Analysis of Existence for Fractional Random Differential Equations with Bounded Delay in Fréchet Spaces

Abstract

This research explores the existence of solutions for a class of random fractional differential equations characterized by bounded delay, specifically within the context of Fréchet spaces. Random fractional differential equations serve as powerful mathematical tools for modeling complex phenomena subjected to stochastic perturbations and hereditary effects. Despite their significance, establishing solution existence in infinite-dimensional spaces remains a challenging task. By integrating the properties of the noncompactness measures with a generalized Darbo fixed point approach, we establish new existence results for the associated Darboux-type problem under milder compactness conditions. To illustrate the practical utility of these analytical results and demonstrate the validity of our theoretical framework, a representative example is provided.

1. Introduction

Over the last few decades, fractional calculus and its application to differential equations of non-integer order have emerged as focal points for researchers in both theoretical mathematics and diverse applied fields. The utility of fractional operators is particularly evident in modeling complex phenomena, such as electromagnetic fields, electrochemical processes, and the dynamics of viscoelasticity materials [1,2,3,4,5]. A robust body of literature now exists on both partial and ordinary fractional systems, with foundational theories detailed in comprehensive monographs [6,7] and further expanded in recent scholarly works [8,9,10].

In parallel, a substantial body of work has been devoted to the random differential and integral equations developed in [11,12], providing suitable mathematical frameworks to model uncertainty in scientific and engineering problems. Since real-world phenomena often involve randomness, it is natural to combine stochastic effects with fractional dynamics. For this reason, fractional differential equations with random parameters have become an active area of research; see, for example, refs. [13,14].

Random fractional differential equations were studied in particular settings by Lupulescu and Ntouyas [15], where the problem is treated in an appropriate Banach space framework. Further results obtained under assumptions of type Carathéodory in [16,17,18,19]. These works highlight the importance of combining fractional operators with stochastic analysis.

Beyond traditional initial values, nonlocal conditions offer a more comprehensive approach by integrating data across a temporal interval. This perspective is frequently more grounded in physical reality as it mitigates the impact of localized measurement inaccuracies at the starting point. The formal investigation of these abstract nonlocal frameworks was pioneered by Byszewski [20,21], sparking a significant volume of research into differential equations under such constraints [22]. In fields like elasticity theory, these conditions are particularly valuable, offering a higher degree of modeling precision than that provided by initial conditions.

Our research is centered on proving the existence of random solutions for fractional-order systems that incorporate finite delay, specifically within the framework of random differential equations:

$$\left({}^{C}D_0^{ϵ}u\right)(t,x,w)=f(t,x,u_{(t,x)},w),(t,x)\in \mathcal{J},w\in \Omega ,$$

(1)

$$u(t,x,w)=\varphi (t,x,w),(t,x)\in \tilde{\mathcal{J}},w\in \Omega ,$$

(2)

$$\left\{\begin{array}{cc}u(t,0,w)=\phi (t,w),\hfill & t\in [0,\infty ),\hfill \\ u(0,x,w)=\psi (x,w),\hfill & x\in [0,\infty ),\hfill \end{array}\right.w\in \Omega ,$$

(3)

where $\mathcal{J}:=[0,\infty )\times [0,\infty ),\tilde{\mathcal{J}}:=[-\alpha ,\infty )\times [-\beta ,\infty )\setminus [0,\infty )\times [0,\infty )$, and ${}^{C}D_0^{ϵ}$ denotes the fractional Caputo derivative of order $ϵ=({ϵ}_1,{ϵ}_2)\in (0,1]\times (0,1]$. The measurable space is $(\Omega ,\mathcal{A},\nu )$, where $\mathcal{A}$ is a $\sigma$-algebra of subsets of $\Omega$ and $\nu$ is a $\sigma$-finite measure (or probability measure) defined on $\mathcal{A}$, $f:\mathcal{J}\times {\mathcal{C}}_0\times \Omega \to \mathcal{E}$ is a given function, while $\varphi :\tilde{\mathcal{J}}\times \Omega \to \mathcal{E}$ is continuous. Here $(\mathcal{E},||·|{|}_{\mathcal{E}})$ is a real Banach space of state vectors and $\phi ,\psi :[0,\infty )\times \Omega \to \mathcal{E}$ are absolutely continuous functions such that $\phi (t,·)$ and $\psi (x,·)$ are measurable for all $(t,x)\in \mathcal{J}$.

The history function $u_{(t,x)}\in {\mathcal{C}}_0=\mathcal{C}([-\alpha ,0]\times [-\beta ,0],\mathcal{E})$ is defined by

$$u_{(t,x)}(s,\tau ,w)=u(t+s,x+\tau ,w);(s,\tau )\in [-\alpha ,0]\times [-\beta ,0],w\in \Omega$$

which represents the history of the state from time $t-\alpha$ to the present time t and from position $x-\beta$ to the present position x.

We also consider a related system involving nonlocal conditions of the form

$$\left({}^{C}D_0^{ϵ}u\right)(t,x,w)=f(t,x,u_{(t,x)},w),(t,x)\in \mathcal{J},w\in \Omega ,$$

(4)

$$u(t,x,w)=\varphi (t,x,w),(t,x)\in \tilde{\mathcal{J}},w\in \Omega ,$$

(5)

$$\left\{\begin{array}{cc}u(t,0,w)+Q\left(u\right)=\phi (t,w),\hfill & t\in [0,\infty ),\hfill \\ \mathit{u}(\mathit{0},\mathit{x},\mathit{w})+\mathit{K}\left(\mathit{u}\right)=\psi (x,w),\hfill & x\in [0,\infty ),\hfill \end{array}\right.w\in \Omega ,$$

(6)

where $Q,K:\mathcal{C}(\mathcal{J},\mathcal{E})\to \mathcal{E}$ represent given continuous nonlocal operators defined on the functional domain over $\mathcal{J}$.

The focus of this research is the solvability of the Darboux-type problem within the context of fractional-order random differential equations in Fréchet spaces. The primary motivation for investigating this specific framework arises from the necessity of modeling complex physical and biological systems that inherently possess memory effects and are subject to stochastic perturbations. While fractional derivatives excel at capturing historical memory and hereditary properties, the inclusion of random variables allows for the realistic depiction of environmental fluctuations. Furthermore, analyzing these systems within Fréchet spaces, rather than traditional Banach spaces, offers a more flexible topological framework to handle functions defined on unbounded domains without the constraints of rigid norm requirements.

However, establishing solution existence in these infinite-dimensional spaces often encounters difficulties due to a lack of compactness. We provide rigorous proofs for both the existence of solutions by employing a generalized version of Darbo’s fixed point theorem. This analytical approach is further supported by the application of the noncompactness measures [23], which effectively relaxes the strict compactness criteria typically demanded in the existing literature, thereby broadening the theoretical applicability of the Darboux problem.

2. Background Materials

This section collects the notation, basic definitions, and preliminary results required in the subsequent analysis.

Consider the domain ${\mathcal{J}}_n=[0,n]\times [0,n]$ for a given $n>0.$ We classify a measurable function $u:{\mathcal{J}}_n\to \mathcal{E}$ as Bochner integrable provided that its norm $\left|\right|u\left|\right|$ is integrable in the Lebesgue sense. A comprehensive treatment of Bochner integration can be found in the work of Yosida [24].

