Solving Fractional Random Differential Equations by Using Fixed Point Methodologies under Mild Boundary Conditions

Abstract

This manuscript aims to study the existence and uniqueness of solutions to a new system of differential equations. This system is a mixture of fractional operators and stochastic variables. The study has been completed under nonlocal functional boundary conditions. In the study, we used the fixed-point method to examine the existence of a solution to the proposed system, mainly focusing on the theorems of Leray, Schauder, and Perov in generalized metric spaces. Finally, an example has been provided to support and underscore our results.

1. Introduction

In 1940, fractional Brownian motion (FBM), often known as a Wiener Helix, was first described by Kolmogorov [1] in the setting of a Hilbert space (HS) and after giving a stochastic integral description of this process in terms of a standard BM, Mandelbrot and Van Ness introduced fractional BM in 1968. Reliability, asymptotic behavior, and character of moderate solutions to fractional BM stochastic delay evolution equations have yet to be well explored in many works. Sadovskii’s fixed point (FP) theorem was utilized by Cui and Yan [2] to investigate the possibility of a moderate solution to neutral stochastic integro-differential equations (DEs) with infinite delay. Following the proof of the existence of a mild solution to a nonlocal fractional stochastic DE by Sakthivel et al. [3], Jingyun and Xiaoyuan [4] provided sufficient conditions for the existence and uniqueness (EU) of mild solutions to a system with nonlocal fractional stochastic BM and Hurst index $H>\frac{1}{2}$. Additional findings about fractional BM in stochastic equations can be discovered in [5,6,7,8,9,10,11].

Differential equations (DEs) incorporating fractional derivatives in time provide a more accurate description of many scientific processes, including those in economics, finance, chemistry, physics, and biology. Many physical phenomena, as represented by evolution equations, show some degree of dependence on previous events. As a result, recent years have seen a notable advancement in the theory of fractional integro-DEs. In particular, state-dependent delays in functional DEs arise frequently as mathematical models in a variety of applications. In recent years, much research has been conducted in this field because of the considerable differences in specific properties between these equations and those with constant or time-dependent delays. To learn more, go to [12,13,14,15].

One practical approach for solving nonlinear engineering issues is the FP technique. By utilizing this technology, engineers can efficiently address complicated systems when typical linear approaches fail. By identifying points that do not change under a given transformation, this technique allows for the iterative determination of solutions. It benefits tasks like structural stability analysis, design optimization, and solving various nonlinear equations commonly encountered in engineering applications. For other diverse applications of this approach, go to [16,17,18,19,20,21].

Recent years have seen a significant interest in the literature for problems pertaining to the presence of solutions of various equation types incorporating nonlocal conditions, as seen by the publications [22,23,24,25,26,27]. This paper investigates the EU of a mild solution to the fractional DE system driven by BM

$$\left\{\begin{array}{c}{ }^C{D}^p\omega \left(s\right)=\left[{\Im }_1\omega \left(s\right)+{¯h}_1\left(s,\omega \left(s\right),\upsilon \left(s\right),\theta \left(s\right)\right)\right]ds+{\rho }_1\left(s\right)d{\Re }_s^{H_1},p\in (\frac{1}{2},1],s\in U=[0,c],\hfill \\ { }^C{D}^p\upsilon \left(s\right)=\left[{\Im }_2\upsilon \left(s\right)+{¯h}_2\left(s,\omega \left(s\right),\upsilon \left(s\right),\theta \left(s\right)\right)\right]ds+{\rho }_2\left(s\right)d{\Re }_s^{H_2},\hfill \\ { }^C{D}^p\theta \left(s\right)=\left[{\Im }_3\theta \left(s\right)+{¯h}_3\left(s,\omega \left(s\right),\upsilon \left(s\right),\theta \left(s\right)\right)\right]ds+{\rho }_3\left(s\right)d{\Re }_s^{H_3},\hfill \\ \omega \left(0\right)=\kappa \left(\omega ,\upsilon ,\theta \right),\upsilon \left(0\right)=\tau \left(\omega ,\upsilon ,\theta \right)\mathrm{and}\theta \left(0\right)=\mu \left(\omega ,\upsilon ,\theta \right),\hfill \end{array}\right.$$

(1)

where $p\in (\frac{1}{2},1]$ is the fractional-order of the Caputo fractional derivative (CFD) formulation ${ }^C{D}^p$ with lower limit 0. On a Banach space (BS) $\left({\mathbf{C}}_u,{∥.∥}_{{\mathbf{C}}_u}\right),$ ${\Im }_u:D\left({\Im }_u\right)\subseteq {\mathbf{C}}_u\to {\mathbf{C}}_u$ $\left(u=1,2,3\right)$ are linear operators, which generates a strongly continuous semigroup of contractions $\{Z_p\left(s\right):s\ge 0\}.$ Assume that $\left(\mho ,{\Theta }_c,P\right)$ is a probability space (PS) and $\mathsf{{\rm Y}}$ is a real Hilbert space (HS), for the stochastic process (SP) ${\left\{S\left(s\right)\right\}}_{s\in U},$ ${¯h}_u:U\times S_1\times S_2\times S_3\to S_u$ and ${\rho }_u:J\to S_u$ $\left(u=1,2,3\right)$ are given functions and $\kappa ,\tau ,\mu :\mathbf{C}\left(U,L^2\left(\mho ,{\Theta }_c;\mathsf{{\rm Y}}\right)\right)\times \mathbf{C}\left(U,L^2\left(\mho ,{\Theta }_c;\mathsf{{\rm Y}}\right)\right)\times \mathbf{C}\left(U,L^2\left(\mho ,{\Theta }_c;\mathsf{{\rm Y}}\right)\right)\to \mathsf{{\rm Y}}$ are given functional, where ${\left\{S\left(s\right)\right\}}_{s\in U}$ takes values in $\mathsf{{\rm Y}}.$ Further, for $u=1,2,3,$ on a real HS W, ${\Re }_s^{H_u}=\left\{{\Re }^{H_u}\left(s\right):s\in U\right\}$ refer to the fractional BM via Hurst index $H_i\in \left(0,\frac{1}{2}\right).$

The spaces below will be used in the next sections.

$$\left\{\begin{array}{c}L\left(W,\mathsf{{\rm Y}}\right)=\left\{z:W\to \mathsf{{\rm Y}}:z\mathrm{represents}\mathrm{a}\mathrm{bounded}\mathrm{linear}\mathrm{operator}\left(\mathrm{BLO}\right)\right\},\hfill \\ L^2\left(\mho ,{\Theta }_c,\mathsf{{\rm Y}}\right)=\left\{\begin{array}{c}\tilde{z}:\mho \to \mathsf{{\rm Y}}:\tilde{z}\mathrm{refers}\mathrm{to}{\Theta }_c\mathrm{is}\mathrm{a}\mathrm{measurable}\mathrm{square}\\ \mathrm{integrable}\mathrm{random}\mathrm{variable}\end{array}\right\},\hfill \\ \mathbf{C}\left(U,L^2\left(\mho ,{\Theta }_c;\mathsf{{\rm Y}}\right)\right)=\left\{\begin{array}{c}S:U\to L^2\left(\mho ,{\Theta }_c;\mathsf{{\rm Y}}\right):S\mathrm{acts}\mathrm{a}\mathrm{continuous}\mathrm{mapping}\\ \mathrm{with}\underset{s\in U}{sup}E\left[{∥S\left(s\right)∥}^2\right]<\infty \end{array}\right\},\hfill \\ \complement =\left\{\begin{array}{c}S:U\times \mho \to \mathsf{{\rm Y}}:S\in \mathbf{C}\left(U,L^2\left(\mho ,{\Theta }_c;\mathsf{{\rm Y}}\right)\right)\mathrm{is}\mathrm{an}\mathrm{adapted}\mathrm{SP}\\ \mathrm{with}{∥S∥}_{\complement }=\sqrt{{sup}_{s\in U}E\left[{∥S\left(s\right)∥}^2\right]}\mathrm{for}S\in \complement \mathrm{and}\left(\complement ,{∥.∥}_{\complement }\right)\mathrm{is}\mathrm{a}\mathrm{BS}\end{array}\right\}.\hfill \end{array}\right.$$

2. Preliminaries

In this part, we will collect most of the definitions and theorems related to the subject of our study, briefly and clearly. We begin with the notation of a generalized metric space (GMS) and its topological properties [28].

Definition 1.

Assume that $S\ne \varnothing$ and the distance mapping $d:S\times S\to {\mathbb{R}}^u$ $(u\in \mathbb{N})$ fulfills the following conditions: for all ${\varpi }_1,{\varpi }_2,{\varpi }_3\in S$

(i) 

$d\left({\varpi }_1,{\varpi }_2\right)\ge 0$ and if $d\left({\varpi }_1,{\varpi }_2\right)=0,$ then ${\varpi }_1={\varpi }_2;$

(ii) 

$d\left({\varpi }_1,{\varpi }_2\right)=d\left({\varpi }_2,{\varpi }_1\right);$

(iii) 

$d\left({\varpi }_1,{\varpi }_2\right)\le d\left({\varpi }_1,{\varpi }_3\right)+d\left({\varpi }_3,{\varpi }_2\right).$

Then, the pair $(S,d)$ is called a GMS.

Here, $\omega ,\nu \in {\mathbb{R}}^u$ means $\omega =\left({\omega }_1,{\omega }_2,\cdots ,{\omega }_u\right)$ and $\nu =\left({\nu }_1,{\nu }_2,\cdots ,{\nu }_u\right)$; $\omega \le \nu$ means ${\omega }_i\le {\nu }_i$ for $i=1,2,\cdots ,u.$ It is clear that

• for any $i\in \{1,2,\cdots ,u\},$

  $${\left(d\left(\omega ,\nu \right)\right)}_i=d_i\left(\omega ,\nu \right)$$

  is a metric space on S,
• for $a=\left(a_1,a_2,\cdots ,a_u\right)\in {\mathbb{R}}_{+}^u,$

  $$\Re \left({\omega }_0,a\right)=\left\{d\left(\omega ,{\omega }_0\right)<a\right\}$$

  represents the open ball with the center at ${\omega }_0$ and radius $a,$ also, the closure of $\Re \left({\omega }_0,a\right)$ is denoted by $\overline{\Re \left({\omega }_0,a\right)},$
• $(S,d)$ is a GMS with

  $$d\left(\omega ,\nu \right)=\left(\begin{array}{c}{d}_1\left(\omega ,\nu \right)\\ ⋮\\ d_u\left(\omega ,\nu \right)\end{array}\right).$$
• d is a GMS if and only if $d_i$ are metrics on S for $i\in \{1,2,\cdots u\},$
• for the vector valued metric $d\left(\omega ,\nu \right)=∥\omega -\nu ∥,$ $\left(S,∥.∥\right)$ is generalized Banach space (BS) if S is complete with respect to (w.r.t.) d.

Definition 2.

Assume that V is a square matrix in $\mathbb{R}$ with a spectral radius $\vartheta \left(V\right).$ We say that V is convergent to zero $\left(V\to 0\right)$ if $\vartheta \left(V\right)\le 1.$

Theorem 1

([29]). Assume that $V\in V_{u\times u}\left({\mathbb{R}}_{+}\right),$ then the axioms below are equivalent

(1) 

$V\to 0,$

(2) 

$V^j\to 0$ as $j\to \infty ,$

(3) 

$det(I-V)\ne 0,$ that is, the matrix $(I-V)$ is nonsingular and

$${\left(I-V\right)}^{-1}=I+V+V^2+\cdots +V^j+\cdots$$

where I denotes the identity matrix in $V_{u\times u}\left({\mathbb{R}}_{+}\right).$

(4) 

${\left(I-V\right)}^{-1}$ has nonnegative elements, provided that $det(I-V)\ne 0.$

Definition 3.

Let ℵ be a self-mapping defined on a GMS $(S,d).$ If there exists a convergent matrix V to zero such that

$$d\left(\aleph \left(\omega \right),\aleph \left(\nu \right)\right)\le Vd\left(\omega ,\nu \right)\mathrm{for}\mathrm{all}\omega ,\nu \in \aleph .$$

Then, ℵ is called a contractive mapping.

It should be noted that, the classical contraction mapping can be obtained if we take $u=1$.

Next, we present two FP theorems, which we used it in our proofs; see [30,31].

Theorem 2

(Perov theorem). Assume that $(S,d)$ is a GMS and ℵ is a contractive mapping, then ℵ has a unique FP ${\omega }^{*}\in S$ and

$$d\left({\aleph }^j\left({\omega }_0\right),{\omega }^{*}\right)\le V^j{\left(I-V\right)}^{-1}d\left(\aleph \left({\omega }_0\right),{\omega }_0\right),\mathrm{for}\mathrm{every}{\omega }_0\in S,j\ge 1.$$

A note was made in [31] regarding the function of spectral radius smaller than one matrices in the analysis of semilinear operator systems and their relation to other abstract ideas from nonlinear functional analysis.

Theorem 3

(Leray–Schauder theorem). Let $\left(S,{\left|.\right|}_S\right)$ be a BS, $Q>0$, and ${\overline{\Re }}_S\left(0,Q\right)$ = $\left\{\omega \in S:{\left|\omega \right|}_S<Q\right\},$ and the operator $\chi :{\overline{\Re }}_S\left(0,Q\right)\to S$ be a completely continuous. If ${\left|\varpi \right|}_S<Q$ for every solution ϖ of the equation $\varpi =\alpha \chi \left(\varpi \right)$ and any $\alpha \in (0,1),$ then χ has at least one FP.

Next, we introduce some basic concepts of one-dimensional fractional BM. For more details about fractional BM, see [32,33,34].

