\in Q$.

Proof. 

A randomly selected point, denoted by $p\left(\mathfrak{m}\right)$, moves within a bounded orbit represented by the set Q. Specifically, the orbit defined by $O_f(p\left(\mathfrak{m}\right);+\infty )$ has a finite diameter $\delta$. Therefore, it yields

$$\begin{array}{c}\hfill \underset{0\le i,j<+\infty }{sup}\left\{∥\rho \left(f^{i}p\left(\mathfrak{m}\right),f^{j}p\left(\mathfrak{m}\right)\right)∥\right\}=\delta <+\infty .\end{array}$$

Furthermore, following the proof of Theorem 3, consider the sequence

$$\begin{array}{cc}\hfill & p_0\left(\mathfrak{m}\right)=p\left(\mathfrak{m}\right),p_1\left(\mathfrak{m}\right)=f^{k\left(p_0\left(\mathfrak{m}\right)\right)}{p}_0\left(\mathfrak{m}\right),p_2\left(\mathfrak{m}\right)=f^{k\left(p_1\left(\mathfrak{m}\right)\right)}{p}_1\left(\mathfrak{m}\right),\dots ,\hfill \\ \hfill & p_{n+1}\left(\mathfrak{m}\right)=f^{k\left(p_n\left(\mathfrak{m}\right)\right)}{p}_n\left(\mathfrak{m}\right),\dots \hfill \end{array}$$

Notice that $\rho \left(p_n\left(\mathfrak{m}\right),p_{n+1}\left(\mathfrak{m}\right)\right)\le {\lambda }^n\rho \left(p_0\left(\mathfrak{m}\right),f^{kn\left(pn\left(\mathfrak{m}\right)\right)}{p}_0\left(\mathfrak{m}\right)\right)$, and for $m>n$, we obtain

$$\begin{array}{cc}\hfill \rho \left(p_n\left(\mathfrak{m}\right),p_m\left(\mathfrak{m}\right)\right)& \le {\lambda }^n\rho \left(p_0\left(\mathfrak{m}\right),f^{k\left(p_n\left(\mathfrak{m}\right)\right)}{p}_0\left(\mathfrak{m}\right)\right)+\lambda \rho \left(p_0\left(\mathfrak{m}\right),f^{k\left(p_{n+1}\left(\mathfrak{m}\right)\right)}{p}_0\left(\mathfrak{m}\right)\right)+\cdots \hfill \\ \hfill & +{\lambda }^{m-n-1}\rho \left(p_0\left(\mathfrak{m}\right),f^{k\left(p_{m-1}\left(\mathfrak{m}\right)\right)}{p}_0\left(\mathfrak{m}\right)\right)\hfill \\ \hfill \hspace{1em}& ={\lambda }^n{u}_{m,n}\left(\mathfrak{m}\right)\left(\lambda ,p_0\left(\mathfrak{m}\right)\right).\hfill \end{array}$$

The sequence $u_{m,n}\left(\lambda ,p_0\left(\mathfrak{m}\right)\right)$ fulfills

$$\begin{array}{cc}\hfill d(u_{m,n}(\lambda ,p_0\left(\mathfrak{m}\right))& \le diam{O}_f\left(p_0\left(\mathfrak{m}\right);+\infty \right)·(1+\lambda +\cdots +\lambda )\hfill \\ \hfill & <{\displaystyle \frac{\delta }{1-\lambda }}.\hfill \end{array}$$

Considering that ${\displaystyle \frac{\delta }{1-\lambda }}$ serves as an upper bound for the sequence $u_{m,n}\left(\mathfrak{m}\right)\left(\lambda ,p_0\left(\mathfrak{m}\right)\right)$, we observe that the sequence of vectors ${\lambda }^n{u}_{m,n}\left(\mathfrak{m}\right)\left(\lambda ,p_0\left(\mathfrak{m}\right)\right)$ converges to zero (in the norm of the space E) as n tends toward infinity. Hence, leveraging the remarks given above yields $\rho \left(p_n\left(\mathfrak{m}\right),p_m\left(\mathfrak{m}\right)\right)\ll \mathfrak{m}$. Thus, $p_n\left(\mathfrak{m}\right)$ forms a Cauchy sequence, implying that it converges to a particular point $u\left(\mathfrak{m}\right)\in Q$.

Next, our objective is to prove the equality $fu\left(\mathfrak{m}\right)=u\left(\mathfrak{m}\right)$. For every $u\left(\mathfrak{m}\right)\in Q$, there exists a mapping $k\left(u\right(\mathfrak{m}\left)\right)$, as defined by Equation (3).

Let $\eta$ with $0\ll \eta$ be given. We select a natural number $n_0\in \mathbb{N}$ such that

$$\rho \left(u\left(\mathfrak{m}\right),pn\left(\mathfrak{m}\right)\right)\ll {\displaystyle \frac{\eta }{3}},$$

holds for every $n\ge n_0$.

In line with Remark, we have

$$\rho \left(f^{k\left(u\right(\mathfrak{m}\left)\right)}u\left(\mathfrak{m}\right),f^{k\left(u\right(\mathfrak{m}\left)\right)}{p}_n\left(\mathfrak{m}\right)\right)\ll {\displaystyle \frac{\eta }{3}}.$$

This implies that

$$\begin{array}{c}\hfill f^{k\left(u\right(\mathfrak{m}\left)\right)}{p}_n\left(\mathfrak{m}\right)\to \rho \left(f^{k\left(u\right(\mathfrak{m}\left)\right)}u\left(\mathfrak{m}\right)\right),\mathrm{as}n\to +\infty .\end{array}$$

Moreover, we have the inequality

$$\begin{array}{c}\hfill \rho \left(f^{k\left(u\right(\mathfrak{m}\left)\right)}{p}_n\left(\mathfrak{m}\right),p_n\left(\mathfrak{m}\right)\right)\le {\lambda }^n\rho \left(f^{k\left(u\right(\mathfrak{m}\left)\right)}{p}_0\left(\mathfrak{m}\right),p_0\left(\mathfrak{m}\right)\right).\end{array}$$

Consequently, there is $n_1\in \mathbb{N}$ such that for any $n\ge n_1$, we have

$$\rho \left(f^{k\left(u\right(\mathfrak{m}\left)\right)}{p}_n\left(\mathfrak{m}\right),p_n\left(\mathfrak{m}\right)\right)\ll {\displaystyle \frac{\eta }{3}}.$$

Consequently, there is $n_2\in \mathbb{N}$ such that for every $n\ge n_2$, it holds that

$$\begin{array}{cc}\hfill & \rho \left(f^{k\left(u\right)}u\left(\mathfrak{m}\right),u\left(\mathfrak{m}\right)\right)\hfill \\ \hfill & \le \rho \left(f^{k\left(u\right(\mathfrak{m}\left)\right)}u\left(\mathfrak{m}\right),f^{k\left(u\right(\mathfrak{m}\left)\right)}{p}_n\left(\mathfrak{m}\right)\right)+\rho \left(f^{k\left(u\right(\mathfrak{m}\left)\right)}{p}_n\left(\mathfrak{m}\right),p_n\left(\mathfrak{m}\right)\right)+\rho \left(p_n\left(\mathfrak{m}\right),u\left(\mathfrak{m}\right)\right)\hfill \\ \hfill & \ll {\displaystyle \frac{\eta }{3}}+{\displaystyle \frac{\eta }{3}}+{\displaystyle \frac{\eta }{3}}=\eta .\hfill \end{array}$$

By applying Remark 1, we have $f^{k\left(u\right(\mathfrak{m}\left)\right)}u\left(\mathfrak{m}\right)=u\left(\mathfrak{m}\right)$. The conclusion of the proof can be derived from Theorem 3. □

Theorem 5.

Let $(Q,\rho )$ be a complete, separable RCMS. Let M be a nonempty, separable, and closed subset of Q, and let f be a continuous random operator defined on M (for $\mathfrak{m}\in \mathcal{M}$). Suppose that for each $\mathfrak{m}\in \mathcal{M}$, there exist $n\left(p\right(\mathfrak{m}\left)\right)\in \mathbb{N}$ and $k\left(\mathfrak{m}\right)\in [0,1)$ as well as

$$f(\mathfrak{m},·):\mathcal{M}\times M⟶M,$$

satisfying

$$\begin{array}{c}\hfill \rho (f^{n\left(p\right(\left(\mathfrak{m}\right)\left)\right)}\left(p\left(\mathfrak{m}\right)\right),f^{n\left(p\right(\mathfrak{m}\left)\right)}\left(q\left(\mathfrak{m}\right)\right))\le k\left(\mathfrak{m}\right)\rho (p\left(\mathfrak{m}\right),q\left(\mathfrak{m}\right));\forall p\left(\mathfrak{m}\right),q\left(\mathfrak{m}\right)\in Q.\end{array}$$

(4)

Then, there is a random fixed point f denoted by $p_0$. Moreover, $f^n\left(p_0\left(\mathfrak{m}\right)\right)⟶\xi \left(\mathfrak{m}\right)$ for each $p_0\left(\mathfrak{m}\right)\in Q.$

Proof. 