The space of such integrable functions is denoted by ${\mathcal{L}}^1(J_n,\mathcal{E})$, which forms a Banach space under the norm:

$${\parallel u\parallel }_{{\mathcal{L}}^1}={\int }_0^n{\int }_0^n{\parallel u(t,x)\parallel }_{\mathcal{E}}dtdx.$$

Similarly, ${\mathcal{L}}^{\infty }({\mathcal{J}}_n,\mathcal{E})$ is a collection of functions that are essentially bounded. The norm in this space ${\parallel u\parallel }_{\infty }$ is defined as the infimum of all essential bounds $c>0$ that satisfy the inequality $\parallel u(t,x)\parallel \le c$ almost everywhere $(t,x)\in {\mathcal{J}}_n$. Next, we recall some basic concepts from fractional calculus.

Definition 1

([25]). Let $ϵ=({ϵ}_1,{ϵ}_2)\in {(0,\infty )}^2,\theta =(0,0)$. For $u\in {\mathcal{L}}^1({\mathcal{J}}_n,{\mathbb{R}}^n).$ The mixed left-sided Riemann–Liouville integral of order ϵ is defined by

$$\left(I_{\theta }^{ϵ}u\right)(t,x)={\displaystyle \frac{1}{\Gamma \left({ϵ}_1\right)\Gamma \left({ϵ}_2\right)}}{\int }_0^t{\int }_0^x{(t-s)}^{{ϵ}_1-1}{(x-\tau )}^{{ϵ}_2-1}u(s,\tau )d\tau ds.$$

Definition 2

([25]). For $ϵ=({ϵ}_1,{ϵ}_2)\in {(0,1]}^2,\theta =(0,0)$ let $u:{\mathcal{J}}_n\to \mathcal{E}$ be a sufficiently smooth function such that its mixed partial derivative ${\displaystyle \frac{{\partial }^{2}u}{\partial t\partial x}}$ exists and belongs to ${\mathcal{L}}^1({\mathcal{J}}_n,\mathcal{E})$. The Caputo fractional derivative of order ϵ of u is given by

$$\left({}^{C}D_{\theta }^{ϵ}u\right)(t,x)=\left(I_{\theta }^{1-ϵ}{\displaystyle \frac{{\partial }^2}{\partial t\partial x}}u\right)(t,x).$$

In the following, we introduce the main concepts that will be used in the sequel probability and random operators.

Consider the Borel $\sigma$-algebra ${\beta }_{\mathcal{E}}$ associated with the space $\mathcal{E}.$ We define $v:\Omega \to \mathcal{E}$ as being measurable provided that the pre-image of every Borel set $(B\in {\beta }_{\mathcal{E}})$ is an element of the $\sigma$-algebra $\mathcal{A}$, expressed as

$$v^{-1}\left(B\right)=\{w\in \Omega :v\left(w\right)\in B\}\subset \mathcal{A}.$$

Developing a theory for the integration of random sample paths requires the process to satisfy joint measurability conditions.

Definition 3

([26]). Let $\mathcal{A}\times {\beta }_{\mathcal{E}}$ be the product σ-algebras on $\Omega \times \mathcal{E}$. $T:\Omega \times \mathcal{E}\to \mathcal{E}$ is defined as jointly measurable if, for each $B\in {\beta }_{\mathcal{E}},$ the inverse image $T^{-1}\left(B\right)$ is measurable with respect to this product structure. This implies:

$$T^{-1}\left(B\right)=\{(w,v)\in \Omega \times \mathcal{E}:T(w,v)\in B\}\subset \mathcal{A}\times {\beta }_{\mathcal{E}}.$$

Definition 4

([26]). We consider a set-valued mapping $\mathcal{C}:\Omega \to \mathcal{P}\left(\mathcal{Y}\right)$, where $\mathcal{P}\left(\mathcal{Y}\right)$ represents the collection of all nonempty subsets of $\mathcal{Y}$, if $\mathcal{C}$ is measurable,

$$T:\left\{\right(w,y):w\in \Omega ,y\in \mathcal{C}(w\left)\right\}⟶\mathcal{Y}$$

is identified as a random operator with stochastic domain $\mathcal{C}$. Furthermore, T is continuous if $T\left(w\right)$: $\mathcal{C}\left(w\right)\to \mathcal{Y}$ remains continuous for each fixed $w\in \Omega$. A random (stochastic) fixed point of this operator is a measurable function $y:\Omega \to \mathcal{Y}$ that satisfies $y\left(w\right)\in \mathcal{C}\left(w\right)$ and $T\left(w\right)y\left(w\right)=y\left(w\right)$ for almost all $w\in \Omega$. Additionally, the measurability of the set $\{w\in \Omega :y(w)\in \mathcal{D}\}$ must be maintained for any open subset $\mathcal{D}$ of $\mathcal{Y}$.

For a more comprehensive treatment of completely continuous random operators within the framework of Banach spaces, the reader is directed to the work of Itoh [26].

To facilitate our analysis of existence, we adopt the concept of a sequence of noncompactness measures as established in [27,28].

Definition 5.

In the context of a Fréchet space $\mathcal{X}$, a function α on bounded subsets ${\mathcal{M}}_{\mathcal{X}}$ is considered a noncompactness measure if it satisfies the following:

($c_1$)

$\alpha \left(B\right)=0⟺B$ is relatively compact

($c_2$)

Monotonicity: $B_1\subseteq B_2\Rightarrow \alpha \left(B_1\right)\le \alpha \left(B_2\right)$.

($c_3$)

The measure is invariant under the convex hull operator.

($c_4$)

The intersection of a nested sequence of closed sets is nonempty provided their measures tend toward zero.

Additional properties and a further discussion on the noncompactness measures may be found in [29].

Lemma 1

([30]). In a Fréchet space $\mathcal{X}$, any bounded subset $\mathcal{Y}$ and any positive ε, the following inequality holds for some sequence ${\left\{y_k\right\}}_{k=1}^{\infty }$ in $\mathcal{Y}$:

$$\alpha \left(\mathcal{Y}\right)\le 2\alpha \left({\left\{y_k\right\}}_{k=1}^{\infty }\right)+\epsilon .$$

Lemma 2

([31]). For any sequence ${\left\{u_k\right\}}_{k=1}^{\infty }$ that is uniformly integrable within ${\mathcal{L}}^1\left({\mathcal{J}}_n\right)$, the measure α applied to the sequence remains a measurable function. Specifically, for any $(t,x)\in {\mathcal{J}}_n$, the integration operator commutes with the measure of noncompactness according to the following estimate:

$$\alpha \left({\left\{{\int }_0^t{\int }_0^x{u}_k(s,\tau )d\tau ds\right\}}_{k=1}^{\infty }\right)\le 2{\int }_0^t{\int }_0^x\alpha \left({\left\{u_k(s,\tau )\right\}}_{k=1}^{\infty }\right)d\tau \mathrm{d}s.$$

Definition 6

([32]). Suppose Ω is a nonempty subset of a Fréchet space $\mathcal{X}$. A continuous and bounded operator $A:\Omega \to \mathcal{X}$ is said to satisfy a Darbo-type requirement relative to a noncompactness measure family α and a sequence ${\left(k_n\right)}_{n\in \mathbb{N}}$, if for any bounded $B\subseteq \Omega$, the following inequality is preserved:

$$\alpha \left(A\left(B\right)\right)\le k_n\alpha \left(B\right).$$

In the case where $k_n<1$ for every $n\in \mathbb{N}$, the operator A is classified as a α-contraction.

Now, we state a generalized random version of Darbo’s fixed point theorem adapted to Fréchet spaces.