Definition 4.

For Hurst index $H\in \left(0,0.5\right),$ the one-dimensional fractional BM ${\Re }_s^H$ = $\left\{{\Re }^H\left(s\right):s\in U\right\}$ is a Gaussian process, that is, continuous, centered, and has a covariance function

$$Q^H(s,r)=E\left[{\Re }^H\left(s\right){\Re }^H\left(r\right)\right]=\frac{1}{2}\left(s^{2H}+r^{2H}-{\left|s-r\right|}^{2H}\right),s,r\in U.$$

(2)

Remark 1.

(i) 

${\Re }_s^{0.5}$ is a standard BM if $H=0.5.$

(ii) 

${\Re }_s^H$ can be described over a finite interval, for $H\in \left(0,0.5\right)$ as

$${\Re }^H\left(s\right)={\int }_0^s{M}^H\left(s,r\right)d\mathsf{{\rm Y}}\left(r\right),$$

where $\mathsf{{\rm Y}}=\left\{\mathsf{{\rm Y}}\left(s\right):s\in U\right\}$ represents a Wiener process (WP),

$$M^H\left(s,r\right)=b_H\left(H-0.5\right)r^{0.5-H}{\int }_r^s{\left(\varpi -r\right)}^{H-1.5}{\varpi }^{H-0.5}d\varpi ,$$

and $b_H$ is a constant depending on $H.$

(iii) 

The process ${\Re }_t$ is a usual BM if $H=1.$

(iv) 

If $H\ne 1$, then (2) takes the form $E{\left[{\Re }_s.{\Re }_r\right]}^2={\left|s-r\right|}^{2H},$ that is, no have independent increments and ${\Re }_s^H$ has α-Hölder continuous paths for $\alpha \in \left(0,H\right).$

In the rest of the manuscript, we shall consider $H\in (0.5,1).$

Assume that $\eta$ be a linear space defined on the following step function in $U:$

$$\psi \left(s\right)=\sum _{i=1}^{u-1}{h}_i{I}_{(s_i,s_{i+1}]}\left(s\right),$$

where $0=s_1<s_2<\cdots <s_u=c,$ $u\in \mathbb{N},$ $h_i\in Q.$ Also, we denote the closure of $\eta$ by $\phi$ with w.r.t. ${⟨I_{[0,s]},I_{[0,r]}⟩}_{\phi }=Q^H(s,r).$ The Wiener integral of $\psi \in \eta$ w.r.t. ${\Re }^H$ is presented by

$$\underset{0}{\overset{c}{\int }}\psi \left(r\right)d{\Re }^H\left(r\right)=\sum _{i=1}^{u-1}{h}_i\left[{\Re }^H(s_i+1)-{\Re }^H\left(s_i\right)\right].$$

In addition, there exists an isometric mapping

$$\psi \to \underset{0}{\overset{c}{\int }}\psi \left(r\right)d{\Re }^H\left(r\right)$$

between $\eta$ and linear space span $\left\{{\Re }^H\left(s\right):s\in U\right\}$ viewed as a subspace of $L^2\left(\mho \right).$ One can expand this mapping to an isometric between $\phi$ and the fractional BM’s first Wiener chaos

$${\overline{span}}^{L^2\left(\mho \right)}\left\{{\Re }^H\left(s\right):s\in U\right\}.$$

Based on this isometric, the image of $f\in \phi$ is called the Wiener integral of f w.r.t. ${\Re }^H\left(s\right).$

Suppose that $M_{\beta }:\eta \to L^2\left(U\right)$ is a linear operator described as

$$M_{\beta }\left(\psi \right)\left(r\right)=\underset{r}{\overset{\beta }{\int }}\psi \left(r\right)\frac{\partial M^H\left(s,r\right)}{\partial s}ds,\mathrm{for}\mathrm{any}\beta \in U.$$

Clearly, an isometric between $\eta$ and $L^2\left(U\right)$ that can be extended to $\phi$ is produced by the operator $M_c.$ The relation between the Itô integral w.r.t. the WP and the Wiener integral w.r.t. fractional BM is as follows:

$$\underset{0}{\overset{c}{\int }}f\left(r\right)d{\Re }^H\left(r\right)=\underset{0}{\overset{c}{\int }}{M}_c\left(f\right)\left(r\right)d\mathsf{{\rm Y}}\left(r\right),f\in \phi ,\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}{M}_{\beta }\left(f\right)\in L^2\left(U\right)$$

Moreover, for $s\in U,$

$$\underset{0}{\overset{s}{\int }}f\left(r\right)d{\Re }^H\left(r\right)=\underset{0}{\overset{s}{\int }}f\left(r\right)I_{[0,s]}\left(s\right)d{\Re }^H\left(r\right).$$

Further, we have

$$\underset{0}{\overset{s}{\int }}f\left(r\right)d{\Re }^H\left(r\right)=\underset{0}{\overset{s}{\int }}{M}_s\left(f\right)\left(r\right)d\mathsf{{\rm Y}}\left(r\right),s\in U,f{I}_{[0,s]}\in \phi ,$$

provided that $M_s\left(f\right)\in L^2\left(U\right).$

According to the results of [35], for $H>0.5$, we get

$$L^{\frac{1}{H}}\left(U\right)\subset L^2\left(U\right).$$

(3)

Next, we provide the definition of the related random integral and the infinite dimensional fractional BM. Assume that $R\in L\left(W,\mathsf{{\rm Y}}\right)$ is a non-negative self-adjoint trace (Tr) class operator described by $R{ϵ}_u={\alpha }_u{ϵ}_u$ such that

$$Tr\left(R\right)=\sum _{u=1}^{\infty }{\alpha }_u<\infty ,$$

where ${\alpha }_u\in {\mathbb{R}}_{+}\cup \left\{0\right\},$ $u\in \mathbb{N}$ and ${ϵ}_u$ represent the orthonormal basis for W.

In the context of the PS $\left(\mho ,\Theta ,P\right)$ with covariance operator R, define the W-valued R-cylindrical fractional BM by

$${\Re }^H\left(s\right)=\sum _{u=1}^{\infty }{R}^{0.5}{ϵ}_u{\Re }_u^H\left(s\right)=\sum _{u=1}^{\infty }\sqrt{{\alpha }_u}{ϵ}_u{\Re }_u^H\left(s\right)$$

where ${\Re }_u^H$ contained in $\mathbb{R}$, independent, one-dimensional fractional BM. This process has zero mean, zero covariance and is a W-valued Gaussian process. It begins at 0. Hence, one can write

$$E\left(⟨{\Re }^H\left(s\right),\omega ⟩⟨{\Re }^H\left(r\right),\nu ⟩\right)=Q(r,s)⟨R\left(\omega \right),\nu ⟩,\forall s,r\in U,\omega ,\nu \in W,$$

and ${\Re }^H\left(s\right)$ possesses W-valued and sample paths over P.

Define the space $L_R^0\left(W,\mathsf{{\rm Y}}\right)$ by

$$L_R^0\left(W,\mathsf{{\rm Y}}\right)=\left\{\zeta :W\to \mathsf{{\rm Y}}:\zeta \mathrm{is}\mathrm{an}\mathrm{R}-\mathrm{Hilbert}-\mathrm{Schmidt}\mathrm{operator}\left(\mathrm{HSO}\right)\right\}.$$

It should be noted that $\zeta \in L_R\left(W,\mathsf{{\rm Y}}\right)$ is called an R-HSO if

$${∥\zeta ∥}_{L_R^0\left(W,\mathsf{{\rm Y}}\right)}^2=\sum _{u=1}^{\infty }{∥\sqrt{{\alpha }_u}\zeta {ϵ}_u∥}^2<\infty .$$

The space $L_R^0\left(W,\mathsf{{\rm Y}}\right)$ endowed with the inner product

$${⟨\zeta ,\varrho ⟩}_{L_R^0\left(W,\mathsf{{\rm Y}}\right)}=\sum _{u=1}^{\infty }⟨\zeta {ϵ}_u,\varrho {ϵ}_u⟩$$

is a separable HS.

Definition 5

([36]). Assume that $\Phi :U\to L_R^0\left(W,\mathsf{{\rm Y}}\right)$ satisfy

$$\sum _{u=1}^{\infty }{∥M_c\left(\Phi R^{0.5}\right){ϵ}_u∥}_{L^2\left(U,\mathsf{{\rm Y}}\right)}<\infty .$$

(4)

Then, the random integral of Φ w.r.t. ${\Re }^H$ is given by

$$\begin{array}{ccc}\hfill \underset{0}{\overset{s}{\int }}\Phi \left(r\right)d{\Re }^H\left(r\right)& =& \sum _{u=1}^{\infty }\underset{0}{\overset{s}{\int }}\Phi \left(r\right)R^{0.5}{ϵ}_{u}d{\Re }_u^H\left(r\right)\hfill \\ & =& \sum _{u=1}^{\infty }\underset{0}{\overset{s}{\int }}\left(M_c\left(\Phi R^{0.5}{ϵ}_u\right)\right)\left(r\right)d\mathsf{{\rm Y}}\left(r\right),r\in U.\hfill \end{array}$$

Clearly, if

$$\sum _{u=1}^{\infty }{∥\Phi R^{0.5}{ϵ}_u∥}_{L^{\frac{1}{H}}\left(U,\mathsf{{\rm Y}}\right)}<\infty ,$$

(5)

then, (4) comes from (3).

Lemma 1

([37]). If $\Phi :U\to L_R^0\left(W,\mathsf{{\rm Y}}\right)$ fulfills (5), then

$$E\left[{∥\underset{r}{\overset{s}{\int }}\Phi \left(\beta \right)d{\Re }^H\left(\beta \right)∥}_{L_R^0\left(W,\mathsf{{\rm Y}}\right)}^2\right]\le B_H{\left(s-r\right)}^{2H-1}\sum _{u=1}^{\infty }\underset{r}{\overset{s}{\int }}{∥\Phi \left(\beta \right)R^{0.5}{ϵ}_u∥}_{L_R^0\left(W,\mathsf{{\rm Y}}\right)}^{2}d\beta ,$$

where $B_H$ is a constant depending on $H.$ Further, if ${\sum }_{u=1}^{\infty }{∥\Phi \left(s\right)R^{0.5}{ϵ}_u∥}_{L_R^0\left(W,\mathsf{{\rm Y}}\right)}$ is uniformly convergent, then

$$E\left[{∥\underset{r}{\overset{s}{\int }}\Phi \left(\beta \right)d{\Re }^H\left(\beta \right)∥}_{L_R^0\left(W,\mathsf{{\rm Y}}\right)}^2\right]\le B_H{\left(s-r\right)}^{2H-1}\underset{r}{\overset{s}{\int }}{∥\Phi \left(\beta \right)∥}_{L_R^0\left(W,\mathsf{{\rm Y}}\right)}^{2}d\beta .$$

From the fractional calculus, we now provide some fundamental definitions and properties. Here, the integer part of p is represented by $\left[p\right]$, and $\Gamma (.)$ is the Gamma function.

Definition 6

([38]). For the function $¯h:{\mathbb{R}}_{+}\to \mathbb{R},$

(i) 

the Riemann–Liouville (RL) integral of order p is described as

$$I_{0^{+}}^p¯h\left(s\right)=\frac{1}{\Gamma \left(p\right)}\underset{0}{\overset{s}{\int }}{\left(s-r\right)}^{p-1}¯h\left(r\right)dr,s,p>0,$$

provided that the integral $\underset{0}{\overset{s}{\int }}{\left(s-r\right)}^{p-1}¯h\left(r\right)dr$ exists on ${\mathbb{R}}_{+},$ where ${\mathbb{R}}_{+}=[0,\infty );$

(ii) 

the RL derivative of order p is given by

$${ }^{RL}{D}_{0^{+}}^p¯h\left(s\right)=\frac{1}{\Gamma (u-p)}\frac{d^u}{d{s}^u}\underset{0}{\overset{s}{\int }}{\left(s-r\right)}^{u-p-1}¯h\left(r\right)dr,$$

for all $s>0$ and all $u=\left[p\right]+1;$

(iii) 

the CFD of order p is defined by

$${ }^C{D}_{0^{+}}^p¯h\left(s\right){=}^{RL}{D}_{0^{+}}^p\left(¯h\left(s\right)-\sum _{j=0}^{u-1}\frac{s^j}{j!}{¯h}^{\left(j\right)}\left(0\right)\right),s>0,u=\left[p\right]+1.$$

In addition, if ${¯h}^{\left(u\right)}\in \mathbf{C}\left({\mathbb{R}}_{+}\right),$ we get

$${ }^C{D}_{0^{+}}^p¯h\left(s\right)=\frac{1}{\Gamma (u-p)}\frac{d^u}{d{s}^u}\underset{0}{\overset{s}{\int }}{\left(s-r\right)}^{u-p-1}{¯h}^{\left(u\right)}\left(r\right)dr,u=\left[p\right]+1.$$

The meaning of a mild solution to the system (1) is then defined. We need the following ideas in order to accomplish this.

Definition 7

([34]). Assume that $\left(\mho ,\Theta ,P\right)$ is a PS. If the filtration $\Theta ={\left({\Theta }_s\right)}_{s\ge 0}$ satisfies the axioms below

(i) 

${\Theta }_r\subset {\Theta }_s,$ if $r\le s;$

(ii) 

${\Theta }_{\infty }=\sigma \left({\cup }_{s\ge 0}{\Theta }_s\right).$

Then Θ is called a family of $\sigma -$ algebras ${\Theta }_s,$ indexed by $s\in [0,\infty ),$ and all belonging to $\Theta .$

Definition 8

([34]). Let $\left(\mho ,\Theta ,P\right)$ be a PS. If the random variable $S_s$ $(s\ge 0)$ is measurable relative to $\Theta ,$ we say that the SP ${\left(S_s\right)}_{s\ge 0}$ is adapted.