Assume that $p_0\left(\mathfrak{m}\right)\in Q$ is arbitrary. Let $m_0=n\left(p_0\left(\mathfrak{m}\right)\right),p_1\left(\left(\mathfrak{m}\right)\right)=f^{m_0}\left(p_0\left(\mathfrak{m}\right)\right)$, and inductively, $m_i=n\left(p_i\left(\mathfrak{m}\right)\right),p_{i+1}\left(\mathfrak{m}\right)=f^{m_i}\left(p_i\left(\mathfrak{m}\right)\right).$

We prove that the sequence $p_n\left(\mathfrak{m}\right)$ converges. With a routine calculation we have

$$\begin{array}{cc}\hfill \rho (p_{n+1}\left(\mathfrak{m}\right),p_n\left(\left(\mathfrak{m}\right)\right))& =\rho (f^{m_n}\left(p_n\left(\left(\mathfrak{m}\right)\right)\right),f^{m_n-1}\left(p_{n-1}\left(\mathfrak{m}\right)\right))\hfill \\ \hfill & =\rho (f^{m_n-1}.f^{m_n}\left(p_{n-1}\left(\left(\mathfrak{m}\right)\right)\right),f^{m_n-1}\left(p_{n-1}\left(\left(\mathfrak{m}\right)\right)\right))\hfill \\ \hfill & \le k\rho (f^{m_n}\left(p_{n-1}\left(\left(\mathfrak{m}\right)\right)\right),p_{n-1}\left(\left(\mathfrak{m}\right)\right))\hfill \\ \hfill & ⋮\hfill \\ \hfill & \le k^n\rho (f^{m_n}\left(p_0\left(\left(\mathfrak{m}\right)\right)\right),p_0\left(\left(\mathfrak{m}\right)\right)).\hfill \end{array}$$

Thus, it can be deduced from Lemma 1 that $\rho (p_{n+1}\left(\left(\mathfrak{m}\right)\right),p_n\left(\left(\mathfrak{m}\right)\right))\le k^{n}r\left(p_0\left(\left(\mathfrak{m}\right)\right)\right).$

Thus, for $m>n$,

$$\begin{array}{c}\hfill \rho (p_m\left(\left(\mathfrak{m}\right)\right),p_n\left(\left(\mathfrak{m}\right)\right))\le \sum _{i=n}^{m-1}\rho (p_{i+1}\left(\left(\mathfrak{m}\right)\right),p_i\left(\left(\mathfrak{m}\right)\right))\le {\displaystyle \frac{k^n}{1-k}}r\left(p_0\left(\left(\mathfrak{m}\right)\right)\right)⟶0\mathrm{as}n\to +\infty .\end{array}$$

(5)

Therefore, the sequence $\left\{p_n\left(\mathfrak{m}\right)\right\}$ forms a Cauchy sequence. Let

$$p_n\left(\left(\mathfrak{m}\right)\right)⟶\xi \left(\mathfrak{m}\right)\in Q.$$

We are going to prove that $f(\mathfrak{m},\xi (\mathfrak{m}\left)\right)=\xi \left(\mathfrak{m}\right)$. Suppose, to the contrary, that

$$f(\mathfrak{m},\xi (\mathfrak{m}\left)\right)\ne \xi \left(\mathfrak{m}\right).$$

In such a case, there exist two disjoint closed neighborhoods U and V such that $\xi \left(\mathfrak{m}\right)\in U$, $f(\mathfrak{m},\xi (\mathfrak{m}\left)\right)\in V$, and there exists a positive value $\delta$ defined as the infimum of

$$\begin{array}{c}\hfill \rho \left(p\right(\left(\mathfrak{m}\right)),q(\left(\mathfrak{m}\right)\left)\right)\mathrm{over}\mathrm{all}p\left(\mathfrak{m}\right)\in U\mathrm{and}q(\prime \mathfrak{m})\in V.\end{array}$$

(6)

Since f is continuous, we conclude that $p_n\left(\left(\mathfrak{m}\right)\right)\in U$ and $f\left(p_n\left(\left(\mathfrak{m}\right)\right)\right)\in \mathfrak{V})$ for all sufficiently large n values. However, we have

$$\begin{array}{cc}\hfill \rho (f\left(p_n\left(\left(\mathfrak{m}\right)\right)\right),p_n\left(\left(\mathfrak{m}\right)\right))& =\rho (f^{m_n-1}.f^{m_n}\left(p_{n-1}\left(\left(\mathfrak{m}\right)\right)\right),f^{m_n-1}\left(p_{n-1}\left(\left(\mathfrak{m}\right)\right)\right))\hfill \\ \hfill & \le k\rho (f^{m_n}\left(p_{n-1}\left(\left(\mathfrak{m}\right)\right)\right),p_{n-1}\left(\left(\mathfrak{m}\right)\right))\hfill \\ \hfill & ⋮\hfill \\ \hfill & \le k^n\rho (f^{m_n}\left(p_0\left(\left(\mathfrak{m}\right)\right)\right),p_0\left(\left(\mathfrak{m}\right)\right))⟶0,\mathrm{as}n⟶+\infty .\hfill \end{array}$$

This contradicts Equation (6). Therefore, $f(\mathfrak{m},\xi (\mathfrak{m}\left)\right)=\xi \left(\mathfrak{m}\right)$, showing the uniqueness of the random fixed point relying on the provided hypothesis.

To demonstrate that $f^n\left(p_0\left(\left(\mathfrak{m}\right)\right)\right)⟶\xi \left(\mathfrak{m}\right)$, we set

$$\begin{array}{c}\hfill \gamma =max\{\rho (f^{m\left(\mathfrak{m}\right)}\left(p_0\left(\left(\mathfrak{m}\right)\right)\right),\xi \left(\mathfrak{m}\right)):m=0,1,\dots ,(n\left(\xi \left(\mathfrak{m}\right)\right)-1)\}.\end{array}$$

If n is large enough, then it can be expressed as $n=t·n\left(\xi \right(\mathfrak{m}\left)\right)+s$, where $0\le s<t·n\left(\xi \right(\mathfrak{m}\left)\right)$ and $t>0$. Therefore, we have

$$\begin{array}{cc}\hfill \rho (f^n(p_0\left(\mathfrak{m}\right),\xi \left(\mathfrak{m}\right))& =\rho (f^{t·n\left(\xi \right(\mathfrak{m}\left)\right)+s}\left(p_0\left(\mathfrak{m}\right)\right),f^{n\left(\xi \right(\mathfrak{m}\left)\right)}\left(\xi \left(\mathfrak{m}\right)\right))\hfill \\ \hfill & \le k\rho (f^{(t-1)·n\left(\xi \right(\mathfrak{m}\left)\right)+s}(p_0\left(\left(\mathfrak{m}\right)\right),\xi \left(\mathfrak{m}\right))\hfill \\ \hfill & ⋮\hfill \\ \hfill & \le k^t\rho (f^s(p_0\left(\mathfrak{m}\right),\xi \left(\mathfrak{m}\right))\le k^t\gamma .\hfill \end{array}$$

Observe that $n⟶+\infty$ implies $r⟶+\infty$, and thus $\rho (f^n(p_0\left(\left(\mathfrak{m}\right)\right),\xi \left(\mathfrak{m}\right))⟶0$ as n tends toward infinity. This completes the proof. □

4. Example

In this section, we give an example to show that our results are distinct from existing ones. It also shows that the random fixed-point problem cannot be solved in metric spaces, as in the random cone metric setting.

Example 1.

Suppose $\mathcal{M}=[0,1]$, and let Σ denote the sigma algebra of Lebesgue measurable subsets of $[0,1]$. Suppose we have $Q=[0,1]$ and a cone $\mathcal{P}\subseteq Q$ equipped with the random cone metric

$${\rho }_{\mathcal{P}}(p\left(\mathfrak{m}\right),q\left(\mathfrak{m}\right))=\alpha \left(\mathfrak{m}\right)·|p\left(\mathfrak{m}\right)-q\left(\mathfrak{m}\right)|$$

for all $p\left(\mathfrak{m}\right),q\left(\mathfrak{m}\right)\in Q$ and $\alpha \left(\mathfrak{m}\right)\in \mathcal{P}$, which is continuous and has a positive value on $\mathcal{M}$. The random mapping $f:\mathcal{M}\times [0,1]⟶[0,1]$ is defined. The set $Q=[0,1]$ can be represented as the union of intervals $\left[{\displaystyle \frac{1}{2^n}},{\displaystyle \frac{1}{2^{n-1}}}\right]$ for natural numbers n, along with the singleton set $\left\{0\right\}$. A random mapping f is defined for each natural number n as follows:

$$\begin{array}{c}\hfill f:\mathcal{M}\times \left[{\displaystyle \frac{1}{2^n}},{\displaystyle \frac{1}{2^{n-1}}}\right]⟶\left[{\displaystyle \frac{1}{2^{n+1}}},{\displaystyle \frac{1}{2^n}}\right].\end{array}$$

(7)

This mapping f operates within the specified intervals for each n, ensuring that the output falls within the range $\left[{\displaystyle \frac{1}{2^{n+1}}},{\displaystyle \frac{1}{2^n}}\right]$, defined by

$$f(\mathfrak{m},p\left(\mathfrak{m}\right))=\left\{\begin{array}{cc}{\displaystyle \frac{n+2}{n+3}}\left({\displaystyle \frac{p\left(\mathfrak{m}\right)-\mathfrak{m}}{4}}-{\displaystyle \frac{1}{2^{n-1}}}\right)+{\displaystyle \frac{1}{2^n}},\hfill & \mathrm{if}{\displaystyle \frac{3n+5}{2^{n+1}(n+2)}}\le p\left(\mathfrak{m}\right)\le {\displaystyle \frac{1}{2^{n-1}}},\hfill \\ {\displaystyle \frac{1}{2^{n+1}}},\hfill & \mathrm{if}{\displaystyle \frac{1}{2^n}}\le p\left(\mathfrak{m}\right)\le {\displaystyle \frac{3n+5}{2^{n+1}(n+2)}},\hfill \end{array}\right.$$

(8)

with $f\left(0\right)=0.$

To check if f is non-decreasing, let $p\left(\mathfrak{m}\right),q\left(\mathfrak{m}\right)\in \left[{\displaystyle \frac{1}{2^n}},{\displaystyle \frac{1}{2^{n+1}(n+2)}}\right]$ with $p\left(\mathfrak{m}\right)<q\left(\mathfrak{m}\right).$

Now, $f(\mathfrak{m},p\left(\mathfrak{m}\right))={\displaystyle \frac{1}{2^{n+1}}}$, $f(\mathfrak{m},q\left(\mathfrak{m}\right))={\displaystyle \frac{1}{2^{n+1}}}$; that is, $f(\mathfrak{m},p(\mathfrak{m}\left)\right)=f(\mathfrak{m},q(\mathfrak{m}\left)\right)$. Observe that f is constant on the segment $[{\displaystyle \frac{1}{2^n}},{\displaystyle \frac{3n+5}{2^{n+1}(n+2)}}]$.