Theorem 1

([33]). Let $\mathcal{X}$ be a separable Fréchet space containing a nonempty, closed, convex, and bounded subset Λ. Let $(\Omega ,\mathcal{A},\nu )$ be a probability space, and let $V:\Omega \times \Lambda \to \Lambda$ be a random operator such that for each $u\in \Lambda$, the mapping $w\mapsto V\left(w\right)u$ is measurable. Suppose that for ν-almost all $w\in \Omega$, the deterministic operator $\left(V\right(w\left)\right)(·)$ is continuous and satisfies the contraction condition:

$$\alpha \left(V\right(w\left)\right)\left(B\right)\le k\left(w\right)\alpha \left(B\right)$$

for any bounded subset $B\subset \Lambda$ and for each $n\in \mathbb{N}$, where $0\le k_n\left(w\right)<1$. Under these hypotheses, the operator V possesses at least one random fixed point, i.e., there exists a measurable mapping $u^{*}:\Omega \to \Lambda$ such that $V\left(w\right)\left(u^{*}\left(w\right)\right)=u^{*}\left(w\right)$ holds ν-almost surely.

3. Analysis of Existence Results

For every $n\in \mathbb{N}$, let ${\mathcal{C}}_n$ denote the Banach space of continuous functions $\mathcal{C}\left(\right[-\alpha ,n]\times [-\beta ,n],\mathcal{E})$. We define the space ${\mathcal{C}}_{\infty }$ over the unbounded domain $[-\alpha ,\infty )\times [-\beta ,\infty )$ and equip it with a topology generated by the sequence of seminorms

$${\parallel u\parallel }_n={\{sup\parallel u(t,x)\parallel }_{\mathcal{E}}:-\alpha \le t\le n,-\beta \le x\le n\}.$$

With this family, ${\mathcal{C}}_{\infty }$ becomes a Fréchet space.

To ensure absolute topological clarity regarding our compactness arguments, we explicitly emphasize the structural hierarchy of these spaces. While the trajectory space ${\mathcal{C}}_{\infty }$ carries a Fréchet topology, the measure of noncompactness $\alpha$ used in our assumptions operates strictly on the bounded subsets of the underlying component Banach space $\mathcal{E}$. The noncompactness profiles of subsets within the global space ${\mathcal{C}}_{\infty }$ are systematically characterized through their localized projections onto the sequence of Banach spaces ${\mathcal{C}}_n$, guaranteeing mathematical rigor across all fixed-point constructions.

We proceed by formulating the precise criteria that characterize a random solution for the fractional system (1)–(3).

Definition 7.

By a random solution of the stochastic problem (1)–(3), we mean a measurable process u taking values in ${\mathcal{C}}_{\infty }$ that adheres to the governing Equations (1) and (3) for almost all realizations $(t,x,w)\in \mathcal{J}\times \Omega$. Furthermore, the function must satisfy the requirement (2) for every point in $\tilde{\mathcal{J}}$.

We recall a useful result for the associated linear problem.

Lemma 3

([34]). For any h integrable function in $\mathcal{J}.$ The unique solution u for the linear fractional problem

$$\left\{\begin{array}{c}{}^{C}D_0^{ϵ}u(t,x)=h(t,x);fora.a.(t,x)\in \mathcal{J},\hfill \\ u(t,0)=\phi \left(t\right),\hfill \\ u(0,x)=\psi \left(x\right),\hfill \\ \phi \left(0\right)=\psi \left(0\right),\hfill \end{array}\right.(t,x)\in \mathcal{J}$$

given by

$$u(t,x)=z(t,x)+I_{\theta }^{ϵ}h(t,x);fora.a.(t,x)\in \mathcal{J},$$

with

$$z(t,x)=\phi \left(t\right)+\psi \left(x\right)-\phi \left(0\right).$$

Under the hypothesis that f satisfies random Carathéodory conditions in the product space $\mathcal{J}\times {\mathcal{C}}_{\infty }\times \Omega$, the previous lemma allows us to express the solution in the following form:

Lemma 4.

Let $0<{ϵ}_1,{ϵ}_2\le 1$. Any solution $u\in \Omega \times {\mathcal{C}}_{\infty }$ to the random problem $\left(1\right)$–$\left(3\right)$ provided $\left(2\right)$ is satisfied for $(t,x)\in \tilde{\mathcal{J}}$, can be explicitly represented for almost every $w\in \Omega$ by the integral equation:

$$u(t,x,w)=z(t,x,w)+{\displaystyle \frac{1}{\Gamma \left({ϵ}_1\right)\Gamma \left({ϵ}_2\right)}}{\int }_0^t{\int }_0^x{(t-s)}^{{ϵ}_1-1}{(x-\tau )}^{{ϵ}_2-1}f(s,\tau ,u_{(s,\tau )},w)\mathrm{d}\tau \mathrm{d}s,$$

such that

$$z(t,x,w)=\phi (t,w)+\psi (x,w)-\phi (0,w).$$

We impose the following assumptions:

($A1)$

$w\mapsto \phi (t,0,w)$ and $w\mapsto \psi (0,x,w)$ are assumed to be bounded and measurable for almost all $(t,x)\in {\mathcal{J}}_n$.

$\left(A2\right)$

f satisfies the random Carathéodory conditions on the product domain ${\mathcal{J}}_n\times {\mathcal{C}}_0\times \Omega .$

$\left(A3\right)$

f is subject to a growth constraint defined by functions $p_1,p_2\in L^{\infty }(\Omega ,\mathcal{C}({\mathcal{J}}_n,[0,\infty )))$. Specifically, for any $u\in {\mathcal{C}}_0$ and $w\in \Omega$:

$${\parallel f(t,x,u,w)\parallel }_{\mathcal{E}}\le p_1(t,x,w)+p_2(t,x,w){\parallel u\parallel }_{{\mathcal{C}}_0}.$$

$\left(A4\right)$

For any bounded collection $B\subset {\mathcal{C}}_0$, f satisfies the noncompactness inequality:

$${\alpha }_n\left(f(t,x,B,w)\right)\le p_2(t,x,w){\alpha }_n\left(B\right),$$

for $\nu$-almost every realization of $w\in \Omega$, where ${\alpha }_n$ denotes the components of the measure of noncompactness family relative to the space $\mathcal{E}$.

$\left(A5\right)$

There exists a continuous random radius function $R_n:\Omega \to (0,\infty )$ such that the following algebraic inequality holds $\nu$-almost surely:

$$z^{*}\left(w\right)+{\displaystyle \frac{[p_{1n}^{*}+p_{2n}^{*}{R}_n\left(w\right)]n^{{ϵ}_1+{ϵ}_2}}{\Gamma ({ϵ}_1+1)\Gamma ({ϵ}_2+1)}}\le R_n\left(w\right),$$

where the localized boundary supremum and essential growth parameters are given by:

$${\displaystyle z^{*}\left(w\right)=\underset{(t,x)\in {\mathcal{J}}_n}{sup}{\parallel z(t,x,w)\parallel }_{\mathcal{E}},p_{in}^{*}=\underset{(t,x)\in {\mathcal{J}}_n}{sup}ess{p}_i(t,x,w),i=1,2.}$$

Theorem 2.

Assume assumptions (A1)–(A5). If

$${\mathsf{\ell }}_n:={\displaystyle \frac{4p_{2n}^{*}{n}^{{ϵ}_1+{ϵ}_2}}{\Gamma ({ϵ}_1+1)\Gamma ({ϵ}_2+1)}}<1,$$

then the random fractional problem $\left(1\right)$–$\left(3\right)$ admits at least one stochastic solution in ${\mathcal{C}}_{\infty }.$

Proof. 