Now, we present our definition about a mild solution in this paper.

Definition 9.

Let $\left(\mho ,\Theta ,P\right)$ be a PS. A real-valued SP $\varpi =\left(\omega ,\upsilon ,\theta \right)\in \complement$ is said to be a mild solution to the system (1) if the following assumptions are true:

(i) 

$\omega \left(0\right)=\kappa \left(\omega ,\upsilon ,\theta \right),$ $\upsilon \left(0\right)=\tau \left(\omega ,\upsilon ,\theta \right)$ and $\omega \left(0\right)=\mu \left(\omega ,\upsilon ,\theta \right),$

(ii) 

for all $s\in U,$ $\varpi \left(s\right)$ is ${\Theta }_s-$ adapted,

(iii) 

for all $s\in U,$ $\varpi \left(s\right)$ owns a limit from the left and is right continuous,

(iv) 

for $s\in U,$ $\varpi \left(s\right)$ fulfills

$$\left\{\begin{array}{c}\omega \left(s\right)=Z_p\left(s\right)\kappa \left(\omega ,\upsilon ,\theta \right)+\underset{0}{\overset{s}{\int }}{\left(s-r\right)}^{p-1}{\chi }_p\left(s-r\right){¯h}_1\left(r,\omega \left(r\right),\upsilon \left(r\right),\theta \left(r\right)\right)dr\hfill \\ +\underset{0}{\overset{s}{\int }}{\left(s-r\right)}^{p-1}{\chi }_p\left(s-r\right){\rho }_1\left(r\right)d{\Re }^{H_1}\left(r\right),\hfill \\ \upsilon \left(s\right)=Z_p\left(s\right)\tau \left(\omega ,\upsilon ,\theta \right)+\underset{0}{\overset{s}{\int }}{\left(s-r\right)}^{p-1}{\chi }_p\left(s-r\right){¯h}_2\left(r,\omega \left(r\right),\upsilon \left(r\right),\theta \left(r\right)\right)dr\hfill \\ +\underset{0}{\overset{s}{\int }}{\left(s-r\right)}^{p-1}{\chi }_p\left(s-r\right){\rho }_2\left(r\right)d{\Re }^{H_2}\left(r\right),\hfill \\ \theta \left(s\right)=Z_p\left(s\right)\mu \left(\omega ,\upsilon ,\theta \right)+\underset{0}{\overset{s}{\int }}{\left(s-r\right)}^{p-1}{\chi }_p\left(s-r\right){¯h}_3\left(r,\omega \left(r\right),\upsilon \left(r\right),\theta \left(r\right)\right)dr,\hfill \\ +\underset{0}{\overset{s}{\int }}{\left(s-r\right)}^{p-1}{\chi }_p\left(s-r\right){\rho }_3\left(r\right)d{\Re }^{H_3}\left(r\right),\hfill \end{array}\right.$$

where

$$Z_p\left(s\right)={\int }_0^{\infty }{\zeta }_p\left(\nu \right)Z\left(s^p\nu \right)d\nu ,{\chi }_p\left(s\right)=p{\int }_0^{\infty }\nu {\zeta }_p\left(\nu \right)Z\left(s^p\nu \right)d\nu ,$$

$\left\{Z\left(s\right):s\in U\right\}$ is our strongly continuous semigroup in $\mathsf{{\rm Y}},$

$$\begin{array}{ccc}\hfill {\zeta }_p\left(\nu \right)& =& \frac{1}{p}{\nu }^{-\left(1+\frac{1}{p}\right)}{\varkappa }_p\left({\nu }^{-\frac{1}{p}}\right)\ge 0,\hfill \\ \hfill {\varkappa }_p\left(\nu \right)& =& \frac{1}{\pi }\sum _{j=0}^{u-1}{\left(-1\right)}^{u-1}{\nu }^{-up-1}\frac{\Gamma \left(up+1\right)}{u!}sin\left(u\pi p\right),\nu \in (0,\infty ),\hfill \end{array}$$

the function ${\zeta }_p$ is a probability density described on $(0,\infty )$ with

$${\zeta }_p\left(\nu \right)\ge 0,and{\int }_0^{\infty }{\zeta }_p\left(\nu \right)d\nu =1.$$

Moreover, we need the following lemma:

Lemma 2.

The following hypotheses are true:

(1) 

for each fixed $s\ge 0,$ $Z_p\left(s\right)$ and ${\chi }_p\left(s\right)$ are BLOs. Particularly, there is a positive constant K such that

$$∥Z_p\left(s\right)\omega ∥\le K∥\omega ∥,and∥{\chi }_p\left(s\right)\omega ∥\le \frac{pK}{\Gamma \left(1+p\right)}∥\omega ∥,\mathrm{for}\omega \in S,$$

(2) 

for all $s\ge 0,$ $Z_p\left(s\right)$ and ${\chi }_p\left(s\right)$ are strongly continuous,

(3) 

the operators $Z_p\left(s\right)$ and ${\chi }_p\left(s\right)$ are compact, provided that $Z\left(s\right)$ is compact for every $s>0.$

We can consider our system as a FP problem in $\mathbf{C}\left(U,S_1\right)\times \mathbf{C}\left(U,S_2\right)\times \mathbf{C}\left(U,S_3\right)$ for the nonlinear operator

$$\beth =\left({\beth }_1,{\beth }_2,{\beth }_3\right):\complement \times \complement \times \complement \to \complement \times \complement \times \complement ,$$

(6)

which is described as

$$\left\{\begin{array}{c}{\beth }_1\left(\omega ,\upsilon ,\theta \right)=Z_p\left(s\right)\kappa \left(\omega ,\upsilon ,\theta \right)+\underset{0}{\overset{s}{\int }}{\left(s-r\right)}^{p-1}{\chi }_p\left(s-r\right){¯h}_1\left(r,\omega \left(r\right),\upsilon \left(r\right),\theta \left(r\right)\right)dr\hfill \\ +\underset{0}{\overset{s}{\int }}{\left(s-r\right)}^{p-1}{\chi }_p\left(s-r\right){\rho }_1\left(r\right)d{\Re }^{H_1}\left(r\right),\hfill \\ {\beth }_2\left(\omega ,\upsilon ,\theta \right)=Z_p\left(s\right)\tau \left(\omega ,\upsilon ,\theta \right)+\underset{0}{\overset{s}{\int }}{\left(s-r\right)}^{p-1}{\chi }_p\left(s-r\right){¯h}_2\left(r,\omega \left(r\right),\upsilon \left(r\right),\theta \left(r\right)\right)dr\hfill \\ +\underset{0}{\overset{s}{\int }}{\left(s-r\right)}^{p-1}{\chi }_p\left(s-r\right){\rho }_2\left(r\right)d{\Re }^{H_2}\left(r\right),\hfill \\ {\beth }_3\left(\omega ,\upsilon ,\theta \right)=Z_p\left(s\right)\mu \left(\omega ,\upsilon ,\theta \right)+\underset{0}{\overset{s}{\int }}{\left(s-r\right)}^{p-1}{\chi }_p\left(s-r\right){¯h}_3\left(r,\omega \left(r\right),\upsilon \left(r\right),\theta \left(r\right)\right)dr,\hfill \\ +\underset{0}{\overset{s}{\int }}{\left(s-r\right)}^{p-1}{\chi }_p\left(s-r\right){\rho }_3\left(r\right)d{\Re }^{H_3}\left(r\right).\hfill \end{array}\right.$$

3. Main Theorems

In this section, we present the existence and uniqueness of the solution to the system (1) under mild conditions. We start with the following assertions:

(A₁) 

There exist the constants $h_j\ge 0$, $c_j\ge 0$ and $q_j\ge 0$ $\left(j=1,2,3\right)$ such that

$$\left\{\begin{array}{c}{∥{¯h}_1\left(s,\omega \left(r\right),\upsilon \left(r\right),\theta \left(r\right)\right)-{¯h}_1\left(s,\overline{\omega }\left(r\right),\overline{\upsilon }\left(r\right),\overline{\theta }\left(r\right)\right)∥}^2\hfill \\ \le h_{1}E\left[{∥\omega -\overline{\omega }∥}^2\right]+c_{1}E\left[{∥\upsilon -\overline{\upsilon }∥}^2\right]+q_{1}E\left[{∥\theta -\overline{\theta }∥}^2\right]\hfill \\ {∥{¯h}_2\left(s,\omega \left(r\right),\upsilon \left(r\right),\theta \left(r\right)\right)-{¯h}_2\left(s,\overline{\omega }\left(r\right),\overline{\upsilon }\left(r\right),\overline{\theta }\left(r\right)\right)∥}^2\hfill \\ \le h_{2}E\left[{∥\omega -\overline{\omega }∥}^2\right]+c_{2}E\left[{∥\upsilon -\overline{\upsilon }∥}^2\right]+q_{2}E\left[{∥\theta -\overline{\theta }∥}^2\right]\hfill \\ {∥{¯h}_2\left(s,\omega \left(r\right),\upsilon \left(r\right),\theta \left(r\right)\right)-{¯h}_2\left(s,\overline{\omega }\left(r\right),\overline{\upsilon }\left(r\right),\overline{\theta }\left(r\right)\right)∥}^2\hfill \\ \le h_{3}E\left[{∥\omega -\overline{\omega }∥}^2\right]+c_{3}E\left[{∥\upsilon -\overline{\upsilon }∥}^2\right]+q_{3}E\left[{∥\theta -\overline{\theta }∥}^2\right],\hfill \end{array}\right.$$

for all $\omega ,\upsilon ,\theta ,\overline{\omega },\overline{\upsilon },\overline{\theta }\in \mathbb{R}$ and $r\in U=[0,c].$

(A₂) 

There exists the constants ${\Im }_j>0$, ${\Re }_j>0$ and ${\gamma }_j>0$ $\left(j=1,2,3\right)$ such that

$$\left\{\begin{array}{c}{∥\kappa \left(\omega ,\upsilon ,\theta \right)-\kappa \left(\overline{\omega },\overline{\upsilon },\overline{\theta }\right)∥}^2\le {\Im }_1{∥\omega -\overline{\omega }∥}^2+{\Re }_1{∥\upsilon -\overline{\upsilon }∥}^2+{\gamma }_1{∥\theta -\overline{\theta }∥}^2,\hfill \\ {∥\tau \left(\omega ,\upsilon ,\theta \right)-\tau \left(\overline{\omega },\overline{\upsilon },\overline{\theta }\right)∥}^2\le {\Im }_2{∥\omega -\overline{\omega }∥}^2+{\Re }_2{∥\upsilon -\overline{\upsilon }∥}^2+{\gamma }_2{∥\theta -\overline{\theta }∥}^2,\hfill \\ {∥\mu \left(\omega ,\upsilon ,\theta \right)-\mu \left(\overline{\omega },\overline{\upsilon },\overline{\theta }\right)∥}^2\le {\Im }_3{∥\omega -\overline{\omega }∥}^2+{\Re }_3{∥\upsilon -\overline{\upsilon }∥}^2+{\gamma }_3{∥\theta -\overline{\theta }∥}^2,\hfill \end{array}\right.$$

for all $\omega ,\upsilon ,\theta ,\overline{\omega },\overline{\upsilon },\overline{\theta }\in \mathbb{R}$.

Here, we apply Perov’s FP theorem to study the existence and uniqueness of the FP, which is consider a unique solution of our system.

Theorem 4.

Via the hypotheses $\left(A_1\right)$ and $\left(A_2\right)$, if the matrix

$$V=\sqrt{3}K\left(\begin{array}{ccc}\sqrt{{\Im }_1+\frac{c^{2p-1}}{{\left(\Gamma \left(p\right)\right)}^2(2p-1)}{h}_1}& \sqrt{{\Re }_1+\frac{c^{2p-1}}{{\left(\Gamma \left(p\right)\right)}^2(2p-1)}{c}_1}& \sqrt{{\gamma }_1+\frac{c^{2p-1}}{{\left(\Gamma \left(p\right)\right)}^2(2p-1)}{q}_1}\\ \sqrt{{\Im }_2+\frac{c^{2p-1}}{{\left(\Gamma \left(p\right)\right)}^2(2p-1)}{h}_2}& \sqrt{{\Re }_2+\frac{c^{2p-1}}{{\left(\Gamma \left(p\right)\right)}^2(2p-1)}{c}_2}& \sqrt{{\gamma }_2+\frac{c^{2p-1}}{{\left(\Gamma \left(p\right)\right)}^2(2p-1)}{q}_2}\\ \sqrt{{\Im }_3+\frac{c^{2p-1}}{{\left(\Gamma \left(p\right)\right)}^2(2p-1)}{h}_3}& \sqrt{{\Re }_3+\frac{c^{2p-1}}{{\left(\Gamma \left(p\right)\right)}^2(2p-1)}{c}_3}& \sqrt{{\gamma }_3+\frac{c^{2p-1}}{{\left(\Gamma \left(p\right)\right)}^2(2p-1)}{q}_3}\end{array}\right)$$

convergence to 0. Then, the system (1) has a unique solution.

Proof. 