On the other hand, when taking $p\left(\mathfrak{m}\right),q\left(\mathfrak{m}\right)\in [{\displaystyle \frac{3n+5}{2^{n+1}(n+2)}},{\displaystyle \frac{1}{2^{n-1}}}]$ with $p\left(\mathfrak{m}\right)<q\left(\mathfrak{m}\right)$, we have

$$\begin{array}{cc}\hfill & {\displaystyle \frac{p\left(\mathfrak{m}\right)-\mathfrak{m}}{4}}<{\displaystyle \frac{q\left(\mathfrak{m}\right)-\mathfrak{m}}{4}},\hfill \\ \hfill & {\displaystyle \frac{p\left(\mathfrak{m}\right)-\mathfrak{m}}{4}}-{\displaystyle \frac{1}{2^{n-1}}}<{\displaystyle \frac{q\left(\mathfrak{m}\right)-\mathfrak{m}}{4}}-{\displaystyle \frac{1}{2^{n-1},}}\hfill \\ \hfill & {\displaystyle \frac{n+2}{n+3}}\left({\displaystyle \frac{p\left(\mathfrak{m}\right)-\mathfrak{m}}{4}}-{\displaystyle \frac{1}{2^{n-1}}}\right)<{\displaystyle \frac{n+2}{n+3}}\left({\displaystyle \frac{q\left(\mathfrak{m}\right)-\mathfrak{m}}{4}}-{\displaystyle \frac{1}{2^{n-1}}}\right),\hfill \\ \hfill & {\displaystyle \frac{n+2}{n+3}}\left({\displaystyle \frac{p\left(\mathfrak{m}\right)-\mathfrak{m}}{4}}-{\displaystyle \frac{1}{2^{n-1}}}\right)+{\displaystyle \frac{1}{2^n}}<{\displaystyle \frac{n+2}{n+3}}\left({\displaystyle \frac{q\left(\mathfrak{m}\right)-\mathfrak{m}}{4}}-{\displaystyle \frac{1}{2^{n-1}}}\right)+{\displaystyle \frac{1}{2^n}},\hfill \\ \hfill & f(\mathfrak{m},p(\mathfrak{m}\left)\right)<f(\mathfrak{m},q(\mathfrak{m}\left)\right),\hfill \end{array}$$

which means that f is non-decreasing.

Now, we show that f is sequentially continuous (and we know that in this case, every sequentially continuous mapping is continuous). Let $p_n\left(\mathfrak{m}\right)\to p\left(\mathfrak{m}\right).$ Then, we have

$$\begin{array}{cc}\hfill & \parallel p_n\left(\mathfrak{m}\right)-p\left(\mathfrak{m}\right)\parallel <{\displaystyle \frac{4(n+5)}{n+4}}·ϵ,\hfill \\ \hfill \mathrm{and}& f(\mathfrak{m},p_n\left(\mathfrak{m}\right))={\displaystyle \frac{n+2}{n+3}}\left({\displaystyle \frac{p_n\left(\mathfrak{m}\right)-\mathfrak{m}}{4}}-{\displaystyle \frac{1}{2^{n-1}}}\right)+{\displaystyle \frac{1}{2^n}},\hfill \\ \hfill & f(\mathfrak{m},p\left(\mathfrak{m}\right))={\displaystyle \frac{n+2}{n+3}}\left({\displaystyle \frac{p_n\left(\mathfrak{m}\right)-\mathfrak{m}}{4}}-{\displaystyle \frac{1}{2^{n-1}}}\right)+{\displaystyle \frac{1}{2^n}}.\hfill \end{array}$$

Therefore, we obtain

$$\begin{array}{cc}\hfill & |f(\mathfrak{m},p_n\left(\mathfrak{m}\right))-f(\mathfrak{m},p\left(\mathfrak{m}\right))|\hfill \\ \hfill & \le {\displaystyle \frac{n+4}{4(n+5)}}|p_n\left(\mathfrak{m}\right)-p\left(\mathfrak{m}\right)|\hfill \\ \hfill & <{\displaystyle \frac{n+4}{4(n+5)}}\left({\displaystyle \frac{4(n+5)}{n+4}}·ϵ\right)=ϵ,\hfill \\ \hfill & \mathrm{i}.\mathrm{e}.,|f(\mathfrak{m},p_n\left(\mathfrak{m}\right))-f(\mathfrak{m},p\left(\mathfrak{m}\right))|<ϵ.\hfill \end{array}$$

This means f is sequentially continuous and hence continuous.

The above explanation shows that f is a non-decreasing and continuous function defined on the interval $[0,1]$.

We will now demonstrate that the given mapping does not meet the existing contraction conditions, but it does satisfy our proposed random Sehgal contraction.

Assume that we select $p\left(\mathfrak{m}\right)\in \left[{\displaystyle \frac{1}{2^n}},{\displaystyle \frac{3n+5}{2^{n+1}(n+2)}}\right]$. Then, $f(\mathfrak{m},p\left(\mathfrak{m}\right))={\displaystyle \frac{1}{2^{n+1}}}$, and for $q\left(\mathfrak{m}\right)\in \left[{\displaystyle \frac{1}{2^{n+2}}},{\displaystyle \frac{3(n+2)+5}{2^{n+3}(n+4)}}\right]$, we have $f(\mathfrak{m},q\left(\mathfrak{m}\right))={\displaystyle \frac{1}{2^{n+3}}}$. From here, we obtain

$$\begin{array}{cc}\hfill |f(\mathfrak{m},p\left(\mathfrak{m}\right))-f(\mathfrak{m},q\left(\mathfrak{m}\right))|=|{\displaystyle \frac{1}{2^{n+1}}}-{\displaystyle \frac{1}{2^{n+3}}}|& =\left|{\displaystyle \frac{3}{2^{n+3}}}\right|,\hfill \end{array}$$

and thus

$$\begin{array}{c}\hfill |p\left(\mathfrak{m}\right)-q\left(\mathfrak{m}\right)|=|{\displaystyle \frac{1}{2^n}}-{\displaystyle \frac{1}{2^{n+1}}}|.\end{array}$$

It is evident that ${\displaystyle \frac{3}{2^{n+3}}}\ge k\left(\mathfrak{m}\right)·{\displaystyle \frac{1}{2^{n+1}}}={\displaystyle \frac{1}{2^{n+2}}}$ holds for $k\left(\mathfrak{m}\right)={\displaystyle \frac{1}{2}}$, which implies that

$$\begin{array}{c}\hfill {\rho }_{\mathcal{P}}(f(\mathfrak{m},q\left(\mathfrak{m}\right)),f(\mathfrak{m},q\left(\mathfrak{m}\right)))\ge k\left(\mathfrak{m}\right){\rho }_{\mathcal{P}}(p\left(\mathfrak{m}\right),q\left(\mathfrak{m}\right)).\end{array}$$

Therefore, we conclude that f is not a random contraction.

Now, we will check the random Sehgal contraction for the given mapping.

Clearly, $f^2\left(p\left(\mathfrak{m}\right)\right)={\displaystyle \frac{1}{2^{n+2}}}$ and $f^2\left(q\left(\mathfrak{m}\right)\right)={\displaystyle \frac{1}{2^{n+4}}},$ which gives

$$\parallel f^2(\mathfrak{m},p\left(\mathfrak{m}\right))-f^2(\mathfrak{m},q\left(\mathfrak{m}\right))\parallel ={\displaystyle \frac{3}{2^{n+4}}}.$$

Consequently, it is evident that

$${\rho }_{\mathcal{P}}(f^2(\mathfrak{m},p\left(\mathfrak{m}\right)),f^2(\mathfrak{m},q\left(\mathfrak{m}\right)))\le k\left(\mathfrak{m}\right){\rho }_{\mathcal{P}}(p\left(\mathfrak{m}\right),q\left(\mathfrak{m}\right)),\mathrm{with}k\left(\mathfrak{m}\right)={\displaystyle \frac{1}{2}}.$$

Hence, $f^2$ satisfies the random Sehgal contraction condition. Consequently, we can assert that f is a random Sehgal contraction, and it possesses a distinct random fixed point in the RCMS, which is zero.