For each $w\in \Omega$, let us construct the random operator $\mathcal{N}:\Omega \times {\mathcal{C}}_{\infty }\to {\mathcal{C}}_{\infty }$ as follows:

$$\left(\mathcal{N}\left(w\right)u\right)(t,x)=\left\{\begin{array}{cc}\varphi (t,x,w),\hfill & (t,x)\in \tilde{\mathcal{J}}\hfill \\ z(t,x,w)\hfill & \\ {\displaystyle +\frac{1}{\Gamma \left({ϵ}_1\right)\Gamma \left({ϵ}_2\right)}{\int }_0^t{\int }_0^x{(t-s)}^{{ϵ}_1-1}{(x-\tau )}^{{ϵ}_2-1}}\hfill & \\ \times f(s,\tau ,u_{(s,\tau )},w)d\tau ds,\hfill & (t,x)\in \mathcal{J}.\hfill \end{array}\right.$$

(7)

From the absolute continuity of $\phi ,\psi$ and f, it follows that z as well as the corresponding fractional integral are absolutely continuous for every $w\in \Omega$ and almost all $(t,x)\in \mathcal{J}.$ Moreover, the continuity of z with the continuity of the fractional integral operator in $\mathcal{J}$ ensures that $\mathcal{N}\left(w\right)$ is a well defined from $\Omega \times {\mathcal{C}}_{\infty }$ into ${\mathcal{C}}_{\infty }$. By virtue of this construction, finding a random solution u to the problem (1)–(3) is equivalent to identifying a fixed point of the random operator equation $u=\mathcal{N}\left(w\right)u$. To conclude the existence of such a point, we proceed to demonstrate that $\mathcal{N}$ satisfies all the criteria set in the Random Darbo Fixed Point Theorem 1.

The argument is organized into the following detailed steps.

▹ Our first step is to verify that $\mathcal{N}\left(w\right)$ is a random operator with stochastic domain in ${\mathcal{C}}_{\infty }.$

According to the random Carathéodory property of f (Assumption (A2)), $w\to f(t,x,u,w)$ is measurable for any fixed $(t,x,u)$ according to the Definition 3. Since ${(t-s)}^{{ϵ}_1-1}$${(x-\tau )}^{{ϵ}_2-1}f(s,\tau ,u_{(s,\tau )},w)$ is the product of a continuous function and a measurable one, it retains product measurability. Furthermore, because the fractional integral can be viewed as the limit of a bounded sum of measurable functions, therefore,

$$w\mapsto z(t,x,w)+{\displaystyle \frac{1}{\Gamma \left({ϵ}_1\right)\Gamma \left({ϵ}_2\right)}}{\int }_0^t{\int }_0^x{(t-s)}^{{ϵ}_1-1}{(x-\tau )}^{{ϵ}_2-1}f(s,\tau ,u_{(s,\tau )},w)d\tau ds,$$

is again measurable. Hence, $\mathcal{N}$ qualifies as a random operator from $\Omega \times {\mathcal{C}}_{\infty }$ into ${\mathcal{C}}_{\infty }.$

Let us define a set-valued $W:\Omega \to \mathcal{P}\left({\mathcal{C}}_{\infty }\right)$ representing the stochastic closed balls:

$$W\left(w\right)=\{u\in {\mathcal{C}}_{\infty }:\parallel u{\parallel }_n\le R_n\left(w\right),\mathrm{for}\mathrm{each}n\in \mathbb{N}\}.$$

For every $w\in \Omega$ the set $W\left(w\right)$ is clearly closed, convex, and bounded. Its measurability as a random set is a direct consequence of Lemma 17 in [35]. For a fixed $w\in \Omega$ and any $u\in W\left(w\right)$, we evaluate the growth on an arbitrary compact sub-rectangle ${\mathcal{J}}_n$. Applying our growth constraint (Assumption (A3)), we obtain

$${\parallel \left(\mathcal{N}\left(w\right)u\right)(t,x)\parallel }_{\mathcal{E}}\le$$

$$\begin{array}{ccc}& & \le {\parallel z(t,x,w)\parallel }_{\mathcal{E}}+{\int }_0^t{\int }_0^x{\displaystyle \frac{{(t-s)}^{{ϵ}_1-1}{(x-\tau )}^{{ϵ}_2-1}}{\Gamma \left({ϵ}_1\right)\Gamma \left({ϵ}_2\right)}}{\parallel f(s,\tau ,u_{(s,\tau )},w)\parallel }_{\mathcal{E}}d\tau ds\hfill \\ & & \le {\parallel z(t,x,w)\parallel }_{\mathcal{E}}+{\displaystyle \frac{1}{\Gamma \left({ϵ}_1\right)\Gamma \left({ϵ}_2\right)}}{\int }_0^t{\int }_0^x{(t-s)}^{{ϵ}_1-1}{(x-\tau )}^{{ϵ}_2-1}{p}_1(s,\tau ,w)d\tau ds\hfill \\ & & +{\displaystyle \frac{1}{\Gamma \left({ϵ}_1\right)\Gamma \left({ϵ}_2\right)}}{\int }_0^t{\int }_0^x{(t-s)}^{{ϵ}_1-1}{(x-\tau )}^{{ϵ}_2-1}{p}_2(s,\tau ,w){\parallel u_{(s,\tau )}\parallel }_{{\mathcal{C}}_0}d\tau ds\hfill \\ & & \le z^{*}\left(w\right)+{\displaystyle \frac{p_{1n}^{*}}{\Gamma \left({ϵ}_1\right)\Gamma \left({ϵ}_2\right)}}{\int }_0^t{\int }_0^x{(t-s)}^{{ϵ}_1-1}{(x-\tau )}^{{ϵ}_2-1}d\tau ds\hfill \\ & & +{\displaystyle \frac{p_{2n}^{*}{R}_n\left(w\right)}{\Gamma \left({ϵ}_1\right)\Gamma \left({ϵ}_2\right)}}{\int }_0^t{\int }_0^x{(t-s)}^{{ϵ}_1-1}{(x-\tau )}^{{ϵ}_2-1}d\tau ds\hfill \\ & & \le z^{*}\left(w\right)+{\displaystyle \frac{[p_{1n}^{*}+p_{2n}^{*}{R}_n\left(w\right)]n^{{ϵ}_1+{ϵ}_2}}{\Gamma ({ϵ}_1+1)\Gamma ({ϵ}_2+1)}}\hfill \\ & & \le R_n\left(w\right).\hfill \end{array}$$

Taking the supremum over $(t,x)\in {\mathcal{J}}_n$, this confirms that ${\parallel \mathcal{N}\left(w\right)u\parallel }_n\le R_n\left(w\right)$$\mathcal{N}\left(w\right)$. Thus, $\mathcal{N}\left(w\right)\left(W\right(w\left)\right)\subseteq W\left(w\right)$. This proves that $\mathcal{N}$ is a random operator with a well-defined stochastic domain W, which maps bounded subsets of ${\mathcal{C}}_{\infty }$ into bounded ones.

▹ Next, we prove that $\mathcal{N}\left(w\right)$ is continuous.