We obtain the proof by fulfilling the assumptions of Perov’s FP theorem. Consider

$$\begin{array}{ccc}\hfill {\daleth }_1& =& E\left[{∥{\beth }_1\left(\omega \left(s\right),\upsilon \left(s\right),\theta \left(s\right)\right)-{\beth }_1\left(\overline{\omega }\left(s\right),\overline{\upsilon }\left(s\right),\overline{\theta }\left(s\right)\right)∥}^2\right]\hfill \\ & \le & 3E\left[{∥Z_p\left(s\right)\left(\kappa \left(\omega ,\upsilon ,\theta \right)-\kappa \left(\overline{\omega },\overline{\upsilon },\overline{\theta }\right)\right)∥}^2\right]\hfill \\ & & +3E\left[{∥\underset{0}{\overset{s}{\int }}{\left(s-r\right)}^{p-1}{\chi }_p\left(s-r\right)\left({¯h}_1\left(s,\omega \left(r\right),\upsilon \left(r\right),\theta \left(r\right)\right)-{¯h}_1\left(r,\overline{\omega }\left(r\right),\overline{\upsilon }\left(r\right),\overline{\theta }\left(r\right)\right)\right)ds∥}^2\right],\hfill \end{array}$$

and applying Lemma 2, the hypotheses $\left(A_1\right)$ and $\left(A_2\right)$, Hölder’s inequality, and Fubini’s theorem, we can write

$$\begin{array}{ccc}\hfill {\daleth }_1& \le & 3K^{2}E\left[{∥\kappa \left(\omega ,\upsilon ,\theta \right)-\kappa \left(\overline{\omega },\overline{\upsilon },\overline{\theta }\right)∥}^2\right]\hfill \\ & & +3\frac{{\left(pK\right)}^2}{{\left(\Gamma \left(1+p\right)\right)}^2}E\left[\underset{0}{\overset{s}{\int }}{∥{¯h}_1\left(s,\omega \left(r\right),\upsilon \left(r\right),\theta \left(r\right)\right)-{¯h}_1\left(r,\overline{\omega }\left(r\right),\overline{\upsilon }\left(r\right),\overline{\theta }\left(r\right)\right)∥}^{2}ds\right]\hfill \\ & \le & 3K^2\left[{\Im }_1{∥\omega -\overline{\omega }∥}_{\mathbf{C}}^2+{\Re }_1{∥\upsilon -\overline{\upsilon }∥}_{\mathbf{C}}^2+{\gamma }_1{∥\theta -\overline{\theta }∥}_{\mathbf{C}}^2\right]\hfill \\ & & +3\frac{{\left(pK\right)}^2{c}^{2p-1}}{{\left(\Gamma \left(1+p\right)\right)}^2\left(2p-1\right)}\left[h_1{∥\omega -\overline{\omega }∥}_{\mathbf{C}}^2+c_1{∥\upsilon -\overline{\upsilon }∥}_{\mathbf{C}}^2+q_1{∥\theta -\overline{\theta }∥}_{\mathbf{C}}^2\right],\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}\hfill {\daleth }_1& =& E\left[{∥{\beth }_1\left(\omega \left(s\right),\upsilon \left(s\right),\theta \left(s\right)\right)-{\beth }_1\left(\overline{\omega }\left(s\right),\overline{\upsilon }\left(s\right),\overline{\theta }\left(s\right)\right)∥}^2\right]\hfill \\ & \le & \left(3K^2{\Im }_1+\frac{3{\left(pK\right)}^2{c}^{2p-1}}{{\left(\Gamma \left(1+p\right)\right)}^2\left(2p-1\right)}{h}_1\right){∥\omega -\overline{\omega }∥}_{\complement }^2\hfill \\ & & +\left(3K^2{\Re }_1+\frac{3{\left(pK\right)}^2{c}^{2p-1}}{{\left(\Gamma \left(1+p\right)\right)}^2\left(2p-1\right)}{c}_1\right){∥\upsilon -\overline{\upsilon }∥}_{\complement }^2\hfill \\ & & +\left(3K^2{\gamma }_1+\frac{3{\left(pK\right)}^2{c}^{2p-1}}{{\left(\Gamma \left(1+p\right)\right)}^2\left(2p-1\right)}{q}_1\right){∥\theta -\overline{\theta }∥}_{\complement }^2,\hfill \end{array}$$

Analogously, we get

$$\begin{array}{ccc}\hfill {\daleth }_2& =& E\left[{∥{\beth }_2\left(\omega \left(s\right),\upsilon \left(s\right),\theta \left(s\right)\right)-{\beth }_2\left(\overline{\omega }\left(s\right),\overline{\upsilon }\left(s\right),\overline{\theta }\left(s\right)\right)∥}^2\right]\hfill \\ & \le & \left(3K^2{\Im }_2+\frac{3{\left(pK\right)}^2{c}^{2p-1}}{{\left(\Gamma \left(1+p\right)\right)}^2\left(2p-1\right)}{h}_2\right){∥\omega -\overline{\omega }∥}_{\complement }^2\hfill \\ & & +\left(3K^2{\Re }_2+\frac{3{\left(pK\right)}^2{c}^{2p-1}}{{\left(\Gamma \left(1+p\right)\right)}^2\left(2p-1\right)}{c}_2\right){∥\upsilon -\overline{\upsilon }∥}_{\complement }^2\hfill \\ & & +\left(3K^2{\gamma }_3+\frac{3{\left(pK\right)}^2{c}^{2p-1}}{{\left(\Gamma \left(1+p\right)\right)}^2\left(2p-1\right)}{q}_3\right){∥\theta -\overline{\theta }∥}_{\complement }^2,\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\daleth }_3& =& E\left[{∥{\beth }_3\left(\omega \left(s\right),\upsilon \left(s\right),\theta \left(s\right)\right)-{\beth }_3\left(\overline{\omega }\left(s\right),\overline{\upsilon }\left(s\right),\overline{\theta }\left(s\right)\right)∥}^2\right]\hfill \\ & \le & \left(3K^2{\Im }_3+\frac{3{\left(pK\right)}^2{c}^{2p-1}}{{\left(\Gamma \left(1+p\right)\right)}^2\left(2p-1\right)}{h}_3\right){∥\omega -\overline{\omega }∥}_{\complement }^2\hfill \\ & & +\left(3K^2{\Re }_3+\frac{3{\left(pK\right)}^2{c}^{2p-1}}{{\left(\Gamma \left(1+p\right)\right)}^2\left(2p-1\right)}{c}_3\right){∥\upsilon -\overline{\upsilon }∥}_{\complement }^2\hfill \\ & & +\left(3K^2{\gamma }_3+\frac{3{\left(pK\right)}^2{c}^{2p-1}}{{\left(\Gamma \left(1+p\right)\right)}^2\left(2p-1\right)}{q}_3\right){∥\theta -\overline{\theta }∥}_{\complement }^2.\hfill \end{array}$$

Hence, one has

$$\begin{array}{ccc}& & {∥\beth \left(\omega ,\upsilon ,\theta \right)-\beth \left(\overline{\omega },\overline{\upsilon },\overline{\theta }\right)∥}_{\complement }\hfill \\ & =& \left(\begin{array}{c}{∥{\beth }_1\left(\omega ,\upsilon ,\theta \right)-{\beth }_2\left(\overline{\omega },\overline{\upsilon },\overline{\theta }\right)∥}_{\complement }\\ {∥{\beth }_2\left(\omega ,\upsilon ,\theta \right)-{\beth }_2\left(\overline{\omega },\overline{\upsilon },\overline{\theta }\right)∥}_{\complement }\\ {∥{\beth }_3\left(\omega ,\upsilon ,\theta \right)-{\beth }_3\left(\overline{\omega },\overline{\upsilon },\overline{\theta }\right)∥}_{\complement }\end{array}\right)\hfill \\ & \le & \sqrt{3}K\left(\begin{array}{ccc}\sqrt{{\Im }_1+\frac{c^{2p-1}}{{\left(\Gamma \left(p\right)\right)}^2(2p-1)}{h}_1}& \sqrt{{\Re }_1+\frac{c^{2p-1}}{{\left(\Gamma \left(p\right)\right)}^2(2p-1)}{c}_1}& \sqrt{{\gamma }_1+\frac{c^{2p-1}}{{\left(\Gamma \left(p\right)\right)}^2(2p-1)}{q}_1}\\ \sqrt{{\Im }_2+\frac{c^{2p-1}}{{\left(\Gamma \left(p\right)\right)}^2(2p-1)}{h}_2}& \sqrt{{\Re }_2+\frac{c^{2p-1}}{{\left(\Gamma \left(p\right)\right)}^2(2p-1)}{c}_2}& \sqrt{{\gamma }_2+\frac{c^{2p-1}}{{\left(\Gamma \left(p\right)\right)}^2(2p-1)}{q}_2}\\ \sqrt{{\Im }_3+\frac{c^{2p-1}}{{\left(\Gamma \left(p\right)\right)}^2(2p-1)}{h}_3}& \sqrt{{\Re }_3+\frac{c^{2p-1}}{{\left(\Gamma \left(p\right)\right)}^2(2p-1)}{c}_3}& \sqrt{{\gamma }_3+\frac{c^{2p-1}}{{\left(\Gamma \left(p\right)\right)}^2(2p-1)}{q}_3}\end{array}\right)\hfill \\ & & \times \left(\begin{array}{c}{∥\omega -\overline{\omega }∥}_{\complement }\\ {∥\upsilon -\overline{\upsilon }∥}_{\complement }\\ {∥\theta -\overline{\theta }∥}_{\complement }\end{array}\right).\hfill \end{array}$$

Therefore,

$${∥\beth \left(\omega ,\upsilon ,\theta \right)-\beth \left(\overline{\omega },\overline{\upsilon },\overline{\theta }\right)∥}_{\complement }\le V\left(\begin{array}{c}{∥\omega -\overline{\omega }∥}_{\complement }\\ {∥\upsilon -\overline{\upsilon }∥}_{\complement }\\ {∥\theta -\overline{\theta }∥}_{\complement }\end{array}\right).$$

According to Theorem 2, ℶ has a unique FP $\left(\omega ,\upsilon ,\theta \right)\in \complement \times \complement \times \complement ,$ which it is a unique solution to the system (1). □

Now, using the nonlinear alternative of the Leray-Schauder type, we shall present an existence result. To get our result, we need the following circumstances:

(A₃) 

${¯h}_1,$ ${¯h}_2,$ and ${¯h}_3$ are $L^1$-Carathédory functions;

(A₄) 

There exist positive constants ${\tilde{\Im }}_j,$ ${\tilde{\Re }}_j,$ ${\tilde{\gamma }}_j$ and ${\tilde{M}}_j$ $\left(j\in \{1,2,3\}\right)$ such that

$$\left\{\begin{array}{c}{∥\kappa \left(\omega ,\upsilon ,\theta \right)∥}^2\le {\tilde{\Im }}_1{∥\omega ∥}_{\mathbf{C}}^2+{\tilde{\Re }}_1{∥\upsilon ∥}_{\mathbf{C}}^2+{\tilde{\gamma }}_1{∥\theta ∥}_{\mathbf{C}}^2+{\tilde{M}}_1,\hfill \\ {∥\tau \left(\omega ,\upsilon ,\theta \right)∥}^2\le {\tilde{\Im }}_2{∥\omega ∥}_{\mathbf{C}}^2+{\tilde{\Re }}_2{∥\upsilon ∥}_{\mathbf{C}}^2+{\tilde{\gamma }}_2{∥\theta ∥}_{\mathbf{C}}^2+{\tilde{M}}_2,\hfill \\ {∥\mu \left(\omega ,\upsilon ,\theta \right)∥}^2\le {\tilde{\Im }}_3{∥\omega ∥}_{\mathbf{C}}^2+{\tilde{\Re }}_3{∥\upsilon ∥}_{\mathbf{C}}^2+{\tilde{\gamma }}_3{∥\theta ∥}_{\mathbf{C}}^2+{\tilde{M}}_3,\hfill \end{array}\right.$$

for all $\omega ,\upsilon ,\theta \in \mathbf{C}\left(U\right).$

(A₅) 

There exist the functions ${𝓁}_j,$ ${\widetilde{𝓁}}_j,$ ${\widehat{𝓁}}_j,$ and ${\overline{𝓁}}_j$ in $L^1\left(U,{\mathbb{R}}_{+}\right),$ for $j\in \{1,2,3\}$ such that

$$\left\{\begin{array}{c}{∥{¯h}_1\left(s,\omega ,\upsilon ,\theta \right)∥}^2\le {𝓁}_1\left(s\right){∥\omega ∥}^2+{\widetilde{𝓁}}_1\left(s\right){∥\upsilon ∥}^2+{\widehat{𝓁}}_1\left(s\right){∥\theta ∥}^2+{\overline{𝓁}}_1\left(s\right),\hfill \\ {∥{¯h}_2\left(s,\omega ,\upsilon ,\theta \right)∥}^2\le {𝓁}_2\left(s\right){∥\omega ∥}^2+{\widetilde{𝓁}}_2\left(s\right){∥\upsilon ∥}^2+{\widehat{𝓁}}_2\left(s\right){∥\theta ∥}^2+{\overline{𝓁}}_2\left(s\right),\hfill \\ {∥{¯h}_3\left(s,\omega ,\upsilon ,\theta \right)∥}^2\le {𝓁}_3\left(s\right){∥\omega ∥}^2+{\widetilde{𝓁}}_3\left(s\right){∥\upsilon ∥}^2+{\widehat{𝓁}}_3\left(s\right){∥\theta ∥}^2+{\overline{𝓁}}_3\left(s\right),\hfill \end{array}\right.$$

for every $\omega ,\upsilon ,\theta \in \mathbf{C}\left(U\right)$, and $s\in U=[0,c].$

(A₆) 

For $j=1,2,3,$ there exist measurable functions ${\rho }^j:U\to L_R^0(W,\mathsf{{\rm Y}})$ and there exist positive constants ${\zeta }_j>0$ such that

(i)

${sup}_{0\le r\le c}{∥{\rho }^j\left(r\right)∥}_{L_R^0(W,\mathsf{{\rm Y}})}^2\le {\zeta }_j,$

(ii)

${\sum }_{u=1}^{\infty }{∥{\rho }^j{R}^{0.5}{ϵ}_u∥}_{L_R^0(W,\mathsf{{\rm Y}})}<\infty$,

(iii)

${\sum }_{u=1}^{\infty }{∥{\rho }^j{R}^{0.5}{ϵ}_u∥}_{\mathsf{{\rm Y}}}$ is uniformly convergent.