5. Applications of Transverse Oscillations in a Random Homogeneous Bar

In this section, our aim is to utilize the established results to explore the existence of a solution to the random BVP governing the transverse oscillations of a random homogeneous bar. This investigation stems from the outcomes we previously established.

Let $Q=C\left[\mathcal{M},\mathbb{R}\right]$ be the collection of continuous functions from $\mathcal{M}$ to $\mathbb{R}$. A random cone metric can be defined as follows:

$$\begin{array}{c}\hfill \rho (p\left(\mathfrak{m}\right),q\left(\mathfrak{m}\right))=\underset{♭\left(\mathfrak{m}\right)\in \mathcal{M}}{sup}|p\left(\mathfrak{m}\right)(♭\left(\mathfrak{m}\right))-q\left(\mathfrak{m}\right)(♭\left(\mathfrak{m}\right))|\end{array}$$

Consequently, the pair $\left(Q,\rho \right)$ constitutes a complete RCMS.

The transverse vibrations of a uniform bar have significant practical importance, which positioned along segment $(0,1)$ of the x axis. At any given moment, we can characterize the ordinary differential equation (ODE) governing the lateral vibrations of the homogeneous bar by examining the displacement:

$$\left\{\begin{array}{c}{\displaystyle \frac{dp\left(\mathfrak{m}\right)(♭(\mathfrak{m}\left)\right)}{d{♭}^4\left(\mathfrak{m}\right)}}=k^4\left(\mathfrak{m}\right)\mathcal{L}(♭\left(\mathfrak{m}\right),p\left(\mathfrak{m}\right)(♭\left(\mathfrak{m}\right)));♭\left(\mathfrak{m}\right)\in \mathcal{M}\hfill \\ p\left(\mathfrak{m}\right)\left(0\right)=p^{\prime }\left(\mathfrak{m}\right)\left(0\right)=p^{\prime \prime }\left(\mathfrak{m}\right)\left(1\right)=p^{\prime \prime \prime }\left(\mathfrak{m}\right)\left(1\right)=0,\hfill \end{array}\right.$$

where $\mathcal{L}$ is a continuous random function defined on $\mathcal{M}\times \mathbb{R}$ and $k\left(\mathfrak{m}\right)>0$ is a constant. Through standard calculus techniques, it can be shown that the previously mentioned problem can be expressed as the following Fredholm integral equation:

$$p\left(\mathfrak{m}\right)(♭\left(\mathfrak{m}\right))=k^4\left(\mathfrak{m}\right){\int }_0^{1}G(♭\left(\mathfrak{m}\right),\varsigma \left(\mathfrak{m}\right))L(\varsigma \left(\mathfrak{m}\right),p\left(\mathfrak{m}\right)\left(\varsigma \left(\mathfrak{m}\right)\right))d\varsigma \left(\mathfrak{m}\right)$$

(9)

where the Green function is defined as

$$G=\left\{\begin{array}{cc}{\displaystyle \frac{3{\varsigma }^2\left(\mathfrak{m}\right)♭\left(\mathfrak{m}\right)-{\varsigma }^3\left(\mathfrak{m}\right)}{6}},\hfill & 0\le \varsigma \left(\mathfrak{m}\right)\le ♭\left(\mathfrak{m}\right)\le 1\hfill \\ {\displaystyle \frac{3{♭}^2\left(\mathfrak{m}\right)\varsigma \left(\mathfrak{m}\right)-{♭}^3\left(\mathfrak{m}\right)}{6}},\hfill & 0\le ♭\left(\mathfrak{m}\right)\le \varsigma \left(\mathfrak{m}\right)\le 1.\hfill \end{array}\right.$$

Next, consider the following conditions $\left(A_1\right)$ and $\left(A_2\right)$ concerning the random mapping $f:\mathcal{M}\times Q\to Q$, which is defined as

$$fp\left(\mathfrak{m}\right)(♭\left(\mathfrak{m}\right))=k^4\left(\mathfrak{m}\right){\int }_0^1\mathcal{R}(♭\left(\mathfrak{m}\right),\varsigma \left(\mathfrak{m}\right))\mathcal{L}(\varsigma \left(\mathfrak{m}\right),p\left(\mathfrak{m}\right)\left(\varsigma \left(\mathfrak{m}\right)\right)d\varsigma \left(\mathfrak{m}\right),\forall ♭\left(\mathfrak{m}\right)\in \mathcal{M}.$$

(10)

$\left({\mathbf{A}}_{\mathbf{1}}\right)$ Suppose a constant $k\left(\mathfrak{m}\right)$ for which

$$\begin{array}{c}\hfill 0\le \underset{♭\left(\mathfrak{m}\right)\in \mathcal{M}}{sup}\nabla (♭\left(\mathfrak{m}\right),k\left(\mathfrak{m}\right))<1\end{array}$$

where

$$\begin{array}{c}\hfill \nabla (♭\left(\mathfrak{m}\right),k\left(\mathfrak{m}\right))=k^4\left(\mathfrak{m}\right)\left({\displaystyle \frac{{♭}^4\left(\mathfrak{m}\right)-4{♭}^3\left(\mathfrak{m}\right)+6{♭}^2\left(\mathfrak{m}\right)}{24}}\right).\end{array}$$

$\left({\mathbf{A}}_{\mathbf{2}}\right)$ For each $♭\left(\mathfrak{m}\right)\in [0,1]$ and $p\left(\mathfrak{m}\right),q\left(\mathfrak{m}\right)\in Q$, the following holds:

$$\left|\mathcal{L}\right(♭\left(\mathfrak{m}\right),fp\left(\mathfrak{m}\right)(♭(\mathfrak{m}\left)\right))-\mathcal{L}(♭\left(\mathfrak{m}\right),fq\left(\mathfrak{m}\right)(♭(\mathfrak{m}\left)\right)\left)\right|\le \left|p\right(♭\left(\mathfrak{m}\right))-q(\mathfrak{m}\left)\right(♭\left(\mathfrak{m}\right)\left)\right|.$$

(11)

Theorem 6.

According to the specified conditions $\left({\mathbf{A}}_{\mathbf{1}}\right)$ and $\left({\mathbf{A}}_{\mathbf{2}}\right)$, and according to Equation (10), the random mapping f suggests a unique solution for the ODE in Equation (1), which describes the lateral vibrations of a uniform bar.

Proof. 

By utilizing a random mapping f with conditions $\left({\mathbf{A}}_{\mathbf{1}}\right)$ and $\left({\mathbf{A}}_{\mathbf{2}}\right)$, we can obtain the following equation:

$$\begin{array}{cc}\hfill & \left|f^2\left(\mathfrak{m}\right)p\left(\mathfrak{m}\right)(♭\left(\mathfrak{m}\right))-f^2\left(\mathfrak{m}\right)q\left(\mathfrak{m}\right)(♭\left(\mathfrak{m}\right))\right|\hfill \\ \hfill & =\left|k^4\left(\mathfrak{m}\right){\int }_0^1\mathcal{R}(♭\left(\mathfrak{m}\right),\varsigma \left(\mathfrak{m}\right))\mathcal{L}(\varsigma \left(\mathfrak{m}\right),fp\left(\varsigma \left(\mathfrak{m}\right)\right))d\varsigma \left(\mathfrak{m}\right)\right.\hfill \\ \hfill & \left.-k^4\left(\mathfrak{m}\right){\int }_0^1\mathcal{R}(♭\left(\mathfrak{m}\right),\varsigma \left(\mathfrak{m}\right))\mathcal{L}(\varsigma \left(\mathfrak{m}\right),fq\left(\mathfrak{m}\right)\left(\varsigma \left(\mathfrak{m}\right)\right))d\varsigma \left(\mathfrak{m}\right)\right|\hfill \\ \hfill & \le k^4\left(\mathfrak{m}\right){\int }_0^1\mathcal{R}(♭\left(\mathfrak{m}\right),\varsigma \left(\mathfrak{m}\right))|\mathcal{L}(♭\left(\mathfrak{m}\right),fp(♭\left(\mathfrak{m}\right)))-\mathcal{L}(♭\left(\mathfrak{m}\right),fq\left(\mathfrak{m}\right)(♭\left(\mathfrak{m}\right)))|ds\left(\mathfrak{m}\right)\hfill \\ \hfill & \le k^4\left(\mathfrak{m}\right){\int }_0^1\mathcal{R}(♭\left(\mathfrak{m}\right),\varsigma \left(\mathfrak{m}\right))|p(♭\left(\mathfrak{m}\right))-q(♭\left(\mathfrak{m}\right))|d\varsigma \left(\mathfrak{m}\right)\hfill \\ \hfill & \le k^4\left(\mathfrak{m}\right)\underset{\xi \left(\mathfrak{m}\right)\in \mathcal{M}}{sup}|p\left(\xi \left(\mathfrak{m}\right)\right)-q\left(\mathfrak{m}\right)\left(\xi \left(\mathfrak{m}\right)\right)|{\int }_0^1\mathcal{R}(♭\left(\mathfrak{m}\right),\varsigma \left(\mathfrak{m}\right))d\varsigma \left(\mathfrak{m}\right)\hfill \\ \hfill & =k^4\left(\mathfrak{m}\right)\left({\displaystyle \frac{{♭}^4\left(\mathfrak{m}\right)-4{♭}^3\left(\mathfrak{m}\right)+6{♭}^2\left(\mathfrak{m}\right)}{24}}\right)\underset{\xi \left(\mathfrak{m}\right)\in \mathcal{M}}{sup}|p\left(\xi \left(\mathfrak{m}\right)\right)-q\left(\mathfrak{m}\right)\left(\xi \left(\mathfrak{m}\right)\right)|,\hfill \end{array}$$