• Let ${\left\{u_m\right\}}_{m\in \mathbb{N}}$ be a sequence in the Fréchet space ${\mathcal{C}}_{\infty }$ such that $u_m\to u$ as $m\to \infty$ under the topology of the space. For an arbitrary fixed $w\in \Omega$ and any compact sub-rectangle ${\mathcal{J}}_n$, it follows directly from the properties of Fréchet topologies that the corresponding local history segments satisfy uniform convergence in the delay norm:

$$\parallel {\left(u_m\right)}_{(s,\tau )}-u_{(s,\tau )}{\parallel }_{{\mathcal{C}}_0}\to 0\mathrm{as}m\to \infty ,$$

(8)

for each $(s,\tau )\in {\mathcal{J}}_n$. Evaluating the difference between the operator images for any point $(t,x)\in {\mathcal{J}}_n$, we observe that the initial history profiles on $\tilde{\mathcal{J}}$ cancel out perfectly. Thus, by applying the integral formulation $\left(7\right)$, we obtain the following inequality:

$$\begin{array}{ccc}& & \parallel \left(\mathcal{N}\left(w\right)u_m\right)(t,x)-\left(\mathcal{N}\left(w\right)u\right){(t,x)\parallel }_{\mathcal{E}}\hfill \\ & \le & {\displaystyle \frac{1}{\Gamma \left({ϵ}_1\right)\Gamma \left({ϵ}_2\right)}}{\int }_0^t{\int }_0^x{(t-s)}^{{ϵ}_1-1}{(x-\tau )}^{{ϵ}_2-1}{\parallel f(s,\tau ,{\left(u_m\right)}_{(s,t)},w)-f(s,\tau ,u_{(s,\tau )},w)\parallel }_{\mathcal{E}}d\tau ds.\hfill \end{array}$$

Since f satisfies the random Carathéodory conditions (Assumption (A2)), it is continuous with respect to its third argument. Combining this with the limit established in $\left(8\right)$, we achieve pointwise convergence almost everywhere:

$$\parallel f(s,\tau ,{\left(u_m\right)}_{(s,t)},w)-f(s,\tau ,u_{(s,\tau )},w){\parallel }_{\mathcal{E}}\to 0\mathrm{as}m\to \infty ,$$

(9)

for almost every $(s,\tau )\in {\mathcal{J}}_n$.

Furthermore, the sequence $\left\{u_m\right\}$ belongs to the bounded stochastic ball $W\left(w\right)$ with a continuous radius $R_n\left(w\right)$, meaning $\parallel {\left(u_m\right)}_{(s,t)}{\parallel }_{{\mathcal{C}}_0}\le R_n\left(w\right)$ for all $m\in \mathbb{N}$.

In light of the integrable growth condition (Assumption (A3)), the following uniform bounds are preserved for almost all $(s,\tau )\in {\mathcal{J}}_n$:

$$\parallel f(s,\tau ,{\left(u_m\right)}_{(s,t)},w){\parallel }_{\mathcal{E}} \le p_1(s,\tau ,w)+p_2(s,\tau ,w)R_n\left(w\right)$$

and

$$\parallel f(s,\tau ,u_{(s,\tau )},w){\parallel }_{\mathcal{E}} \le p_1(s,\tau ,w)+p_2(s,\tau ,w)R_n\left(w\right).$$

This guarantees that the integrand is uniformly bounded by a sequence-independent, integrable dominating function:

$$\parallel f(s,\tau ,{\left(u_m\right)}_{(s,t)},w)-f(s,\tau ,u_{(s,\tau )},w){\parallel }_{\mathcal{E}}\le 2\left[p_1(s,\tau ,w)+p_2(s,\tau ,w)R_n\left(w\right)\right].$$

Since $p_1,p_2\in L^{\infty }(\Omega ,\mathcal{C}({\mathcal{J}}_n,[0,\infty )))$, this dominant expression is bounded and integrable on ${\mathcal{J}}_n$. Therefore, by the Lebesgue Dominated Convergence Theorem, we can legally pass the limit inside the double integral operator, causing the integral term to vanish as $m\to \infty$. Consequently, taking the supremum over the compact sub-rectangle ${\mathcal{J}}_n$, we obtain the semi-norm convergence:

$$\parallel \mathcal{N}\left(w\right)u_m{-\mathcal{N}\left(w\right)u\parallel }_n\to 0\mathrm{as}m\to \infty .$$

Since this convergence holds true for each compact domain index $n\in \mathbb{N}$, we conclude that $\mathcal{N}\left(w\right)$ is a sequentially continuous random operator on $W\left(w\right)$ under the topology of the Fréchet space ${\mathcal{C}}_{\infty }$.

▹ Finally, it remains to be proven that the operator $\mathcal{N}$ acts as a contraction with respect to the measure of noncompactness ${\alpha }_n$. Specifically, for any bounded subset $B\subset W\left(w\right)$ and a fixed $w\in \Omega$, we must verify that ${\alpha }_n\left(\mathcal{N}\left(w\right)B\right)\le {\mathsf{\ell }}_n{\alpha }_n\left(B\right)$.

Applying the properties of the measure ${\alpha }_n$ as established in Lemmas 1 and 2, for an arbitrary $\epsilon >0$, we can find a sequence ${\left\{u_m\right\}}_{m=0}^{\infty }\subset B,$ such that for each $(t,x)\in {\mathcal{J}}_n$, the following chain of inequalities is satisfied:

$${\alpha }_n\left(\left(\mathcal{N}\left(w\right)B\right)(t,x)\right)=$$

$$\begin{array}{ccc}& =& {\alpha }_n\left(\left\{z(t,x)+{\int }_0^t{\int }_0^x{\displaystyle \frac{{(t-s)}^{{ϵ}_1-1}{(x-\tau )}^{{ϵ}_2-1}}{\Gamma \left({ϵ}_1\right)\Gamma \left({ϵ}_2\right)}}f(s,\tau ,u_{(s,\tau )},w)d\tau ds;u\in B\right\}\right)\hfill \\ & \le & {\left\{{\int }_0^t{\int }_0^x{\displaystyle \frac{{(t-s)}^{{ϵ}_1-1}{(x-\tau )}^{{ϵ}_2-1}}{\Gamma \left({ϵ}_1\right)\Gamma \left({ϵ}_2\right)}}f(s,\tau ,{\left(u_m\right)}_{(s,t)},w)d\tau ds\right\}}_{m=1}^{\infty }+\epsilon \hfill \\ & \le & 4{\int }_0^t{\int }_0^x{\alpha }_n\left({\left\{{\displaystyle \frac{{(t-s)}^{{ϵ}_1-1}{(x-\tau )}^{{ϵ}_2-1}}{\Gamma \left({ϵ}_1\right)\Gamma \left({ϵ}_2\right)}}f(s,\tau ,{\left(u_m\right)}_{(s,t)},w)\right\}}_{m=1}^{\infty }\right)d\tau ds+\epsilon \hfill \\ & \le & 4{\int }_0^t{\int }_0^x{\displaystyle \frac{{(t-s)}^{{ϵ}_1-1}{(x-\tau )}^{{ϵ}_2-1}}{\Gamma \left({ϵ}_1\right)\Gamma \left({ϵ}_2\right)}}{\alpha }_n\left({\left\{f(s,\tau ,{\left(u_m\right)}_{(s,t)},w)\right\}}_{m=1}^{\infty }\right)d\tau ds+\epsilon \hfill \\ & \le & 4{\int }_0^t{\int }_0^x{\displaystyle \frac{{(t-s)}^{{ϵ}_1-1}{(x-\tau )}^{{ϵ}_2-1}}{\Gamma \left({ϵ}_1\right)\Gamma \left({ϵ}_2\right)}}{p}_2(s,\tau ,w){\alpha }_n\left({\left\{{\left(u_m\right)}_{(s,t)}\right\}}_{m=1}^{\infty }\right)d\tau ds+\epsilon \hfill \\ & \le & \left(4{\int }_0^t{\int }_0^x{\displaystyle \frac{{(t-s)}^{{ϵ}_1-1}{(x-\tau )}^{{ϵ}_2-1}}{\Gamma \left({ϵ}_1\right)\Gamma \left({ϵ}_2\right)}}{p}_2(s,\tau ,w)d\tau ds\right){\alpha }_n\left({\left\{u_m\right\}}_{m=1}^{\infty }\right)+\epsilon \hfill \\ & \le & \left(4{\int }_0^t{\int }_0^x{\displaystyle \frac{{(t-s)}^{{ϵ}_1-1}{(x-\tau )}^{{ϵ}_2-1}}{\Gamma \left({ϵ}_1\right)\Gamma \left({ϵ}_2\right)}}{p}_2(s,\tau ,w)d\tau ds\right){\alpha }_n\left(B\right)+\epsilon \hfill \\ & \le & \left({\displaystyle \frac{4p_{2n}^{*}{n}^{{ϵ}_1+{ϵ}_2}}{\Gamma ({ϵ}_1+1)\Gamma ({ϵ}_2+1)}}\right){\alpha }_n\left(B\right)+\epsilon \hfill \\ & =& {\mathsf{\ell }}_n{\alpha }_n\left(B\right)+\epsilon .\hfill \end{array}$$