Theorem 5.

Under the hypotheses $\left(A_3\right)–\left(A_6\right)$, the problem (1) has at least one solution on U, provided that

$$K^{*}=max\left(3K^2\left({\tilde{\Im }}_j+{\tilde{\Re }}_j+{\tilde{\gamma }}_j\right)\right)<1,j\in \left\{1,2,3\right\}$$

is true.

Proof. 

It is evident that the system (1) can be solved by the FPs of the operator ℶ stated in (6). We first demonstrate that ℶ is completely continuous in order to apply Theorem 2. We split the proof onto the following stages:

Stage 1: 

$\beth =\left({\beth }_1,{\beth }_2,{\beth }_3\right)$ is continuous. Assume that $\left({\omega }_u,{\upsilon }_u,{\theta }_u\right)$ is a sequence such that $\left({\omega }_u,{\upsilon }_u,{\theta }_u\right)\to \left(\tilde{\omega },\tilde{\upsilon },\tilde{\theta }\right)\in \complement \times \complement \times \complement$ as $u\to \infty .$ Then,

$$\begin{array}{ccc}& & E\left[{∥{\beth }_1\left({\omega }_u,{\upsilon }_u,{\theta }_u\right)-{\beth }_1\left(\tilde{\omega },\tilde{\upsilon },\tilde{\theta }\right)∥}^2\right]\hfill \\ & \le & E\left[∥Z_p\left(s\right)\kappa \left({\omega }_u,{\upsilon }_u,{\theta }_u\right)+\underset{0}{\overset{s}{\int }}{\left(s-r\right)}^{p-1}{\chi }_p\left(s-r\right){¯h}_1\left(r,{\omega }_u\left(r\right),{\upsilon }_u\left(r\right),{\theta }_u\left(r\right)\right)dr\right.\hfill \\ & & +\underset{0}{\overset{s}{\int }}{\left(s-r\right)}^{p-1}{\chi }_p\left(s-r\right){\rho }_1\left(r\right)d{\Re }^{H_1}\left(r\right)-Z_p\left(s\right)\kappa \left(\tilde{\omega },\tilde{\upsilon },\tilde{\theta }\right)\hfill \\ & & -\underset{0}{\overset{s}{\int }}{\left(s-r\right)}^{p-1}{\chi }_p\left(s-r\right){¯h}_1\left(r,\tilde{\omega }\left(r\right),\tilde{\upsilon }\left(r\right),\tilde{\theta }\left(r\right)\right)dr\hfill \\ & & \left.{-\underset{0}{\overset{s}{\int }}{\left(s-r\right)}^{p-1}{\chi }_p\left(s-r\right){\rho }_1\left(r\right)d{\Re }^{H_1}\left(r\right)∥}^2\right]\hfill \\ & \le & 3E\left[{∥Z_p\left(s\right)\left[\kappa \left({\omega }_u,{\upsilon }_u,{\theta }_u\right)-\kappa \left(\tilde{\omega },\tilde{\upsilon },\tilde{\theta }\right)\right]∥}^2\right]\hfill \\ & & +3E\left[{∥\underset{0}{\overset{s}{\int }}{\left(s-r\right)}^{p-1}{\chi }_p\left(s-r\right)\left[{¯h}_1\left(r,{\omega }_u\left(r\right),{\upsilon }_u\left(r\right),{\theta }_u\left(r\right)\right)-{¯h}_1\left(r,\tilde{\omega }\left(r\right),\tilde{\upsilon }\left(r\right),\tilde{\theta }\left(r\right)\right)\right]dr∥}^2\right]\hfill \\ & =& {\Delta }_1+{\Delta }_2.\hfill \end{array}$$

Using the hypothesis $\left(A_2\right),$ we have

$${\Delta }_1\le 2K^{2}E\left[\left({\Im }_1{∥{\omega }_u-\tilde{\omega }∥}^2+{\Re }_1{∥{\upsilon }_u-\tilde{\upsilon }∥}^2+{\gamma }_1{∥{\theta }_u-\tilde{\theta }∥}^2\right)\right]$$

Since $\left({\omega }_u,{\upsilon }_u,{\theta }_u\right)\to \left(\tilde{\omega },\tilde{\upsilon },\tilde{\theta }\right)$ as $u\to \infty ,$ then, by the Lebesgue dominated convergence theorem (LDCT), we have ${\Delta }_1\to 0$ as $u\to \infty .$ Again, from Hölder’s inequality and Lemma 2, one has

$${\Delta }_2\le \frac{2K^2{c}^{2p-1}}{{\left(\Gamma \left(p\right)\right)}^2(2p-1)}E\left[\underset{0}{\overset{s}{\int }}∥{¯h}_1\left(r,{\omega }_u\left(r\right),{\upsilon }_u\left(r\right),{\theta }_u\left(r\right)\right)-{¯h}_1\left(r,\tilde{\omega }\left(r\right),\tilde{\upsilon }\left(r\right),\tilde{\theta }\left(r\right)\right)∥\right]$$

Applying the hypothesis $\left(A_1\right)$ and using the LDCT, we have ${\Delta }_2\to 0$ as $u\to \infty$ because ${¯h}_1$ is $L^1$-Carathédory (Hypothesis $\left(A_3\right))$ and $\left({\omega }_u,{\upsilon }_u,{\theta }_u\right)\to \left(\tilde{\omega },\tilde{\upsilon },\tilde{\theta }\right)$ as $u\to \infty .$ Similarly, one can write

$$\begin{array}{ccc}& & E\left[{∥{\beth }_2\left({\omega }_u,{\upsilon }_u,{\theta }_u\right)-{\beth }_2\left(\tilde{\omega },\tilde{\upsilon },\tilde{\theta }\right)∥}^2\right]\hfill \\ & \le & 2E\left[{∥Z_p\left(s\right)\left[\tau \left({\omega }_u,{\upsilon }_u,{\theta }_u\right)-\tau \left(\tilde{\omega },\tilde{\upsilon },\tilde{\theta }\right)\right]∥}^2\right]\hfill \\ & & +2E\left[{∥\underset{0}{\overset{s}{\int }}{\left(s-r\right)}^{p-1}{\chi }_p\left(s-r\right)\left[{¯h}_2\left(r,{\omega }_u\left(r\right),{\upsilon }_u\left(r\right),{\theta }_u\left(r\right)\right)-{¯h}_2\left(r,\tilde{\omega }\left(r\right),\tilde{\upsilon }\left(r\right),\tilde{\theta }\left(r\right)\right)\right]dr∥}^2\right]\hfill \\ & =& {\tilde{\Delta }}_1+{\tilde{\Delta }}_2\to 0\mathrm{as}u\to \infty ,\hfill \end{array}$$

and

$$\begin{array}{ccc}& & E\left[{∥{\beth }_3\left({\omega }_u,{\upsilon }_u,{\theta }_u\right)-{\beth }_3\left(\tilde{\omega },\tilde{\upsilon },\tilde{\theta }\right)∥}^2\right]\hfill \\ & \le & 2E\left[{∥Z_p\left(s\right)\left[\mu \left({\omega }_u,{\upsilon }_u,{\theta }_u\right)-\mu \left(\tilde{\omega },\tilde{\upsilon },\tilde{\theta }\right)\right]∥}^2\right]\hfill \\ & & +2E\left[{∥\underset{0}{\overset{s}{\int }}{\left(s-r\right)}^{p-1}{\chi }_p\left(s-r\right)\left[{¯h}_3\left(r,{\omega }_u\left(r\right),{\upsilon }_u\left(r\right),{\theta }_u\left(r\right)\right)-{¯h}_3\left(r,\tilde{\omega }\left(r\right),\tilde{\upsilon }\left(r\right),\tilde{\theta }\left(r\right)\right)\right]dr∥}^2\right]\hfill \\ & =& {\widehat{\Delta }}_1+{\widehat{\Delta }}_2\to 0\mathrm{as}u\to \infty .\hfill \end{array}$$

This proves that ℶ is continuous.

Stage 2: 

ℶ maps bounded sets into bounded sets in $\complement \times \complement \times \complement .$ It’s enough to demonstrate that for any $M>0,$ there exists a constant $N=\left(N_1,N_2,N_3\right)>0$ such that, for $\left(\omega ,\upsilon ,\theta \right)\in {\Re }_N=\left\{\left(\omega ,\upsilon ,\theta \right)\in \complement \times \complement \times \complement :{∥\omega ∥}_{\complement }\le N,{∥\upsilon ∥}_{\complement }\le N,{∥\theta ∥}_{\complement }\le N,\right\},$ we have

$${∥\beth \left(\omega ,\upsilon ,\theta \right)∥}_{\complement }\le N.$$

(7)

For each $s\in U,$ from $\left(A_4\right)$, we can write

$$\begin{array}{ccc}\hfill E\left[{∥{\beth }_1\left(\omega \left(s\right),\upsilon \left(s\right),\theta \left(s\right)\right)∥}^2\right]& =& E\left[∥Z_p\left(s\right)\kappa \left(\omega ,\upsilon ,\theta \right)+\underset{0}{\overset{s}{\int }}{\left(s-r\right)}^{p-1}{\chi }_p\left(s-r\right){¯h}_1\left(r,\omega \left(r\right),\upsilon \left(r\right),\theta \left(r\right)\right)dr\right.\hfill \\ & & \left.{+\underset{0}{\overset{s}{\int }}{\left(s-r\right)}^{p-1}{\chi }_p\left(s-r\right){\rho }_1\left(r\right)d{\Re }^{H_1}\left(r\right)∥}^2\right]\hfill \\ & \le & 3E\left[{∥Z_p\left(s\right)\kappa \left(\omega ,\upsilon ,\theta \right)∥}^2\right]\hfill \\ & & +3E\left[{∥\underset{0}{\overset{s}{\int }}{\left(s-r\right)}^{p-1}{\chi }_p\left(s-r\right){¯h}_1\left(r,\omega \left(r\right),\upsilon \left(r\right),\theta \left(r\right)\right)dr∥}^2\right]\hfill \\ & & +3E\left[{∥\underset{0}{\overset{s}{\int }}{\left(s-r\right)}^{p-1}{\chi }_p\left(s-r\right){\rho }_1\left(r\right)d{\Re }^{H_1}\left(r\right)∥}^2\right]\hfill \\ & =& {\nabla }_1+{\nabla }_2+{\nabla }_3,\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill {\nabla }_1& =& 3E\left[{∥Z_p\left(s\right)\kappa \left(\omega ,\upsilon ,\theta \right)∥}^2\right]\hfill \\ & \le & 3K^2\left({\tilde{\Im }}_1{∥\omega ∥}_{\mathbf{C}}^2+{\tilde{\Re }}_1{∥\upsilon ∥}_{\mathbf{C}}^2+{\tilde{\gamma }}_1{∥\theta ∥}_{\mathbf{C}}^2+{\tilde{M}}_1\right)=N_{11},\hfill \end{array}$$

Applying the hypothesis $\left(A_5\right),$ Hölder’s inequality and Lemma 2, we have

$$\begin{array}{ccc}\hfill {\nabla }_2& =& 3E\left[{∥\underset{0}{\overset{s}{\int }}{\left(s-r\right)}^{p-1}{\chi }_p\left(s-r\right){¯h}_1\left(r,\omega \left(r\right),\upsilon \left(r\right),\theta \left(r\right)\right)dr∥}^2\right]\hfill \\ & \le & \frac{3K^2}{{\left(\Gamma \left(p\right)\right)}^2}E\left[{∥\underset{0}{\overset{s}{\int }}{\left(s-r\right)}^{p-1}{¯h}_1\left(r,\omega \left(r\right),\upsilon \left(r\right),\theta \left(r\right)\right)dr∥}^2\right]\hfill \\ & \le & \frac{3K^2}{{\left(\Gamma \left(p\right)\right)}^2}{\left(\underset{0}{\overset{s}{\int }}{\left(s-r\right)}^{p-1}dr\right)}^{2}E\left[{∥{¯h}_1\left(r,\omega \left(r\right),\upsilon \left(r\right),\theta \left(r\right)\right)∥}^{2}dr\right]\hfill \\ & \le & \frac{3K^2{c}^{2p-1}}{(2p-1){\left(\Gamma \left(p\right)\right)}^2}E\left[\underset{0}{\overset{s}{\int }}\left({𝓁}_1\left(s\right){∥\omega ∥}^2+{\widetilde{𝓁}}_1\left(s\right){∥\upsilon ∥}^2+{\widehat{𝓁}}_1\left(s\right){∥\theta ∥}^2+{\overline{𝓁}}_1\left(s\right)\right)dr\right]\hfill \\ & \le & \frac{3K^2{c}^{2p-1}}{(2p-1){\left(\Gamma \left(p\right)\right)}^2}\left(N{∥{𝓁}_1∥}_{L^1}+N{∥{\widetilde{𝓁}}_1∥}_{L^1}+N{∥{\widehat{𝓁}}_1∥}_{L^1}+{∥{\overline{𝓁}}_1∥}_{L^1}\right)=N_{21}.\hfill \end{array}$$