(12)

which implies the following:

$$\begin{array}{cc}\hfill & \underset{\xi \left(\mathfrak{m}\right)\in \mathcal{M}}{sup}\left|f^{2}p\left(\mathfrak{m}\right)(♭\left(\mathfrak{m}\right))-f^{2}q\left(\mathfrak{m}\right)(♭\left(\mathfrak{m}\right))\right|\hfill \\ \hfill & \le \underset{\xi \left(\mathfrak{m}\right)\in \mathcal{M}}{sup}\nabla (\xi \left(\mathfrak{m}\right),k\left(\mathfrak{m}\right))·\underset{\xi \left(\mathfrak{m}\right)\in \mathcal{M}}{sup}|p\left(\xi \left(\mathfrak{m}\right)\right)-q\left(\mathfrak{m}\right)\left(\xi \left(\mathfrak{m}\right)\right)|.\hfill \end{array}$$

Equivalently, we have

$$\begin{array}{c}\hfill \rho \left(f^{2}p\left(\mathfrak{m}\right),f^{2}q\left(\mathfrak{m}\right)\right)\le \eta \rho (p\left(\mathfrak{m}\right),q\left(\mathfrak{m}\right)),\end{array}$$

for $0\le \eta \left(\mathfrak{m}\right)={sup}_{\xi \left(\mathfrak{m}\right)\in \mathcal{M}}\nabla (\xi \left(\mathfrak{m}\right),k\left(\mathfrak{m}\right))<1$.

Therefore, all of the conditions stated in Theorem 5 are valid for $n=2$. Consequently, a unique solution exists for Equation (1) within the set Q. □

6. Application of a Random Solution for the Nonlinear FDE

We present here a random solution for the nonlinear Caputo-type FDE. Aydi [35] examined Ri’s discoveries, particularly focusing on investigating a nonlinear FDE of the Caputo type. Baleanu [36,37] concentrated on proving the existence of solutions for FDEs.

The Caputo derivative in the random sense is given as follows:

$$\begin{array}{c}\hfill {}^{\mathit{cf}}D^{\delta \left(\mathfrak{m}\right)}\left(p\left(\mathfrak{m}\right)\right)(♭\left(\mathfrak{m}\right))):={\displaystyle \frac{1}{\Gamma (n-\delta (\mathfrak{m}\left)\right)}}{\int }_0^{♭\left(\mathfrak{m}\right)}{(♭\left(\mathfrak{m}\right)-s\left(\mathfrak{m}\right))}^{n-\delta \left(\mathfrak{m}\right)-1}p{\left(\mathfrak{m}\right)}^{\left(n\right(\mathfrak{m}\left)\right)}\left(s\left(\mathfrak{m}\right)\right)ds\left(\mathfrak{m}\right),\end{array}$$

where the value of $\delta \left(\mathfrak{m}\right)$ falls between $(n-1)$ and n, with $n=\delta \left(\mathfrak{m}\right)+1$.

Here, we explore a nonlinear, random integrodifferential equation of the random Caputo type

$$\begin{array}{c}\hfill {}^{\mathit{cf}}D^{\delta }\left(p\left(\mathfrak{m}\right)(♭\left(\mathfrak{m}\right))\right)=H(♭\left(\mathfrak{m}\right),p\left(\mathfrak{m}\right)(♭\left(\mathfrak{m}\right)))\end{array}$$

(13)

subject to the specified conditions, $p\left(\mathfrak{m}\right)\left(0\right)=0$, and

$$p\left(\mathfrak{m}\right)\left(1\right)={\int }_0^{\beta }p\left(s\left(\mathfrak{m}\right)\right)ds\left(\mathfrak{m}\right).$$

Here, H represents a random mapping which is continuous, and $p\in C(\mathcal{M},[0,+\infty \left)\right)$ satisfies the conditions $0<\beta <1$ and $1<\delta \le 2$ (for more details, see Baleanu et al. [36]). Solving Equation (13) leads to a random fixed point of the subsequent random integral equation:

$$\begin{array}{cc}\hfill & fp\left(\mathfrak{m}\right)(♭\left(\mathfrak{m}\right))={\displaystyle \frac{1}{\Gamma \left(\delta \right(\mathfrak{m}\left)\right)}}{\int }_0^{♭\left(\mathfrak{m}\right)}{(♭\left(\mathfrak{m}\right)-s\left(\mathfrak{m}\right))}^{\delta \left(\mathfrak{m}\right)-1}H(s\left(\mathfrak{m}\right),p\left(s\left(\mathfrak{m}\right)\right))ds\left(\mathfrak{m}\right)\hfill \\ \hfill & -{\displaystyle \frac{2♭\left(\mathfrak{m}\right)}{\left(2-{\beta }^2\right)\Gamma \left(\delta \left(\mathfrak{m}\right)\right)}}{\int }_0^1{(1-s\left(\mathfrak{m}\right))}^{\delta \left(\mathfrak{m}\right)-1}p(s\left(\mathfrak{m}\right),p)\left(s\left(\mathfrak{m}\right)\right))ds\left(\mathfrak{m}\right)\hfill \\ \hfill & +{\displaystyle \frac{2♭\left(\mathfrak{m}\right)}{\left(2-{\beta }^2\right)\Gamma \left(\delta \left(\mathfrak{m}\right)\right)}}{\int }_0^{\beta }\left({\int }_0^{s\left(\mathfrak{m}\right)}{(s\left(\mathfrak{m}\right)-r\left(\mathfrak{m}\right))}^{\delta \left(\mathfrak{m}\right)-1}p(r\left(\mathfrak{m}\right),p\left(\mathfrak{m}\right)\left(r\left(\mathfrak{m}\right)\right))dr\left(\mathfrak{m}\right)\right)d\left(\mathfrak{m}\right)\hfill \end{array}$$

(14)

Theorem 7.

When allowing nonlinearity of the random FDE in Equation (13), we suggest the following hypothesis:

$$\begin{array}{c}\hfill |H(s\left(\mathfrak{m}\right),p\left(\mathfrak{m}\right)\left(s\left(\mathfrak{m}\right)\right))-H(\left(\mathfrak{m}\right),q\left(\mathfrak{m}\left(\mathfrak{m}\right)\right))|\le {\displaystyle \frac{\Gamma \left(\delta \right(\mathfrak{m})+1)}{5}}{e}^{-\xi \left(\mathfrak{m}\right)}\left|\sqrt{\left|p\right(\mathfrak{m}\left)\right(\left(\mathfrak{m}\right)\left)\right|}-\sqrt{\left|q\right(\left(\mathfrak{m}\right)\left)\right|}\right|\end{array}$$

and

$$\begin{array}{c}\hfill |H(s\left(\mathfrak{m}\right),p\left(\mathfrak{m}\right)\left(s\left(\mathfrak{m}\right)\right))+H(\left(\mathfrak{m}\right),q\left(\mathfrak{m}\right)\left(\left(\mathfrak{m}\right)\right))|\le {\displaystyle \frac{\Gamma \left(\delta \right(\mathfrak{m})+1)}{5}}{e}^{-\xi \left(\mathfrak{m}\right)}\left|\sqrt{\left|p\right(\mathfrak{m}\left)\right(\left(\mathfrak{m}\right)\left)\right|}+\sqrt{\left|q\right(\left(\mathfrak{m}\right)\left)\right|}\right|\end{array}$$

for any $s\left(\mathfrak{m}\right)$ within the interval $\mathcal{M}$, where $\xi \left(\mathfrak{m}\right)>0$ and for all functions, we have

$$p\left(\mathfrak{m}\right),q\left(\mathfrak{m}\right)\in C[\mathcal{M},\mathbb{R}].$$

Under the stipulated conditions, a unique random solution to Equation (13) exists.

Proof. 

Consider $\mathcal{M}=[0,1]$, and denote Q as the set of all continuous functions from $\mathcal{M}$ to $\mathbb{R}$, with the random metric defined as follows:

$$\begin{array}{c}\hfill \rho (p\left(\mathfrak{m}\right),q\left(\mathfrak{m}\right))=\underset{♭\left(\mathfrak{m}\right)\in \mathcal{M}}{sup}|p\left(\mathfrak{m}\right)(♭\left(\mathfrak{m}\right))-q\left(\mathfrak{m}\right)(♭\left(\mathfrak{m}\right))|.\end{array}$$