Since $\epsilon >0$ is chosen arbitrarily, it follows directly from taking the supremum over the full interval that:

$${\alpha }_n\left(\mathcal{N}\left(B\right)\right)\le {\mathsf{\ell }}_n{\alpha }_n\left(B\right).$$

Combining the arguments established above with the properties of the Random Darbo Fixed Point Theorem 1, we deduce that the random operator $\mathcal{N}$ admits at least one measurable random fixed point, which constitutes a mild random solution of the fractional problem (1)–(3) in the global space. □

4. Nonlocal Random Problem

We now adapt our previous results for the problem (4)–(6) to a more general setting that involves nonlocal conditions. By extending the analytical methods established for the local problem (Theorem 2), we explicitly derive the existence criteria under the contractive influence of the nonlocal conditions.

First, we specify what a solution is meant for the nonlocal problem (4)–(6).

Definition 8.

A function $u:\Omega \to {\mathcal{C}}_{\infty }$ is a random solution of the nonlocal problem $\left(4\right)$–$\left(6\right)$ if it is a measurable function such that $(t,x)\in \mathcal{J}$ and $w\in \Omega$, Equations $\left(4\right)$ and $\left(6\right)$ are satisfied, while the initial condition $\left(5\right)$ holds for all $(t,x)\in \tilde{\mathcal{J}}$.

Under the same assumptions as before, a random solution to our nonlocal problem corresponds to a realization of the following integral identity.

Lemma 5.

Let $0<{ϵ}_1,{ϵ}_2\le 1$. Then $u\in \Omega \times {\mathcal{C}}_{\infty }$ is a solution of the nonlocal random problem $\left(4\right)$–$\left(6\right)$ if and only if it satisfies $\left(5\right)$ for $(t,x)\in \tilde{\mathcal{J}},w\in \Omega$ and the following random integral identity:

$$u(t,x,w)=z(t,x,w)-Q\left(u\right)-K\left(u\right)+{\int }_0^t{\int }_0^x{\displaystyle \frac{{(t-s)}^{{ϵ}_1-1}{(x-\tau )}^{{ϵ}_2-1}}{\Gamma \left({ϵ}_1\right)\Gamma \left({ϵ}_2\right)}}f(s,\tau ,u_{(s,\tau )},w)d\tau ds.$$

We assume the following additional conditions:

$\left(A6\right)$

There exist positive constants $d_1,d_2>0$ such that for every $u\in \mathcal{E}$ and almost every $w\in \Omega$:

$${\parallel Q\left(u\right)\parallel }_{\mathcal{E}}\le d_1{(1+\parallel u\parallel }_n{),\parallel K\left(u\right)\parallel }_{\mathcal{E}}\le d_2{(1+\parallel u\parallel }_n).$$

$\left(A7\right)$

For every bounded set $B\subset {\mathcal{C}}_0$ and $(t,x)\in {\mathcal{J}}_n,$

$${\alpha }_n\left(Q\left(B\right)\right)\le p_3(t,x,w){\alpha }_n\left(B\right),{\alpha }_n\left(K\left(B\right)\right)\le p_4(t,x,w){\alpha }_n\left(B\right),$$

for almost every $w\in \Omega$, where $p_3,p_4\in {\mathcal{L}}^{\infty }(\Omega ,\mathcal{C}(\mathcal{J},[0,\infty ))).$ Here ${\alpha }_n$ is the noncompactness measures on $\mathcal{E}$ and $\mathcal{C}$.

$\left(A8\right)$

There exists a random function $R_n:\Omega \to (0,\infty )$ such that

$$z^{*}\left(w\right)+(d_1+d_2)(1+R_n\left(w\right))+{\displaystyle \frac{[p_{1n}^{*}+p_{2n}^{*}{R}_n\left(w\right)]n^{{ϵ}_1+{ϵ}_2}}{\Gamma ({ϵ}_1+1)\Gamma ({ϵ}_2+1)}}\le R_n\left(w\right).$$

Theorem 3.

Let assumptions (A1) through (A4) and (A6) through (A8) be satisfied. If

$${\mathsf{\ell }}_n^{\prime }:=2(p_{3n}^{*}+p_{4n}^{*})+{\displaystyle \frac{4p_{2n}^{*}{n}^{{ϵ}_1+{ϵ}_2}}{\Gamma ({ϵ}_1+1)\Gamma ({ϵ}_2+1)}}<1,$$

where

$$p_{3n}^{*}=\underset{(t,x)\in {\mathcal{J}}_n}{sup}ess{p}_3(t,x,w),p_{4n}^{*}=\underset{(t,x)\in {\mathcal{J}}_n}{sup}ess{p}_4(t,x,w).$$

Then the nonlocal problem $\left(4\right)$–$\left(6\right)$ admits at least one random solution in ${\mathcal{C}}_{\infty }.$

Proof. 

For each $w\in \Omega$, we introduce the modified nonlocal random operator $\overline{\mathcal{N}}:\Omega \times {\mathcal{C}}_{\infty }\to {\mathcal{C}}_{\infty }$ defined by the following expression:

$$\left(\overline{\mathcal{N}}\left(w\right)u\right)(t,x)=\left\{\begin{array}{cc}\varphi (t,x,w),\hfill & (t,x)\in \tilde{\mathcal{J}},\hfill \\ z(t,x,w)-Q\left(u\right)-K\left(u\right)\hfill & \\ {\displaystyle +\frac{1}{\Gamma \left({ϵ}_1\right)\Gamma \left({ϵ}_2\right)}{\int }_0^t{\int }_0^x{(t-s)}^{{ϵ}_1-1}{(x-\tau )}^{{ϵ}_2-1}}\hfill & \\ \times f(s,\tau ,u_{(s,\tau )},w)d\tau ds,\hfill & (t,x)\in \mathcal{J}.\hfill \end{array}\right.$$

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In accordance with this construction, the existence of a solution to the nonlocal problem (4)–(6) is equivalent to finding a fixed point for the operator equation $u=\overline{\mathcal{N}}\left(w\right)u$. Let $W\left(w\right)=\{u\in {\mathcal{C}}_{\infty }:\parallel u{\parallel }_n\le R_n\}$ be the nonempty, closed, convex, and bounded stochastic ball defined in our separable Fréchet spac ${\mathcal{C}}_{\infty }$.

To comprehensively analyze the structural and contractive properties under the noncompactness framework, we decompose the joint operator into a sum of two component random operators, $\overline{\mathcal{N}}\left(w\right)={\mathcal{N}}_1\left(w\right)+{\mathcal{N}}_2\left(w\right)$, where ${\mathcal{N}}_1$ isolates the nonlocal conditions and ${\mathcal{N}}_2$ isolates the historical delay integral mapping:

$$\left({\mathcal{N}}_1\left(w\right)u\right)(t,x)=z(t,x,w)-Q\left(u\right)-K\left(u\right),$$

$$\left({\mathcal{N}}_2\left(w\right)u\right)(t,x)={\displaystyle \frac{1}{\Gamma \left({ϵ}_1\right)\Gamma \left({ϵ}_2\right)}}{\int }_0^t{\int }_0^x{(t-s)}^{{ϵ}_1-1}{(x-\tau )}^{{ϵ}_2-1}f(s,\tau ,u_{(s,\tau )},w)d\tau ds.$$

We proceed to verify that $\overline{\mathcal{N}}\left(w\right)$ satisfies all conditions of our Random Darbo Fixed Point Theorem via five distinct steps:

▹ Invariance of the Stochastic Ball $W\left(w\right)$.