To estimate ${\nabla }_3,$ we use the hypothesis $\left(A_6\right)$ and Lemma 2 as follows:

$$\begin{array}{ccc}\hfill {\nabla }_3& =& 3E\left[{∥\underset{0}{\overset{s}{\int }}{\left(s-r\right)}^{p-1}{\chi }_p\left(s-r\right){\rho }_1\left(r\right)d{\Re }^{H_1}\left(r\right)∥}^2\right]\hfill \\ & \le & 3{\mathbf{C}}_H{s}^{2H-1}\underset{0}{\overset{s}{\int }}{∥{\chi }_p\left(s-r\right){\rho }_1\left(r\right)∥}_{L_R^0(W,\mathsf{{\rm Y}})}^{2}dr\le \frac{3{\mathbf{C}}_H{K}^2{\zeta }_1{b}^{2H+2p+2}}{(2p-1){\left(\Gamma \left(p\right)\right)}^2}=N_{31}.\hfill \end{array}$$

Therefore,

$${∥{\beth }_1\left(\omega ,\upsilon ,\theta \right)∥}_{\complement }^2=E\left[{∥{\beth }_1\left(\omega \left(s\right),\upsilon \left(s\right),\theta \left(s\right)\right)∥}^2\right]\le N_{11}+N_{21}+N_{31}=N_1.$$

Analogously,

$${∥{\beth }_2\left(\omega ,\upsilon ,\theta \right)∥}_{\complement }^2=E\left[{∥{\beth }_2\left(\omega \left(s\right),\upsilon \left(s\right),\theta \left(s\right)\right)∥}^2\right]\le N_{12}+N_{22}+N_{32}=N_2,$$

and

$${∥{\beth }_3\left(\omega ,\upsilon ,\theta \right)∥}_{\complement }^2=E\left[{∥{\beth }_3\left(\omega \left(s\right),\upsilon \left(s\right),\theta \left(s\right)\right)∥}^2\right]\le N_{13}+N_{23}+N_{33}=N_3,$$

where

$$\begin{array}{ccc}\hfill N_{12}& =& 3K^2\left({\tilde{\Im }}_2{∥\omega ∥}_{\mathbf{C}}^2+{\tilde{\Re }}_2{∥\upsilon ∥}_{\mathbf{C}}^2+{\tilde{\gamma }}_2{∥\theta ∥}_{\mathbf{C}}^2+{\tilde{M}}_2\right),\hfill \\ \hfill N_{22}& =& \frac{3K^2{c}^{2p-1}}{(2p-1){\left(\Gamma \left(p\right)\right)}^2}\left(N{∥{𝓁}_2∥}_{L^1}+N{∥{\widetilde{𝓁}}_2∥}_{L^1}+N{∥{\widehat{𝓁}}_2∥}_{L^1}+{∥{\overline{𝓁}}_2∥}_{L^1}\right),\hfill \\ \hfill N_{32}& =& \frac{3{\mathbf{C}}_H{K}^2{\zeta }_2{b}^{2H+2p+2}}{(2p-1){\left(\Gamma \left(p\right)\right)}^2},\hfill \\ \hfill N_{13}& =& 3K^2\left({\tilde{\Im }}_3{∥\omega ∥}_{\mathbf{C}}^2+{\tilde{\Re }}_3{∥\upsilon ∥}_{\mathbf{C}}^2+{\tilde{\gamma }}_3{∥\theta ∥}_{\mathbf{C}}^2+{\tilde{M}}_3\right)\hfill \\ \hfill N_{23}& =& \frac{3K^2{c}^{2p-1}}{(2p-1){\left(\Gamma \left(p\right)\right)}^2}\left(N{∥{𝓁}_3∥}_{L^1}+N{∥{\widetilde{𝓁}}_3∥}_{L^1}+N{∥{\widehat{𝓁}}_3∥}_{L^1}+{∥{\overline{𝓁}}_3∥}_{L^1}\right)\hfill \\ \hfill N_{33}& =& \frac{3{\mathbf{C}}_H{K}^2{\zeta }_3{b}^{2H+2p+2}}{(2p-1){\left(\Gamma \left(p\right)\right)}^2}.\hfill \end{array}$$

This implies that the inequality (7) holds.

Stage 3: 

ℶ maps bounded sets into equicontinuous sets in $\complement \times \complement \times \complement .$ Assume that ${\Re }_N$ is a bounded set defined in Stage 2. Also, assume that $\varpi =\left(\omega ,\upsilon ,\theta \right)\in {\Re }_N$ and $a=\left(a_1,a_2\right)\in U$ with $a_1<a_2,$ we have

$$\begin{array}{ccc}\hfill {\Pi }_1& =& E\left[{∥{\beth }_1\left(\omega \left(a_2\right),\upsilon \left(a_2\right),\theta \left(a_2\right)\right)-{\beth }_1\left(\omega \left(a_1\right),\upsilon \left(a_1\right),\theta \left(a_1\right)\right)∥}^2\right]\hfill \\ & \le & 3E\left[{∥\left(Z_p\left(a_2\right)-Z_p\left(a_1\right)\right)\kappa \left(\omega ,\upsilon ,\theta \right)∥}^2\right]\hfill \\ & & +3E\left[∥\underset{0}{\overset{a_2}{\int }}{\left(a_2-r\right)}^{p-1}{\chi }_p\left(a_2-r\right){¯h}_1\left(r,\omega \left(r\right),\upsilon \left(r\right),\theta \left(r\right)\right)dr\right.\hfill \\ & & \left.{-\underset{0}{\overset{a_1}{\int }}{\left(a_1-r\right)}^{p-1}{\chi }_p\left(a_1-r\right){¯h}_1\left(r,\omega \left(r\right),\upsilon \left(r\right),\theta \left(r\right)\right)dr∥}^2\right]\hfill \\ & & +3E\left[∥\underset{0}{\overset{a_2}{\int }}{\left(a_2-r\right)}^{p-1}{\chi }_p\left(a_2-r\right){\rho }_1\left(r\right)d{\Re }^{H_1}\left(r\right)\right.\hfill \\ & & \left.{-\underset{0}{\overset{a_1}{\int }}{\left(a_1-r\right)}^{p-1}{\chi }_p\left(a_1-r\right){\rho }_1\left(r\right)d{\Re }^{H_1}\left(r\right)∥}^2\right]\hfill \\ & =& {\varphi }_1+{\varphi }_2+{\varphi }_3.\hfill \end{array}$$

Since $Z_p\left(s\right)$ is strongly continuous, then

$$\underset{a_2\to a_1}{lim}\left(Z_p\left(a_2\right)-Z_p\left(a_1\right)\right)\kappa \left(\omega ,\upsilon ,\theta \right)=0.$$

Applying the hypothesis $\left(A_4\right)$ and Lemma 2, we have

$$\begin{array}{ccc}& & E\left[{∥\left(Z_p\left(a_2\right)-Z_p\left(a_1\right)\right)\kappa \left(\omega ,\upsilon ,\theta \right)∥}^2\right]\hfill \\ & \le & \left(Z_p\left(a_2\right)-Z_p\left(a_1\right)\right)\left[{\tilde{\Im }}_1{∥\omega ∥}_{\mathbf{C}}^2+{\tilde{\Re }}_1{∥\upsilon ∥}_{\mathbf{C}}^2+{\tilde{\gamma }}_1{∥\theta ∥}_{\mathbf{C}}^2+{\tilde{M}}_1\right].\hfill \end{array}$$

By the LDCT, we conclude that

$$\underset{a_2\to a_1}{lim}{\varphi }_1=3\underset{a_2\to a_1}{lim}E\left[{∥\left(Z_p\left(a_2\right)-Z_p\left(a_1\right)\right)\kappa \left(\omega ,\upsilon ,\theta \right)∥}^2\right]=0.$$

Now,

$$\begin{array}{ccc}\hfill {\varphi }_2& =& 3E\left[∥\underset{0}{\overset{a_2}{\int }}{\left(a_2-r\right)}^{p-1}{\chi }_p\left(a_2-r\right){¯h}_1\left(r,\omega \left(r\right),\upsilon \left(r\right),\theta \left(r\right)\right)dr\right.\hfill \\ & & \left.{-\underset{0}{\overset{a_1}{\int }}{\left(a_1-r\right)}^{p-1}{\chi }_p\left(a_1-r\right){¯h}_1\left(r,\omega \left(r\right),\upsilon \left(r\right),\theta \left(r\right)\right)dr∥}^2\right]\hfill \\ & \le & 6E\left[∥\underset{0}{\overset{a_1}{\int }}{\left(a_2-r\right)}^{p-1}{\chi }_p\left(a_2-r\right){¯h}_1\left(r,\omega \left(r\right),\upsilon \left(r\right),\theta \left(r\right)\right)dr\right.\hfill \\ & & +\underset{a_1}{\overset{a_2}{\int }}{\left(a_2-r\right)}^{p-1}{\chi }_p\left(a_2-r\right){¯h}_1\left(r,\omega \left(r\right),\upsilon \left(r\right),\theta \left(r\right)\right)dr\hfill \\ & & \left.{-\underset{0}{\overset{a_1}{\int }}{\left(a_1-r\right)}^{p-1}{\chi }_p\left(a_1-r\right){¯h}_1\left(r,\omega \left(r\right),\upsilon \left(r\right),\theta \left(r\right)\right)dr∥}^2\right]\hfill \\ & \le & 6E\left[{∥\underset{0}{\overset{a_1}{\int }}\left[{\left(a_2-r\right)}^{p-1}{\chi }_p\left(a_2-r\right)-{\left(a_1-r\right)}^{p-1}{\chi }_p\left(a_1-r\right)\right]{¯h}_1\left(r,\omega \left(r\right),\upsilon \left(r\right),\theta \left(r\right)\right)dr∥}^2\right]\hfill \\ & & +6E\left[{∥\underset{a_1}{\overset{a_2}{\int }}{\left(a_2-r\right)}^{p-1}{\chi }_p\left(a_2-r\right){¯h}_1\left(r,\omega \left(r\right),\upsilon \left(r\right),\theta \left(r\right)\right)dr∥}^2\right]\hfill \\ & \le & {\varphi }_{21}+{\varphi }_{22}.\hfill \end{array}$$

Using the hypothesis $\left(A_5\right)$, Hölder’s inequality, Lemma 2, and Fubini’s stochastic theorem, we can write

$$\begin{array}{ccc}\hfill {\varphi }_{21}& \le & 6E\left[{\left(\underset{0}{\overset{a_1}{\int }}\left[{\left(a_2-r\right)}^{p-1}{\chi }_p\left(a_2-r\right)-{\left(a_1-r\right)}^{p-1}{\chi }_p\left(a_1-r\right)\right]dr\right)}^2\right]\hfill \\ & & \times E\left[\underset{0}{\overset{a_1}{\int }}{∥{¯h}_1\left(r,\omega \left(r\right),\upsilon \left(r\right),\theta \left(r\right)\right)∥}^{2}dr\right]\hfill \\ & \le & 6{\left(\underset{0}{\overset{a_1}{\int }}\left[{\left(a_2-r\right)}^{p-1}{\chi }_p\left(a_2-r\right)-{\left(a_1-r\right)}^{p-1}{\chi }_p\left(a_1-r\right)\right]dr\right)}^2\hfill \\ & & E\left[\underset{0}{\overset{a_1}{\int }}\left[{𝓁}_1\left(s\right){∥\omega ∥}^2+{\widetilde{𝓁}}_1\left(s\right){∥\upsilon ∥}^2+{\widehat{𝓁}}_1\left(s\right){∥\theta ∥}^2+{\overline{𝓁}}_1\left(s\right)\right]dr\right]\hfill \\ & \le & \frac{6K^2\left[N{∥{𝓁}_1∥}_{L^1}+N{∥{\widetilde{𝓁}}_1∥}_{L^1}+N{∥{\widehat{𝓁}}_1∥}_{L^1}+{∥{\overline{𝓁}}_1∥}_{L^1}\right]}{{\Gamma }^2\left(p\right){(p-2)}^2}\times \left[{\left(a_2-a_1\right)}^{p-1}+a_2^{p-2}-a_1^{p-2}\right]\hfill \\ & \to & 0\mathrm{as}{a}_2\to a_1.\hfill \end{array}$$

Hence, ${lim}_{a_2\to a_1}{\varphi }_{21}=0.$ Again, by using the hypothesis $\left(A_5\right)$, Hölder’s inequality, Lemma 2, we get

$$\begin{array}{ccc}\hfill {\varphi }_{22}& \le & 6E\left[{∥\underset{a_1}{\overset{a_2}{\int }}{\left(a_2-r\right)}^{p-1}{\chi }_p\left(a_2-r\right){¯h}_1\left(r,\omega \left(r\right),\upsilon \left(r\right),\theta \left(r\right)\right)dr∥}^2\right]\hfill \\ & \le & 6\left[{\left(\underset{a_1}{\overset{a_2}{\int }}{\left(a_2-r\right)}^{p-1}{\chi }_p\left(a_2-r\right)dr\right)}^2\right]E\left[\underset{a_1}{\overset{a_2}{\int }}{∥{¯h}_1\left(r,\omega \left(r\right),\upsilon \left(r\right),\theta \left(r\right)\right)∥}^{2}dr\right]\hfill \\ & \le & \frac{6K^2{c}^{2p-1}{\zeta }_1\left[N{∥{𝓁}_1∥}_{L^1}+N{∥{\widetilde{𝓁}}_1∥}_{L^1}+N{∥{\widehat{𝓁}}_1∥}_{L^1}+{∥{\overline{𝓁}}_1∥}_{L^1}\right]}{{\Gamma }^2\left(p\right)(2p-1)}{\left(a_2-a_1\right)}^{2p-1}\hfill \\ & \to & 0\mathrm{as}{a}_2\to a_1.\hfill \end{array}$$