Subsequently, the combination of $(Q,\rho )$ constitutes a complete CRMS. Furthermore, we demonstrate that f fulfills all of the conditions of Theorem 5 for $n=2$. Within this framework, we contemplate $p\left(\mathfrak{m}\right)$ and $q\left(\mathfrak{m}\right)$ as elements of Q, while $♭\left(\mathfrak{m}\right)$ ranges over the interval $\mathcal{M}$. The outcome is presented as follows:

$$\begin{array}{cc}\hfill & \left|fp\right(♭\left(\mathfrak{m}\right))-fq(\mathfrak{m}\left)\right(♭\left(\mathfrak{m}\right)\left)\right|\hfill \\ \hfill =& \left|{\displaystyle \frac{1}{\Gamma \left(\delta \right(\mathfrak{m}\left)\right)}}{\int }_0^{♭\left(\mathfrak{m}\right)}{(♭\left(\mathfrak{m}\right)-s)}^{\delta -1}H(s\left(\mathfrak{m}\right),p\left(s\left(\mathfrak{m}\right)\right))ds\left(\mathfrak{m}\right)\right.\hfill \\ \hfill & -{\displaystyle \frac{2♭\left(\mathfrak{m}\right)}{\left(2-{\beta }^2\right)\Gamma \left(\delta \left(\mathfrak{m}\right)\right)}}{\int }_0^1{(1-s\left(\mathfrak{m}\right))}^{\delta \left(\mathfrak{m}\right)-1}H(s,p\left(s\right))ds\left(\mathfrak{m}\right)\hfill \\ \hfill & +{\displaystyle \frac{2♭\left(\mathfrak{m}\right)}{\left(2-{\beta }^2\right)\Gamma \left(\delta \left(\mathfrak{m}\right)\right)}}{\int }_0^{\beta }\left({\int }_0^{s\left(\mathfrak{m}\right)}{(\left(\mathfrak{m}\right)-r\left(\mathfrak{m}\right))}^{\delta \left(\mathfrak{m}\right)-1}H(r,p\left(\mathfrak{m}\right)\left(r\left(\mathfrak{m}\right)\right))dr\left(\mathfrak{m}\right)\right)ds\left(\mathfrak{m}\right)\hfill \\ \hfill & -{\displaystyle \frac{1}{\Gamma \left(\delta \right(\mathfrak{m}\left)\right)}}{\int }_0^{♭\left(\mathfrak{m}\right)}{(♭\left(\mathfrak{m}\right)-s\left(\mathfrak{m}\right))}^{\delta \left(\mathfrak{m}\right)-1}H(s\left(\mathfrak{m}\right),q\left(s\right))ds\left(\mathfrak{m}\right)\hfill \\ \hfill & +{\displaystyle \frac{2♭\left(\mathfrak{m}\right)}{\left(2-{\beta }^2\right)\Gamma \left(\delta \left(\mathfrak{m}\right)\right)}}{\int }_0^1{(1-s\left(\mathfrak{m}\right))}^{\delta \left(\mathfrak{m}\right)-1}H(s\left(\mathfrak{m}\right),q\left(s\right))ds\left(\mathfrak{m}\right)\hfill \\ \hfill & \left.-{\displaystyle \frac{2♭\left(\mathfrak{m}\right)}{\left(2-{\beta }^2\right)\Gamma \left(\delta \left(\mathfrak{m}\right)\right)}}{\int }_0^{\beta }\left({\int }_0^{s\left(\mathfrak{m}\right)}{(s\left(\mathfrak{m}\right)-m\left(\mathfrak{m}\right))}^{\delta \left(\mathfrak{m}\right)-1}H(r\left(\mathfrak{m}\right),q\left(\mathfrak{m}\right)\left(r\left(\mathfrak{m}\right)\right))dr\left(\mathfrak{m}\right)\right)ds\left(\mathfrak{m}\right)\right|.\hfill \end{array}$$

(15)

Consequently, we have

$$\begin{array}{cc}\hfill & \left|fp\right(\mathfrak{m}\left)\right(♭\left(\mathfrak{m}\right))-fq(\mathfrak{m}\left)\right(♭\left(\mathfrak{m}\right)\left)\right|\hfill \\ \hfill \le & {\displaystyle \frac{1}{\Gamma \left(\delta \right(\mathfrak{m}\left)\right)}}{\int }_0^{♭\left(\mathfrak{m}\right)}{|♭\left(\mathfrak{m}\right)-s\left(\mathfrak{m}\right)|}^{\delta \left(\mathfrak{m}\right)-1}{\displaystyle \frac{\Gamma \left(\delta \right(\mathfrak{m})+1)}{5}}\hfill \\ \hfill & \left[e^{-\xi \left(\mathfrak{m}\right)}\underset{s\left(\mathfrak{m}\right)\in \mathcal{M}}{sup}\mid \sqrt{\left|p\right(s\left(\mathfrak{m}\right)\left)\right|}-\sqrt{\left|q\right(s\left(\mathfrak{m}\right)\left)\right|}\mid \right]ds\left(\mathfrak{m}\right)\hfill \\ \hfill & +{\displaystyle \frac{2♭\left(\mathfrak{m}\right)}{\left(2-{\beta }^2\right)\Gamma \left(\delta \left(\mathfrak{m}\right)\right)}}{\int }_0^1{(1-s\left(\mathfrak{m}\right))}^{\delta \left(\mathfrak{m}\right)-1}{\displaystyle \frac{\Gamma \left(\delta \right(\mathfrak{m})+1)}{5}}\hfill \\ \hfill & \left[e^{-\xi }\underset{s\left(\mathfrak{m}\right)\in \mathcal{M}}{sup}|\sqrt{\left|p\right(\mathfrak{m}\left)\right(s\left(\mathfrak{m}\right)\left)\right|}-\sqrt{\left|q\right(\mathfrak{m}\left)\right(s\left(\mathfrak{m}\right)\left)\right|}|\right]ds\left(\mathfrak{m}\right)+{\displaystyle \frac{2♭\left(\mathfrak{m}\right)}{\left(2-{\beta }^2\right)\Gamma \left(\delta \left(\mathfrak{m}\right)\right)}}\hfill \\ \hfill & {\int }_0^{\beta }\left|{\int }_0^s{\left(\mathfrak{m}\right)-r)}^{\delta -1}{\displaystyle \frac{\Gamma \left(\delta \right(\mathfrak{m})+1)}{5}}\left[e^{-\zeta \left(\mathfrak{m}\right)}\underset{\left(\mathfrak{m}\right)\in \mathcal{M}}{sup}\mid \sqrt{\left|p\right(s\left(\mathfrak{m}\right)\left)\right|}-\sqrt{\left|q\right(s\left(\mathfrak{m}\right)\left)\right|}\right]dr\right|ds\left(\mathfrak{m}\right)\hfill \\ \hfill \le & {\displaystyle \frac{\Gamma \left(\delta \right(\mathfrak{m})+1)}{5}}\left[e^{-\tilde{\zeta }}\underset{s\left(\mathfrak{m}\right)\in \mathcal{M}}{sup}\mid \sqrt{p\left(s\right(\mathfrak{m}\left)\right)}-\sqrt{q\left(s\right(\mathfrak{m}\left)\right)\mid }\right]\times \hfill \\ \hfill & \underset{♭\left(\mathfrak{m}\right)\in \mathcal{M}}{sup}\left({\displaystyle \frac{1}{\Gamma \left(\delta \right(\mathfrak{m}\left)\right)}}{\int }_0^1{|♭\left(\mathfrak{m}\right)-s\left(\mathfrak{m}\right)|}^{\delta -1}ds\left(\mathfrak{m}\right)\right.\hfill \\ \hfill & +{\displaystyle \frac{2♭\left(\mathfrak{m}\right)}{\left(2-{\beta }^2\right)\Gamma \left(\delta \left(\mathfrak{m}\right)\right)}}{\int }_0^1{(1-s\left(\mathfrak{m}\right))}^{\delta \left(\mathfrak{m}\right)-1}ds\left(\mathfrak{m}\right)\hfill \\ & \left.+{\displaystyle \frac{2♭}{\left(2-{\beta }^2\right)\Gamma \left(\delta \left(\mathfrak{m}\right)\right)}}{\int }_0^{\beta }{\int }_0^{s\left(\mathfrak{m}\right)}{|s\left(\mathfrak{m}\right)-r\left(\mathfrak{m}\right)|}^{\delta \left(\mathfrak{m}\right)-1}dr\left(\mathfrak{m}\right)ds\left(\mathfrak{m}\right)\right)\hfill \\ \hfill \le & e^{-\widetilde{\zeta \left(\mathfrak{m}\right)}}\underset{s\left(\mathfrak{m}\right)\in \mathcal{M}}{sup}\mid \sqrt{\left|p\right(\mathfrak{m}\left)\right(s\left(\mathfrak{m}\right)\left)\right|}-\sqrt{\left|q\right(\mathfrak{m}\left)\right(s\left(\mathfrak{m}\right)\left)\right|}.\hfill \end{array}$$