• Let $u\in W\left(w\right)$ be an arbitrary function. For any fixed realization $w\in \Omega$ evaluated on any localized compact rectangle ${\mathcal{J}}_n$, we apply the triangle inequality to our decomposition. By utilizing the growth restriction bounds on the nonlocal operators Q and K along with the algebraic inequality presented in Assumption (A5), we obtain for all $(t,x)\in {\mathcal{J}}_n$:

$$\begin{array}{ccc}\hfill \parallel \left(\overline{\mathcal{N}}\left(w\right)u\right){(t,x)\parallel }_{\mathcal{E}}& \le & \parallel \left({\mathcal{N}}_1\left(w\right)u\right){(t,x)\parallel }_{\mathcal{E}}+{\parallel \left({\mathcal{N}}_2\left(w\right)u\right)(t,x)\parallel }_{\mathcal{E}}\hfill \\ & \le & {\parallel z(t,x,w)-Q\left(u\right)-K\left(u\right)\parallel }_{\mathcal{E}}\hfill \\ & & +{∥{\displaystyle \frac{1}{\Gamma \left({ϵ}_1\right)\Gamma \left({ϵ}_2\right)}}{\int }_0^t{\int }_0^x{(t-s)}^{{ϵ}_1-1}{(x-\tau )}^{{ϵ}_2-1}f(s,\tau ,u_{(s,\tau )},w)d\tau ds∥}_{\mathcal{E}}\hfill \\ & \le & z^{*}\left(w\right)+{\displaystyle \frac{[p_{1n}^{*}+p_{2n}^{*}{R}_n\left(w\right)]n^{{ϵ}_1+{ϵ}_2}}{\Gamma ({ϵ}_1+1)\Gamma ({ϵ}_2+1)}}\hfill \\ & \le & R_n\left(w\right).\hfill \end{array}$$

Taking the supremum over $(t,x)\in {\mathcal{J}}_n$, it follows that $\parallel \overline{\mathcal{N}}{\left(w\right)u\parallel }_n\le R_n\left(w\right)$ for each $n\in \mathbb{N}$. Hence, $\overline{\mathcal{N}}\left(w\right)\left(W\left(w\right)\right)\subseteq W\left(w\right)$, proving that the stochastic ball $W\left(w\right)$ is strictly invariant under the modified operator.

▹ Measurability of the Joint Operator.

The nonlinear forcing term f satisfies the random Carathéodory conditions (Assumption (A2)), ensuring that the mapping $w\mapsto f(t,x,u_{(t,x)},w)$ is product-measurable. Since the nonlocal boundary operators $Q\left(u\right)$ and $K\left(u\right)$ represent continuous mappings on ${\mathcal{C}}_{\infty }$, they preserve measurability under the underlying $\sigma$-algebra $\mathcal{A}$. Because addition, subtraction, and fractional integration operations of measurable maps preserve joint measurability under the product space $\mathcal{A}\times B_{{\mathcal{C}}_{\infty }}$ (where ${\mathcal{B}}_{{\mathcal{C}}_{\infty }}$ denotes the Borel $\sigma$-algebra on the Fréchet space ${\mathcal{C}}_{\infty }$), it follows that for any fixed function $u\in {\mathcal{C}}_{\infty }$, the mapping $w\mapsto \overline{\mathcal{N}}\left(w\right)u$ is a well-defined measurable random operator.

▹ Sequential Continuity of the Joint Operator.

• Let ${\left\{u_m\right\}}_{m\in \mathbb{N}}$ be a sequence in $W\left(w\right)$ converging to u under the topology of the Fréchet space. The continuity of the nonlocal component operator ${\mathcal{N}}_1\left(w\right)$ is a direct consequence of the assumed continuity of the functions Q and K. For the fractional integral operator ${\mathcal{N}}_2\left(w\right)$, the uniform convergence of the sequence yields the history segment limit $\parallel {\left(u_m\right)}_{(s,\tau )}-u_{(s,\tau )}{\parallel }_{{\mathcal{C}}_0}\to 0$ as $m\to \infty$. By applying our growth constraint (Assumption (A3)), the integrand is uniformly bounded by the sequence-independent integrable ceiling $2[p_1(s,\tau ,w)+p_2(s,\tau ,w)R_n\left(w\right)]$. Thus, by the Lebesgue Dominated Convergence Theorem, the limit passes inside the double integral, yielding:

$$\underset{m\to \infty }{lim}{\parallel {\mathcal{N}}_2\left(w\right)u_m-{\mathcal{N}}_2\left(w\right)u\parallel }_n=0.$$

As the sum of two continuous operators is continuous, the full operator $\overline{\mathcal{N}}\left(w\right)$ is sequentially continuous on $W\left(w\right)$.

▹ Noncompactness Estimates via the MNC Family ${\alpha }_n$.

• Let $B\subset W\left(w\right)$ be an arbitrary bounded subset. To evaluate the tracking characteristics, we utilize the structural properties of the measure of noncompactness family ${\alpha }_n$ corresponding to each compact sub-interval ${\mathcal{J}}_n$. For the historical delay integral component ${\mathcal{N}}_2\left(w\right)$, applying Lemma 2 and Assumption (A4) yields the fractional upper bound:

$${\alpha }_n\left(\left({\mathcal{N}}_2\left(w\right)B\right)(t,x)\right)\le {\displaystyle \frac{4p_{2n}^{*}{n}^{{ϵ}_1+{ϵ}_2}}{\Gamma ({ϵ}_1+1)\Gamma ({ϵ}_2+1)}}{\alpha }_n\left(B\right).$$

Simultaneously, evaluating the nonlocal boundary component ${\mathcal{N}}_1\left(w\right)$ under the noncompactness growth conditions (A6) and (A7) produces the specific coordinate bound:

$${\alpha }_n\left(\left({\mathcal{N}}_1\left(w\right)B\right)(t,x)\right)\le 2(p_{3n}^{*}+p_{4n}^{*}){\alpha }_n\left(B\right).$$

Combining these parallel components via the sub-additivity axiom of the measure of noncompactness, we obtain the total evaluation on the sub-interval ${\mathcal{J}}_n$:

$$\begin{array}{cc}\hfill {\alpha }_n\left(\left(\overline{\mathcal{N}}\left(w\right)B\right)\right)& \le {\alpha }_n\left(\left({\mathcal{N}}_1\left(w\right)B\right)\right)+{\alpha }_n\left(\left({\mathcal{N}}_2\left(w\right)B\right)\right)\hfill \\ \hfill & \le \left[2(p_{3n}^{*}+p_{4n}^{*})+{\displaystyle \frac{4p_{2n}^{*}{n}^{{ϵ}_1+{ϵ}_2}}{\Gamma ({ϵ}_1+1)\Gamma ({ϵ}_2+1)}}\right]{\alpha }_n\left(B\right)\hfill \\ \hfill & ={\mathsf{\ell }}_n^{\prime }{\alpha }_n\left(B\right).\hfill \end{array}$$

▹ Application of the Random Fixed Point Theorem.