Hence, ${lim}_{a_2\to a_1}{\varphi }_{22}=0.$ Now,

$$\begin{array}{ccc}\hfill {\varphi }_3& =& 3E\left[∥\underset{0}{\overset{a_2}{\int }}{\left(a_2-r\right)}^{p-1}{\chi }_p\left(a_2-r\right){\rho }_1\left(r\right)d{\Re }^{H_1}\left(r\right)\right.\hfill \\ & & \left.{-\underset{0}{\overset{a_1}{\int }}{\left(a_1-r\right)}^{p-1}{\chi }_p\left(a_1-r\right){\rho }_1\left(r\right)d{\Re }^{H_1}\left(r\right)∥}^2\right]\hfill \\ & \le & 3E\left[∥\underset{0}{\overset{a_1}{\int }}{\left(a_2-r\right)}^{p-1}{\chi }_p\left(a_2-r\right){\rho }_1\left(r\right)d{\Re }^{H_1}\left(r\right)-\underset{0}{\overset{a_1}{\int }}{\left(a_1-r\right)}^{p-1}{\chi }_p\left(a_2-r\right){\rho }_1\left(r\right)d{\Re }^{H_1}\left(r\right)\right.\hfill \\ & & \left.{+\underset{a_1}{\overset{a_2}{\int }}{\left(a_2-r\right)}^{p-1}{\chi }_p\left(a_2-r\right){\rho }_1\left(r\right)d{\Re }^{H_1}\left(r\right)∥}^2\right]\hfill \\ & =& {\varphi }_{31}+{\varphi }_{32}.\hfill \end{array}$$

Using the condition $\left(A_5\right)$ and Lemma 2, we get

$$\begin{array}{ccc}\hfill {\varphi }_{31}& =& 6E\left[{∥\underset{0}{\overset{a_1}{\int }}\left[{\left(a_2-r\right)}^{p-1}{\chi }_p\left(a_2-r\right)-{\left(a_1-r\right)}^{p-1}{\chi }_p\left(a_2-r\right)\right]{\rho }_1\left(r\right)d{\Re }^{H_1}\left(r\right)∥}^2\right]\hfill \\ & \le & 6{\mathbf{C}}_H{s}^{2H-1}\underset{0}{\overset{a_1}{\int }}{∥\left[{\left(a_2-r\right)}^{p-1}-{\left(a_1-r\right)}^{p-1}\right]{\rho }_1\left(r\right)∥}_{L_R^0(W,\mathsf{{\rm Y}})}^{2}dr\hfill \\ & \le & \frac{6{\mathbf{C}}_H{s}^{2H-1}}{{\Gamma }^2\left(p\right)(2p-1)}\left[a_1^{2p-1}+{\left(a_2-a_1\right)}^{2q-1}-a_2^{2p-1}\right]\hfill \\ & \to & 0\mathrm{as}{a}_2\to a_1.\hfill \end{array}$$

Therefore, ${lim}_{a_2\to a_1}{\varphi }_{31}=0.$ Also,

$$\begin{array}{ccc}\hfill {\varphi }_{32}& =& 3E\left[{∥\underset{a_1}{\overset{a_2}{\int }}{\left(a_2-r\right)}^{p-1}{\chi }_p\left(a_2-r\right){\rho }_1\left(r\right)d{\Re }^{H_1}\left(r\right)∥}^2\right]\hfill \\ & \le & \frac{6{\mathbf{C}}_H{s}^{2H-1}{\zeta }_1{K}^2{\left(a_2-a_1\right)}^{2q-1}}{{\Gamma }^2\left(p\right)(2p-1)}\to 0,\mathrm{as}{a}_2\to a_1.\hfill \end{array}$$

Therefore, ${lim}_{a_2\to a_1}{\varphi }_{32}=0.$ Thus, ${\varphi }_1\to 0,$ ${\varphi }_2\to 0,$ and ${\varphi }_3\to 0,$ as $a_2\to a_1.$ Consequently,

$$\underset{a_2\to a_1}{lim}E\left[∥{\beth }_1\left(\omega \left(a_2\right),\upsilon \left(a_2\right),\theta \left(a_2\right)\right)-{\beth }_1\left(\omega \left(a_1\right),\upsilon \left(a_1\right),\theta \left(a_1\right)\right)∥\right]=0.$$

Similarly,

$$\underset{a_2\to a_1}{lim}E\left[∥{\beth }_2\left(\omega \left(a_2\right),\upsilon \left(a_2\right),\theta \left(a_2\right)\right)-{\beth }_{21}\left(\omega \left(a_1\right),\upsilon \left(a_1\right),\theta \left(a_1\right)\right)∥\right]=0.$$

and

$$\underset{a_2\to a_1}{lim}E\left[∥{\beth }_3\left(\omega \left(a_2\right),\upsilon \left(a_2\right),\theta \left(a_2\right)\right)-{\beth }_3\left(\omega \left(a_1\right),\upsilon \left(a_1\right),\theta \left(a_1\right)\right)∥\right]=0.$$

Hence,

$$\underset{a_2\to a_1}{lim}E\left[∥\beth \left(\omega \left(a_2\right),\upsilon \left(a_2\right),\theta \left(a_2\right)\right)-\beth \left(\omega \left(a_1\right),\upsilon \left(a_1\right),\theta \left(a_1\right)\right)∥\right]=0.$$

Therefore, the function $s\to \beth \left(\omega \left(s\right),\upsilon \left(s\right),\theta \left(s\right)\right)$ is continuous on $U=[0,c].$ By Arzelá-Ascoli theorem, $\beth :{\Re }_N\to \complement \times \complement \times \complement$ is completely continuous.

Stage 4: 

Solutions are a priori bounded. For $s\in U,$ using the hypotheses $\left(A_4\right),$ $\left(A_5\right),$ and $\left(A_6\right)\left(i\right),$ we get

$$\begin{array}{ccc}& & E\left[{∥\omega \left(s\right)∥}^2\right]\hfill \\ & \le & 3E\left[{∥Z_p\left(s\right)\kappa \left(\omega ,\upsilon ,\theta \right)∥}^2\right]\hfill \\ & & +3E\left[{∥\underset{0}{\overset{s}{\int }}{\left(s-r\right)}^{p-1}{\chi }_p\left(s-r\right){¯h}_1\left(r,\omega \left(r\right),\upsilon \left(r\right),\theta \left(r\right)\right)dr∥}^2\right]\hfill \\ & & +3E\left[{∥\underset{0}{\overset{s}{\int }}{\left(s-r\right)}^{p-1}{\chi }_p\left(s-r\right){\rho }_1\left(r\right)d{\Re }^{H_1}\left(r\right)∥}^2\right]\hfill \\ & \le & 3K^2\left[{\tilde{\Im }}_{1}E\left[{∥\omega \left(s\right)∥}^2\right]+{\tilde{\Re }}_{1}E\left[{∥\upsilon \left(s\right)∥}^2\right]+{\tilde{\gamma }}_{1}E\left[{∥\theta \left(s\right)∥}^2\right]+{\tilde{M}}_1\right]\hfill \\ & & +\frac{3K^2{s}^{2p-1}}{{\Gamma }^2\left(p\right)(2p-1)}\underset{0}{\overset{s}{\int }}\left({𝓁}_1\left(r\right)E\left[{∥\omega \left(r\right)∥}^2\right]+{\widetilde{𝓁}}_1\left(r\right)E\left[{∥\upsilon \left(r\right)∥}^2\right]+{\widehat{𝓁}}_1\left(r\right)E\left[{∥\theta \left(r\right)∥}^2\right]+{∥{\overline{𝓁}}_1\left(r\right)∥}^2\right)dr\hfill \\ & & +\frac{3K^2{\mathbf{C}}_H{s}^{2H+2p-1}{\zeta }_1}{{\Gamma }^2\left(p\right)(2p-1)}.\hfill \end{array}$$

Similarly,

$$\begin{array}{ccc}& & E\left[{∥\upsilon \left(s\right)∥}^2\right]\hfill \\ & \le & 3K^2\left[{\tilde{\Im }}_{2}E\left[{∥\omega \left(s\right)∥}^2\right]+{\tilde{\Re }}_{2}E\left[{∥\upsilon \left(s\right)∥}^2\right]+{\tilde{\gamma }}_{2}E\left[{∥\theta \left(s\right)∥}^2\right]+{\tilde{M}}_2\right]\hfill \\ & & +\frac{3K^2{s}^{2p-1}}{{\Gamma }^2\left(p\right)(2p-1)}\underset{0}{\overset{s}{\int }}\left({𝓁}_2\left(r\right)E\left[{∥\omega \left(r\right)∥}^2\right]+{\widetilde{𝓁}}_2\left(r\right)E\left[{∥\upsilon \left(r\right)∥}^2\right]+{\widehat{𝓁}}_2\left(r\right)E\left[{∥\theta \left(r\right)∥}^2\right]+{∥{\overline{𝓁}}_2\left(r\right)∥}^2\right)dr\hfill \\ & & +\frac{3K^2{\mathbf{C}}_H{s}^{2H+2p-1}{\zeta }_2}{{\Gamma }^2\left(p\right)(2p-1)}.\hfill \end{array}$$

and

$$\begin{array}{ccc}& & E\left[{∥\theta \left(s\right)∥}^2\right]\hfill \\ & \le & 3K^2\left[{\tilde{\Im }}_{3}E\left[{∥\omega \left(s\right)∥}^2\right]+{\tilde{\Re }}_{3}E\left[{∥\upsilon \left(s\right)∥}^2\right]+{\tilde{\gamma }}_{3}E\left[{∥\theta \left(s\right)∥}^2\right]+{\tilde{M}}_3\right]\hfill \\ & & +\frac{3K^2{s}^{2p-1}}{{\Gamma }^2\left(p\right)(2p-1)}\underset{0}{\overset{s}{\int }}\left({𝓁}_3\left(r\right)E\left[{∥\omega \left(r\right)∥}^2\right]+{\widetilde{𝓁}}_3\left(r\right)E\left[{∥\upsilon \left(r\right)∥}^2\right]+{\widehat{𝓁}}_3\left(r\right)E\left[{∥\theta \left(r\right)∥}^2\right]+{∥{\overline{𝓁}}_3\left(r\right)∥}^2\right)dr\hfill \\ & & +\frac{3K^2{\mathbf{C}}_H{s}^{2H+2p-1}{\zeta }_3}{{\Gamma }^2\left(p\right)(2p-1)}.\hfill \end{array}$$