Similarly, we obtain

$$\begin{array}{cc}\hfill & \left|fp\right(\mathfrak{m}\left)\right(♭\left(\mathfrak{m}\right))+fq(\mathfrak{m}\left)\right(♭\left(\mathfrak{m}\right)\left)\right|\hfill \\ \hfill =& \left|{\displaystyle \frac{1}{\Gamma \left(\delta \right(\mathfrak{m}\left)\right)}}{\int }_0^{♭\left(\mathfrak{m}\right)}{(♭\left(\mathfrak{m}\right)-s\left(\mathfrak{m}\right))}^{\delta \left(\mathfrak{m}\right)-1}H(s\left(\mathfrak{m}\right),p\left(\mathfrak{m}\right)\left(s\left(\mathfrak{m}\right)\right))ds\left(\mathfrak{m}\right)\right.\hfill \\ \hfill & -{\displaystyle \frac{2♭\left(\mathfrak{m}\right)}{\left(2-{\beta }^2\right)\Gamma \left(\delta \left(\mathfrak{m}\right)\right)}}{\int }_0^1{(1-s\left(\mathfrak{m}\right))}^{\delta \left(\mathfrak{m}\right)-1}H(\left(\mathfrak{m}\right),p\left(\mathfrak{m}\right)\left(\mathfrak{m}\right)))d\left(\mathfrak{m}\right)\hfill \\ \hfill & \left.+{\displaystyle \frac{2♭\left(\mathfrak{m}\right)}{\left(2-{\beta }^2\right)\Gamma \left(\delta \left(\mathfrak{m}\right)\right)}}{\int }_0^{\beta }\left({\int }_0^{s\left(\mathfrak{m}\right)}{(\left(\mathfrak{m}\right)-r)}^{\delta \left(\mathfrak{m}\right)-1}H(r,p\left(m\right))dr\right)ds\left(\mathfrak{m}\right)\right|\hfill \\ \hfill & +\left|{\displaystyle \frac{1}{\Gamma \left(\delta \right(\mathfrak{m}\left)\right)}}{\int }_0^{♭\left(\mathfrak{m}\right)}{(♭\left(\mathfrak{m}\right)-s\left(\mathfrak{m}\right))}^{\delta \left(\mathfrak{m}\right)-1}H(s,q\left(s\left(\mathfrak{m}\right)\right))ds\left(\mathfrak{m}\right)\right.\hfill \\ \hfill & -{\displaystyle \frac{2♭\left(\mathfrak{m}\right)}{\left(2-{\beta }^2\right)\Gamma \left(\delta \left(\mathfrak{m}\right)\right)}}{\int }_0^1{(1-s\left(\mathfrak{m}\right))}^{\delta -1}H(s,q\left(s\left(\mathfrak{m}\right)\right))ds\left(\mathfrak{m}\right)\hfill \\ \hfill & \left.+{\displaystyle \frac{2♭\left(\mathfrak{m}\right)}{\left(2-{\beta }^2\right)\Gamma \left(\delta \left(\mathfrak{m}\right)\right)}}{\int }_0^{\beta }\left({\int }_0^{s\left(\mathfrak{m}\right)}{(s\left(\mathfrak{m}\right)-r\left(\mathfrak{m}\right))}^{\delta \left(\mathfrak{m}\right)-1}H(r\left(\mathfrak{m}\right),q\left(\mathfrak{m}\right)\left(r\left(\mathfrak{m}\right)\right))dr\left(\mathfrak{m}\right)\right)ds\left(\mathfrak{m}\right)\right|.\hfill \end{array}$$

(16)

Therefore, we have

$$\begin{array}{cc}\hfill & \left|fp\right(\mathfrak{m}\left)\right(♭\left(\mathfrak{m}\right)\left)\right|+\left|fq\right(♭\left(\mathfrak{m}\right)\left)\right|\hfill \\ \hfill \le & {\displaystyle \frac{1}{\Gamma \left(\delta \right(\mathfrak{m}\left)\right)}}{\int }_0^{♭\left(\mathfrak{m}\right)}{|♭\left(\mathfrak{m}\right)-s\left(\mathfrak{m}\right)|}^{\delta \left(\mathfrak{m}\right)-1}\left(\left.{\displaystyle \frac{\Gamma \left(\delta \right(\mathfrak{m})+1)}{5}}{e}^{-\widetilde{\xi \left(\mathfrak{m}\right)}}\right|\sqrt{\left|p\right(\mathfrak{m}\left)\right(s\left(\mathfrak{m}\right)\left)\right|}+\sqrt{\left|q\right(s\left(\mathfrak{m}\right)\left)\right|\mid }\right)ds\left(\mathfrak{m}\right)\hfill \\ \hfill & +{\displaystyle \frac{2♭\left(\mathfrak{m}\right)}{\left(2-{\beta }^2\right)\Gamma \left(\delta \left(\mathfrak{m}\right)\right)}}{\int }_0^1{(1-s\left(\mathfrak{m}\right))}^{\delta -1}\left(\left|{\displaystyle \frac{\Gamma \left(\delta \right(\mathfrak{m})+1)}{5}}{e}^{-\xi }\right|\sqrt{p\left(\mathfrak{m}\right)\left(s\right(\mathfrak{m}\left)\right)}+\sqrt{q\left(\mathfrak{m}\right)\left(s\right(\mathfrak{m}\left)\right)\mid }\right)ds\left(\mathfrak{m}\right)\hfill \\ \hfill & \left.+{\displaystyle \frac{2♭\left(\mathfrak{m}\right)}{\left(2-{\beta }^2\right)\Gamma \left(\delta \left(\mathfrak{m}\right)\right)}}{\int }_0^{\beta }\left|{\int }_0^{s\left(\mathfrak{m}\right)}{(s\left(\mathfrak{m}\right)-r\left(\mathfrak{m}\right))}^{\delta \left(\mathfrak{m}\right)-1}{\displaystyle \frac{\Gamma \left(\delta \right(\mathfrak{m})+1)}{5}}{e}^{-\xi }\right|\sqrt{p\left(r\right)}+\sqrt{q\left(\mathfrak{m}\right)\left(r\right(\mathfrak{m}\left)\right)\mid }d\right|ds\left(\mathfrak{m}\right)\hfill \\ \hfill \le & \left.{\displaystyle \frac{\Gamma \left(\delta \right(\mathfrak{m})+1)}{5}}\underset{s\left(\mathfrak{m}\right)\in \mathcal{M}}{sup}\right|\sqrt{\left|p\right(s\left(\mathfrak{m}\right)\left)\right|}+\sqrt{\left|q\right(s\left(\mathfrak{m}\right)\left)\right|}\times \hfill \\ \hfill & \underset{♭\left(\mathfrak{m}\right)\in \mathcal{M}}{sup}\left({\displaystyle \frac{1}{\Gamma \left(\delta \right(\mathfrak{m}\left)\right)}}{\int }_0^{♭\left(\mathfrak{m}\right)}{|♭\left(\mathfrak{m}\right)-s\left(\mathfrak{m}\right)|}^{\delta \left(\mathfrak{m}\right)-1}ds\left(\mathfrak{m}\right)+{\displaystyle \frac{2♭\left(\mathfrak{m}\right)}{\left(2-{\beta }^2\right)\Gamma \left(\delta \left(\mathfrak{m}\right)\right)}}{\int }_0^1{(1-s\left(\mathfrak{m}\right))}^{\delta -1}ds\left(\mathfrak{m}\right)\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{2♭\left(\mathfrak{m}\right)}{\left(2-{\beta }^2\right)\Gamma \left(\delta \left(\mathfrak{m}\right)\right)}}{\int }_0^{\beta }\left|{\int }_0^{s\left(\mathfrak{m}\right)}{(s\left(\mathfrak{m}\right)-r\left(\mathfrak{m}\right))}^{\delta \left(\mathfrak{m}\right)-1}dr\left(\mathfrak{m}\right)\right|ds\left(\mathfrak{m}\right)\right)\hfill \\ \hfill \le & \underset{s\left(\mathfrak{m}\right)\in \mathcal{M}}{sup}\mid \sqrt{\left|p\right(\mathfrak{m}\left)\right(s\left(\mathfrak{m}\right)\left)\right|}+\sqrt{\left|q\right(\mathfrak{m}\left)\right(s\left(\mathfrak{m}\right)\left)\right|}.\hfill \end{array}$$

(17)

This gives rise to the following:

$$\begin{array}{c}\hfill \underset{♭\left(\mathfrak{m}\right)\in \mathcal{M}}{sup}\left(\right|fp\left(\mathfrak{m}\right)♭\left(\mathfrak{m}\right))|+|fq\left(\mathfrak{m}\right)(♭\left(\mathfrak{m}\right))\left|\right)\le \underset{s\left(\mathfrak{m}\right)\in \mathcal{M}}{sup}\mid \sqrt{\left|p\right(\mathfrak{m}\left)\right(s\left(\mathfrak{m}\right)\left)\right|}+\sqrt{\left|q\right(\mathfrak{m}\left)\right(s\left(\mathfrak{m}\right)\left)\right|}.\end{array}$$