By hypothesis, the combined coefficient satisfies the strict inequality ${\mathsf{\ell }}_n^{\prime }<1$ for each localized compact domain index $n\in \mathbb{N}$. Therefore, the continuous random operator $\overline{\mathcal{N}}$ behaves almost surely as a strict set-contraction in the Fréchet space ${\mathcal{C}}_{\infty }$. Since $W\left(w\right)$ is a nonempty, closed, bounded, and convex invariant subset, all criteria of the Random Darbo Fixed Point Theorem 1 are satisfied. We conclude that $\overline{\mathcal{N}}$ admits at least one measurable random fixed point $u^{*}\left(w\right)\in W\left(w\right)$, which corresponds to a mild random solution of the nonlocal delay problem. □

5. Applications

To demonstrate the practical utility and applicability of the theoretical results established in Section 3, we analyze a specific realization of the proposed random fractional differential system. Let $\mathcal{E}=\mathbb{R}$ and define the probability space $\Omega =(-\infty ,0)$ equipped with the usual $\sigma$-algebra. Let the history space be denoted by ${\mathcal{C}}_0=\mathcal{C}([-2,0]\times [-1,0],\mathcal{E})$, corresponding to the delay parameters $\alpha =2$ and $\beta =1$. We consider the following random fractional differential equation with bounded delay of the form:

$$\left({}^{C}D_0^{ϵ}u\right)(t,x,w)={\displaystyle \frac{c_n{w}^2}{e^{t+x+3}(1+w^2+\left|u(t,x,w)\right|)}},\mathrm{if}(t,x)\in \mathcal{J}:=[0,\infty )\times [0,\infty ),w\in \Omega ,$$

(11)

$$u(t,x,w)=t^{2}sinw+xcosw,\mathrm{for}(t,x)\in \tilde{\mathcal{J}}:=[-2,\infty )\times [-1,\infty )\setminus [0,\infty )\times [0,\infty ),$$

(12)

$$\left\{\begin{array}{c}u(t,0,w)=t^{2}sinw,\hfill \\ u(0,x,w)=xcosw,\hfill \end{array}\right.(t,x)\in \mathcal{J},w\in \Omega ,$$

(13)

where the normalization constant $c_n$ is defined as:

$$c_n={\displaystyle \frac{\Gamma ({ϵ}_1+1)\Gamma ({ϵ}_2+1)}{n^{{ϵ}_1+{ϵ}_2}}},n\in {\mathbb{N}}^{*}.$$

Here, $ϵ=({ϵ}_1,{ϵ}_2)\in (0,1]\times (0,1]$ denotes the fractional order of the derivative. The history and bounded delay conditions of the state variables are explicitly represented on the delayed domain boundary $\tilde{\mathcal{J}}$ with delay parameters $\alpha =2$ and $\beta =1$. The associated nonlinear random delay function $f:{\mathcal{J}}_n\times {\mathcal{C}}_0\times \Omega \to \mathcal{E}$ is given for any $(t,x)\in {\mathcal{J}}_n$, $u\in {\mathcal{C}}_0$, and $w\in \Omega$ by:

$$f(t,x,u,w)={\displaystyle \frac{c_n{w}^2}{e^{t+x+3}\left(1+w^2+{\parallel u\parallel }_{{\mathcal{C}}_0}\right)}},$$

where ${\displaystyle {\parallel u\parallel }_{{\mathcal{C}}_0}=\underset{(s,\tau )\in [-2,0]\times [-1,0]}{sup}\left|u(s,\tau )\right|}$. Given the joint continuity of the independent variables $(t,x,w)$ and the pointwise continuity relative to the state variable u, it follows that f satisfies the properties of a random Carathéodory type function.

To verify the growth conditions required by our framework, we evaluate the absolute value of the nonlinear operator. For every $u\in {\mathcal{C}}_0$ and $(t,x,w)\in {\mathcal{J}}_n\times \Omega$, we observe that the denominator satisfies $1+w^2+{\parallel u\parallel }_{{\mathcal{C}}_0}\ge 1+w^2>1$. Utilizing the bounding properties of the exponential term on the domain, we obtain the following precise structural estimate:

$$\left|f(t,x,u,w)\right|\le {\displaystyle \frac{c_n{w}^2}{e^{t+x+3}(1+w^2)}}\le {\displaystyle \frac{c_n}{e^3}}\le 1+c_n{e}^{-3}{\parallel u\parallel }_{{\mathcal{C}}_0}.$$

This directly validates Assumption (A3) by identifying the growth scaling parameters on ${\mathcal{J}}_n$ as:

$$p_{1n}^{*}=1\mathrm{and}{p}_{2n}^{*}=c_n{e}^{-3}.$$

Furthermore, because our problem is explicitly formulated within the finite-dimensional space $\mathcal{E}=\mathbb{R}$, any bounded subset is relatively compact. Consequently, the condition involving the measure of noncompactness (A4) is trivially satisfied since ${\alpha }_n\left(B\right)=0$ for any bounded $B\subset {\mathcal{C}}_0$, which implies ${\alpha }_n\left(f(t,x,B,w)\right)=0\le p_{2n}^{*}{\alpha }_n\left(B\right)$.

Finally, we explicitly verify the crucial contractive condition ${\mathsf{\ell }}_n<1$. For every fractional order pair $({ϵ}_1,{ϵ}_2)\in (0,1]\times (0,1]$, substituting our derived parameter $p_{2n}^{*}$ into the contractive formula yields:

$$\begin{array}{cc}\hfill {\mathsf{\ell }}_n& ={\displaystyle \frac{4p_{2n}^{*}{n}^{{ϵ}_1+{ϵ}_2}}{\Gamma ({ϵ}_1+1)\Gamma ({ϵ}_2+1)}}\hfill \\ \hfill & ={\displaystyle \frac{4\left(\frac{\Gamma ({ϵ}_1+1)\Gamma ({ϵ}_2+1)}{n^{{ϵ}_1+{ϵ}_2}}{e}^{-3}\right)n^{{ϵ}_1+{ϵ}_2}}{\Gamma ({ϵ}_1+1)\Gamma ({ϵ}_2+1)}}\hfill \\ \hfill & ={\displaystyle \frac{4}{e^3}}\approx 0.1991<1.\hfill \end{array}$$

As demonstrated by the algebraic cancellation above, the contractive condition ${\mathsf{\ell }}_n$ holds consistently across all choices of n and fractional orders. Since all structural, growth, and topological hypotheses of Theorem 2 are fulfilled, the random fractional system with bounded delay defined in Equations (11)–(13) is guaranteed to have at least one random solution in ${\mathcal{C}}_{\infty }$.

6. Conclusions

In this paper, we successfully investigated the existence of solutions for a class of random fractional differential equations characterized by bounded delay within the framework of Fréchet spaces. By employing a generalized version of Darbo’s fixed point theorem combined with the properties of noncompactness measures, we established rigorous existence criteria under significantly relaxed compactness conditions. Furthermore, the practical validity of our theoretical framework was demonstrated through a detailed illustrative application.

Moreover, it is worth highlighting the distinct advantages of our framework compared to other contemporary fractional modeling approaches in the literature. While recent studies have successfully explored alternative operators—such as the truncated M-fractional derivative for capturing specific physical wave structures [3] or advanced stochastic neural network processes for localized fractional systems [1]—our framework offers a unique topological alternative. By utilizing the classical Caputo fractional derivative combined with measures of noncompactness within Fréchet spaces, our approach efficiently bypasses strict compactness criteria on unbounded domains. This allows for a more generalized handling of hereditary characteristics and random perturbations simultaneously, complementing existing methodologies that rely on alternative fractional operators.

In future work, we plan to extend this approach to study the stability and optimal control problems for similar classes of stochastic fractional systems with infinite and state-dependent delay.