In conclusion, we have

$$\begin{array}{cc}\hfill \rho \left(f^{2}p\left(\mathfrak{m}\right),f^{2}q\left(\mathfrak{m}\right)\right)& =\underset{♭\left(\mathfrak{m}\right)\in \mathcal{M}}{sup}\left|f^{2}p\left(\mathfrak{m}\right)-f^{2}q\left(\mathfrak{m}\right)\right|\hfill \\ \hfill & =\underset{♭\left(\mathfrak{m}\right)\in \mathcal{M}}{sup}\left(\right|fp\left(\mathfrak{m}\right)(♭\left(\mathfrak{m}\right))-fq\left(\mathfrak{m}\right)(♭\left(\mathfrak{m}\right))|\times |fp\left(\mathfrak{m}\right)(♭\left(\mathfrak{m}\right))+fq\left(\mathfrak{m}\right)(♭\left(\mathfrak{m}\right))\left|\right)\hfill \\ \hfill & =\underset{♭\left(\mathfrak{m}\right)\in \mathcal{M}}{sup}\left(\right|fp\left(\mathfrak{m}\right)(♭\left(\mathfrak{m}\right))-fq\left(\mathfrak{m}\right)(♭\left(\mathfrak{m}\right))\left|\right)\times \underset{♭\left(\mathfrak{m}\right)\in \mathcal{M}}{sup}\left(\right|fp\left(\mathfrak{m}\right)(♭\left(\mathfrak{m}\right))\hfill \\ & +fq\left(\mathfrak{m}\right)(♭(\mathfrak{m}\left)\right)\left|\right)\hfill \\ \hfill & \le \underset{♭\left(\mathfrak{m}\right)\in \mathcal{M}}{sup}\left(\right|fp(♭\left(\mathfrak{m}\right))-q\left(\mathfrak{m}\right)(♭\left(\mathfrak{m}\right))\left|\right)\times \underset{♭\left(\mathfrak{m}\right)\in \mathcal{M}}{sup}\left(\right|fp\left(\mathfrak{m}\right)(♭\left(\mathfrak{m}\right))|\hfill \\ & +\left|fq\right(\mathfrak{m}\left)\right(♭\left(\mathfrak{m}\right)\left)\right|)\hfill \\ \hfill & \le e^{-\xi \left(\mathfrak{m}\right)}\underset{s\left(\mathfrak{m}\right)\in \mathcal{M}}{sup}\left|\sqrt{\left|p\right(\mathfrak{m}\left)\right(s\left(\mathfrak{m}\right)\left)\right|}-\sqrt{\left|q\right(\mathfrak{m}\left)\right(s\left(\mathfrak{m}\right)\left)\right|}\right|\times \hfill \\ \hfill & \underset{s\left(\mathfrak{m}\right)\in \mathcal{M}}{sup}\sqrt{\left|p\right(\mathfrak{m}\left)\right(s\left(\mathfrak{m}\right)\left)\right|}+\sqrt{\left|q\right(\mathfrak{m}\left)\right(s\left(\mathfrak{m}\right)\left)\right|}\hfill \\ \hfill & =e^{-\xi \left(\mathfrak{m}\right)}\underset{s\left(\mathfrak{m}\right)\in \mathcal{M}}{sup}\left|p\left(s\left(\mathfrak{m}\right)\right)\right|-\left|q\right(\mathfrak{m}\left(s\left(\mathfrak{m}\right)\right)|\mid \hfill \\ \hfill & \le e^{-\zeta \left(\mathfrak{m}\right)}\underset{s\left(\mathfrak{m}\right)\in \mathcal{M}}{sup}\left|p\right(\mathfrak{m}\left(s\left(\mathfrak{m}\right)\right)-q\left(\mathfrak{m}\right)\left(s\left(\mathfrak{m}\right)\right)|\hfill \\ \hfill & =k\left(\mathfrak{m}\right)\rho \left(p\right(\mathfrak{m},q\left(\mathfrak{m}\right)).\hfill \end{array}$$

For $p\left(\mathfrak{m}\right)$ and $q\left(\mathfrak{m}\right)$ belonging to Q and $k\left(\mathfrak{m}\right)=e^{-\xi \left(\mathfrak{m}\right)}$, all conditions are satisfied, affirming the validity of the criteria delineated in Theorem 5, particularly for $n=2$. Consequently, the existence of a unique random fixed point is ensured for f, implying the presence of a unique random solution for the nonlinear Caputo-type FDE in Equation (13). □

7. Application of a Random Solution for the Integral Equation

Theorem 8.

Consider the random integral equation

$$p\left(\mathfrak{m}\right)\left(w\right)={\int }_u^{v}q(w,s,p\left(\mathfrak{m}\right)\left(s\right)ds+g\left(w\right),w\in [u,v].$$

(18)

Suppose that the following are true:

(1) 

$q:[u,v]\times [u,v]\times {\mathbb{R}}^n\to {\mathbb{R}}^n$ and $p:[u,v]\to {\mathbb{R}}^n$;

(2) 

$q(w,s,·):{\mathbb{R}}^n\to {\mathbb{R}}^n$ is increasing for every $w,s\in [u,v]$;

(3) 

There exists a continuous function $k:[u,v]\times [u,v]\to {\mathbb{R}}^{+}$ such that

$$\begin{array}{cc}\hfill & \left(\right|q(w,s,x(\mathfrak{m}\left)\right)-q(w,s,y(\mathfrak{m}\left)\right)|,\alpha |q(w,s,x(\mathfrak{m}\left)\right)-q(w,s),y\left(\mathfrak{m}\right)\left)\right|)\hfill \\ \hfill & ⩽\left(k\right(w,s),rk(w,s\left)\right)\theta \left(\rho \right(x\left(\mathfrak{m}\right),y\left(\mathfrak{m}\right)\left)\right),\hfill \end{array}$$

for every $w,s\in [u,v],x,y\in {\mathbb{R}}^n;$

(4)

${sup}_{w\in [u,v]}{\int }_u^v(k(w,s),\alpha k(w,s))\rho s=1$.

Then, the random integral in Equation (18) has a unique random solution $p^{*}\left(\mathfrak{m}\right)$ in $C\left([u,v]\right)$.

Proof. 

Let $M=Q=C\left([u,v]\right)$, and define

$$\rho (f\left(\mathfrak{m}\right),g\left(\mathfrak{m}\right))=\left({\parallel f\left(\mathfrak{m}\right)-g\left(\mathfrak{m}\right)\parallel }_{+\infty },r\parallel f\left(\mathfrak{m}\right)-\right.\left.{g\left(\mathfrak{m}\right)\parallel }_{+\infty }\right),$$

for every $f\left(\mathfrak{m}\right),g\left(\mathfrak{m}\right)\in Q$. Define a transformation $\phi :p\left(\mathfrak{m}\right)\to {\phi }^{n\left(p\right(\mathfrak{m}\left)\right)}\left(p\left(\mathfrak{m}\right)\right)$ on Q as

$${\phi }^{n\left(p\right(\mathfrak{m}\left)\right)}\left(p\left(\mathfrak{m}\right)\left(w\right)\right):={\int }_u^{v}g\left(w\right)\left(s\right),p\left(\mathfrak{m}\right)\left(s\right))\rho \mathcal{s}+g\left(\mathfrak{m}\right)\left(\mathcal{w}\right),w\in [u,v]$$

(19)

For every $p\left(\mathfrak{m}\right),q\left(\mathfrak{m}\right)\in Q$, we have

$$\begin{array}{cc}\hfill (\mid {\phi }^{n\left(p\right(\mathfrak{m}\left)\right)}& p\left(\mathfrak{m}\right)\left(w\right)-{\phi }^{n\left(p\right(\mathfrak{m}\left)\right)}q\left(\mathfrak{m}\right)\left(w\right)|,r|{\phi }^{n\left(p\right(\mathfrak{m}\left)\right)}p\left(\mathfrak{m}\right)\left(w\right)-{\phi }^{n\left(p\right(\mathfrak{m}\left)\right)}q\left(\mathfrak{m}\right)\left(w\right)\mid )\hfill \\ \hfill & =\left(\left|{\int }_u^v[q(w,s,p\left(\mathfrak{m}\right)\left(s\right))-q(w,s,q\left(\mathfrak{m}\right)\left(s\right))]ds\right|,r\left|{\int }_u^v[q(w,s,p\left(\mathfrak{m}\right)\left(s\right))-q(w,s,q\left(\mathfrak{m}\right)\left(s\right))]ds\right|\right)\hfill \\ \hfill & ⩽\left({\int }_u^v|q(w,s,p\left(\mathfrak{m}\right)\left(s\right))-q(w,s,q\left(\mathfrak{m}\right)\left(s\right))|ds,{\int }_u^{v}rq(w,s,p\left(\mathfrak{m}\right)\left(s\right))-q(w,s,q\left(\mathfrak{m}\right)\left(s\right))|ds\right)\hfill \\ \hfill & ⩽{\int }_u^v(k(w,s),rk(w,s))\theta \left(\right|p\left(\mathfrak{m}\right)\left(s\right)-q\left(\mathfrak{m}\right)\left(s\right)|,r|p\left(\mathfrak{m}\right)\left(s\right)-q\left(\mathfrak{m}\right)\left(s\right)\left|\right)ds\hfill \\ \hfill & ⩽\theta \left({\parallel p\left(\mathfrak{m}\right)-q\left(\mathfrak{m}\right)\parallel }_{+\infty },r{\parallel p\left(\mathfrak{m}\right)-q\left(\mathfrak{m}\right)\parallel }_{+\infty }\right){\int }_u^v(k(w,s),rk(w,s))ds\hfill \\ \hfill & =\theta \left({\parallel p\left(\mathfrak{m}\right)-q\left(\mathfrak{m}\right)\parallel }_{+\infty },r{\parallel p\left(\mathfrak{m}\right)-q\left(\mathfrak{m}\right)\parallel }_{+\infty }\right).\hfill \end{array}$$

Hence, $\rho ({\phi }^{n\left(p\right(\mathfrak{m}\left)\right)}p\left(\mathfrak{m}\right),{\phi }^{n\left(p\right(\mathfrak{m}\left)\right)}q\left(\mathfrak{m}\right))⩽\theta \left(\rho (p\left(\mathfrak{m}\right),q\left(\mathfrak{m}\right))\right)$ for every $p\left(\mathfrak{m}\right),q\left(\mathfrak{m}\right)\in Q$. The conclusion follows now from Theorem 5. □

8. Conclusions

This study established random Sehgal contractions to be applied to randomly generated operators within a complete RCMS as a superior approach compared with traditional Sehgal and Banach contractions in metric spaces. The strength of the current work on random Sehgal contractions in an RCMS is its ability to handle uncertainty and randomness more effectively than traditional Sehgal and Banach contractions in metric spaces. The tailored framework of random Sehgal contractions offers accelerated convergence properties, which are particularly evident in solving nonlinear random FDEs and random BVPs. This innovative methodology outshines the existing literature by providing unique solutions and faster convergence rates, solidifying its position as a leading and powerful technique in modern fixed-point theory within stochastic environments. Our theoretical insights are reinforced by a detailed example, offering empirical validation for our main results.