Hence,

$$\begin{array}{ccc}& & E\left[{∥\omega \left(s\right)∥}^2\right]+E\left[{∥\upsilon \left(s\right)∥}^2\right]+E\left[{∥\theta \left(s\right)∥}^2\right]\hfill \\ & \le & 3K^2\left[{\tilde{\Im }}_{1}E\left[{∥\omega \left(s\right)∥}^2\right]+{\tilde{\Re }}_{1}E\left[{∥\upsilon \left(s\right)∥}^2\right]+{\tilde{\gamma }}_{1}E\left[{∥\theta \left(s\right)∥}^2\right]+{\tilde{M}}_1\right]\hfill \\ & & +3K^2\left[{\tilde{\Im }}_{2}E\left[{∥\omega \left(s\right)∥}^2\right]+{\tilde{\Re }}_{2}E\left[{∥\upsilon \left(s\right)∥}^2\right]+{\tilde{\gamma }}_{2}E\left[{∥\theta \left(s\right)∥}^2\right]+{\tilde{M}}_2\right]\hfill \\ & & +3K^2\left[{\tilde{\Im }}_{3}E\left[{∥\omega \left(s\right)∥}^2\right]+{\tilde{\Re }}_{3}E\left[{∥\upsilon \left(s\right)∥}^2\right]+{\tilde{\gamma }}_{3}E\left[{∥\theta \left(s\right)∥}^2\right]+{\tilde{M}}_3\right]\hfill \\ & & +\frac{3K^2{s}^{2p-1}}{{\Gamma }^2\left(p\right)(2p-1)}\underset{0}{\overset{s}{\int }}\left({𝓁}_1\left(r\right)E\left[{∥\omega \left(r\right)∥}^2\right]+{\widetilde{𝓁}}_1\left(r\right)E\left[{∥\upsilon \left(r\right)∥}^2\right]+{\widehat{𝓁}}_1\left(r\right)E\left[{∥\theta \left(r\right)∥}^2\right]+{∥{\overline{𝓁}}_1\left(r\right)∥}^2\right)dr\hfill \\ & & +\frac{3K^2{s}^{2p-1}}{{\Gamma }^2\left(p\right)(2p-1)}\underset{0}{\overset{s}{\int }}\left({𝓁}_2\left(r\right)E\left[{∥\omega \left(r\right)∥}^2\right]+{\widetilde{𝓁}}_2\left(r\right)E\left[{∥\upsilon \left(r\right)∥}^2\right]+{\widehat{𝓁}}_2\left(r\right)E\left[{∥\theta \left(r\right)∥}^2\right]+{∥{\overline{𝓁}}_2\left(r\right)∥}^2\right)dr\hfill \\ & & +\frac{3K^2{s}^{2p-1}}{{\Gamma }^2\left(p\right)(2p-1)}\underset{0}{\overset{s}{\int }}\left({𝓁}_3\left(r\right)E\left[{∥\omega \left(r\right)∥}^2\right]+{\widetilde{𝓁}}_3\left(r\right)E\left[{∥\upsilon \left(r\right)∥}^2\right]+{\widehat{𝓁}}_3\left(r\right)E\left[{∥\theta \left(r\right)∥}^2\right]+{∥{\overline{𝓁}}_3\left(r\right)∥}^2\right)dr\hfill \\ & & +\frac{3K^2{\mathbf{C}}_H{s}^{2H+2p-1}{\zeta }_1}{{\Gamma }^2\left(p\right)(2p-1)}+\frac{3K^2{\mathbf{C}}_H{s}^{2H+2p-1}{\zeta }_2}{{\Gamma }^2\left(p\right)(2p-1)}+\frac{3K^2{\mathbf{C}}_H{s}^{2H+2p-1}{\zeta }_3}{{\Gamma }^2\left(p\right)(2p-1)}\hfill \\ & \le & \left(3K^2\sum _{j=1}^3{\tilde{\Im }}_j\right)E\left[{∥\omega \left(s\right)∥}^2\right]+\left(3K^2\sum _{j=1}^3{\tilde{\Re }}_j\right)E\left[{∥\upsilon \left(s\right)∥}^2\right]+\left(3K^2\sum _{j=1}^3{\tilde{\gamma }}_j\right)E\left[{∥\theta \left(s\right)∥}^2\right]\hfill \\ & & +3K^2\sum _{j=1}^3{\tilde{M}}_j+\frac{3K^2{s}^{2p-1}}{{\Gamma }^2\left(p\right)(2p-1)}\hfill \\ & & \times \underset{0}{\overset{s}{\int }}\left(\sum _{j=1}^3{𝓁}_j\left(r\right)E\left[{∥\omega \left(r\right)∥}^2\right]+\sum _{j=1}^3{\widetilde{𝓁}}_j\left(r\right)E\left[{∥\upsilon \left(r\right)∥}^2\right]+\sum _{j=1}^3{\widehat{𝓁}}_j\left(r\right)E\left[{∥\theta \left(r\right)∥}^2\right]+\sum _{j=1}^3∥{\overline{𝓁}}_j\left(r\right)∥\right)dr\hfill \\ & & +\frac{3K^2{\mathbf{C}}_H{s}^{2H+2p-1}}{{\Gamma }^2\left(p\right)(2p-1)}\left({\zeta }_1+{\zeta }_2+{\zeta }_3\right),\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}& & E\left[{∥\omega \left(s\right)∥}^2\right]+E\left[{∥\upsilon \left(s\right)∥}^2\right]+E\left[{∥\theta \left(s\right)∥}^2\right]\hfill \\ & \le & \frac{3K^2}{1-K^{*}}\left(\sum _{j=1}^3{\tilde{M}}_j+\frac{{\mathbf{C}}_H{c}^{2H+2p-1}}{{\Gamma }^2\left(p\right)(2p-1)}\left({\zeta }_1+{\zeta }_2+{\zeta }_3\right)+\frac{c^{2p-1}{\sum }_{j=1}^3∥{\overline{𝓁}}_j\left(r\right)∥}{{\Gamma }^2\left(p\right)(2p-1)}\right)\hfill \\ & & +\frac{3K^2{c}^{2p-1}}{\left(1-K^{*}\right){\Gamma }^2\left(p\right)(2p-1)}\left(\sum _{j=1}^3{𝓁}_j\left(r\right)+\sum _{j=1}^3{\widetilde{𝓁}}_j\left(r\right)+\sum _{j=1}^3{\widehat{𝓁}}_j\left(r\right)\right)\hfill \\ & & \times \underset{0}{\overset{s}{\int }}\left(E\left[{∥\omega \left(r\right)∥}^2\right]+E\left[{∥\upsilon \left(r\right)∥}^2\right]+E\left[{∥\theta \left(r\right)∥}^2\right]\right)dr\hfill \\ & =& \widehat{K}+\underset{0}{\overset{s}{\int }}\wp \left(s\right)\left(E\left[{∥\omega \left(r\right)∥}^2\right]+E\left[{∥\upsilon \left(r\right)∥}^2\right]+E\left[{∥\theta \left(r\right)∥}^2\right]\right)dr,\hfill \end{array}$$

(8)

where

$$\widehat{K}=\frac{3K^2}{1-K^{*}}\left(\sum _{j=1}^3{\tilde{M}}_j+\frac{{\mathbf{C}}_H{c}^{2H+2p-1}}{{\Gamma }^2\left(p\right)(2p-1)}\left({\zeta }_1+{\zeta }_2+{\zeta }_3\right)+\frac{c^{2p-1}{\sum }_{j=1}^3∥{\overline{𝓁}}_j\left(r\right)∥}{{\Gamma }^2\left(p\right)(2p-1)}\right)$$

and

$$\wp \left(s\right)=\frac{3K^2{c}^{2p-1}}{\left(1-K^{*}\right){\Gamma }^2\left(p\right)(2p-1)}\left(\sum _{j=1}^3{𝓁}_j\left(r\right)+\sum _{j=1}^3{\widetilde{𝓁}}_j\left(r\right)+\sum _{j=1}^3{\widehat{𝓁}}_j\left(r\right)\right).$$

By using Gronwall’s inequality on (7), there is $K>0$ such that

$$E\left[{∥\omega \left(s\right)∥}^2\right]+E\left[{∥\upsilon \left(s\right)∥}^2\right]+E\left[{∥\theta \left(s\right)∥}^2\right]\le K,\mathrm{for}\mathrm{all}s\in U.$$

Setting

$$J=\left\{\left(\omega ,\upsilon ,\theta \right)\in \complement \times \complement \times \complement :∥\omega \left(s\right)∥<\sqrt{K}+1,∥\upsilon \left(s\right)∥<\sqrt{K}+1,∥\theta \left(s\right)∥<\sqrt{K}+1\right\},$$

and consider the operator $\chi :\overline{J}\to \complement \times \complement \times \complement .$ From the definition of $\chi$ there is no $\left(\omega ,\upsilon ,\theta \right)\in \partial J$ such that $\left(\omega ,\upsilon ,\theta \right)=\alpha \chi \left(\omega ,\upsilon ,\theta \right)$ for some $\alpha \in (0,1).$ Theorem 3 leads to $\chi$ having a FP $\left(\omega ,\upsilon ,\theta \right)$ in J, which is a solution to the problem (1).

□

4. Numerical Example

In this section, we conclude our results by giving an illustrative example and as an application to support the theoretical results, which is finding a solution to the partial neutral stochastic functional differential system. A stochastic process can be used to model phenomena that change randomly over time, like traffic flow in a network or financial markets. Specific examples discussed including using continuous-time Markov chains to model packet networks, Poisson processes to model message generations in telecommunications systems, the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Stochastic deferential equations have many applications throughout pure mathematics and are used to model various behaviors of stochastic models such as stock prices, random grows models or physical systems that are subjected to thermal fluctuations.

Example 1.

Consider the following stochastic problem:

$$\left\{\begin{array}{c}d{\varpi }_1\left(s,\zeta \right)+\frac{{\partial }^2}{\partial {\zeta }^2}{\varpi }_1\left(s,\zeta \right)={ᘐ}_1\left(s,{\varpi }_1\left(s,\zeta \right),{\varpi }_2\left(s,\zeta \right),{\varpi }_3\left(s,\zeta \right)\right)+\rho \left(s\right)\frac{d{\Re }^H}{ds},s\in U,\zeta \in [0,\pi ]\hfill \\ d{\varpi }_2\left(s,\zeta \right)+\frac{{\partial }^2}{\partial {\zeta }^2}{\varpi }_2\left(s,\zeta \right)={ᘐ}_2\left(s,{\varpi }_1\left(s,\zeta \right),{\varpi }_2\left(s,\zeta \right),{\varpi }_3\left(s,\zeta \right)\right)+\rho \left(s\right)\frac{d{\Re }^H}{ds},s\in U,\zeta \in [0,\pi ]\hfill \\ d{\varpi }_3\left(s,\zeta \right)+\frac{{\partial }^2}{\partial {\zeta }^2}{\varpi }_3\left(s,\zeta \right)={ᘐ}_3\left(s,{\varpi }_1\left(s,\zeta \right),{\varpi }_2\left(s,\zeta \right),{\varpi }_3\left(s,\zeta \right)\right)+\rho \left(s\right)\frac{d{\Re }^H}{ds},s\in U,\zeta \in [0,\pi ]\hfill \\ {\varpi }_1\left(s,0\right)={\varpi }_1\left(s,\pi \right)=0,{\varpi }_2\left(s,0\right)={\varpi }_2\left(s,\pi \right)=0{\varpi }_3\left(s,0\right)={\varpi }_3\left(s,\pi \right)=0,s\in U\hfill \\ {\varpi }_1\left(0,\zeta \right)={\int }_0^{\pi }r\left(\zeta ,{\varpi }_2\right){\varpi }_1\left(s,{\varpi }_2\right)d{\varpi }_2,\zeta \in [0,\pi ]\hfill \\ {\varpi }_2\left(0,\zeta \right)={\int }_0^{\pi }r\left(\zeta ,{\varpi }_3\right){\varpi }_2\left(s,{\varpi }_3\right)d{\varpi }_3,\zeta \in [0,\pi ],\hfill \\ {\varpi }_3\left(0,\zeta \right)={\int }_0^{\pi }r\left(\zeta ,{\varpi }_1\right){\varpi }_3\left(s,{\varpi }_1\right)d{\varpi }_1,\zeta \in [0,\pi ],\hfill \end{array}\right.$$

(9)

where $U=[0,c],$ ${\Re }^H$ represents a fractional BM, and the functions ${ᘐ}_1,{ᘐ}_1,{ᘐ}_3:U\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ and $r:U\times [0,\pi ]\to \mathbb{R}$ are continuous. Consider

$$\left\{\begin{array}{c}\omega \left(s\right)\left(\zeta \right)={\varpi }_1\left(s,\zeta \right),\upsilon \left(s\right)\left(\zeta \right)={\varpi }_2\left(s,\zeta \right),\theta \left(s\right)\left(\zeta \right)={\varpi }_3\left(s,\zeta \right),\zeta \in [0,\pi ]\hfill \\ {¯h}_1\left(s,\omega \left(s\right),\upsilon \left(s\right),\theta \left(s\right)\right)\left(\zeta \right)={ᘐ}_1\left(s,{\varpi }_1\left(s,\zeta \right),{\varpi }_2\left(s,\zeta \right),{\varpi }_3\left(s,\zeta \right)\right),\zeta \in [0,\pi ]\hfill \\ {¯h}_2\left(s,\omega \left(s\right),\upsilon \left(s\right),\theta \left(s\right)\right)\left(\zeta \right)={ᘐ}_2\left(s,{\varpi }_1\left(s,\zeta \right),{\varpi }_2\left(s,\zeta \right),{\varpi }_3\left(s,\zeta \right)\right),\zeta \in [0,\pi ]\hfill \\ {¯h}_3\left(s,\omega \left(s\right),\upsilon \left(s\right),\theta \left(s\right)\right)\left(\zeta \right)={ᘐ}_3\left(s,{\varpi }_1\left(s,\zeta \right),{\varpi }_2\left(s,\zeta \right),{\varpi }_3\left(s,\zeta \right)\right),\zeta \in [0,\pi ],\hfill \end{array}\right.$$

for each $s\in U.$ Take $M=\phi =L^2\left([0,\pi ]\right)$ and define an operator ℑ as $\Im {\varpi }_1={\omega }_1^{″}$ with domain

$$D\left(\Im \right)=\left\{{\varpi }_1\in \phi :{\omega }_1^{\prime },{\omega }_1^{″}\in \phi and{\varpi }_1\left(0\right)={\varpi }_1\left(\pi \right)=0\right\}.$$

It is clear that

$$\Im g=-\sum _{u=1}^{\infty }{e}^{-u^{2s}}⟨g,e_u⟩e_u,g\in \phi .$$

Define an analytic semigroup ${\left\{Z\left(s\right)\right\}}_{s\ge 0}$ by

$$Z\left(s\right){\varpi }_1=\sum _{u=1}^{\infty }{e}^{-u^{2s}}⟨{\varpi }_1,e_u⟩e_u,{\varpi }_1\in \phi .$$

Clearly, the operator ℑ is an infinitesimal generator of ${\left\{Z\left(s\right)\right\}}_{s\ge 0},$ where $e_u\left({\varpi }_1\right)=\sqrt{\frac{2}{\pi }}sin\left(u{\varpi }_1\right)$ is the orthogonal set of eigenvectors of ℑ.

Further, ${\left\{Z\left(s\right)\right\}}_{s\ge 0}$ is compact, and there is a constant $K\ge 1$ such that ${∥Z\left(s\right)∥}^2\le K.$ Thus, the system (9) reduces to

$$\left\{\begin{array}{c}d\omega \left(s\right)=\left[{\Im }_1\omega \left(s\right)+{¯h}_1\left(s,\omega ,\upsilon ,\theta \right)\right]ds+{\rho }_1\left(s\right)d{\Re }_s^{H_1},s\in U,\hfill \\ d\upsilon \left(s\right)=\left[{\Im }_2\upsilon \left(s\right)+{¯h}_2\left(s,\omega ,\upsilon ,\theta \right)\right]ds+{\rho }_2\left(s\right)d{\Re }_s^{H_2},s\in U,\hfill \\ d\theta \left(s\right)=\left[{\Im }_3\theta \left(s\right)+{¯h}_3\left(s,\omega ,\upsilon ,\theta \right)\right]ds+{\rho }_3\left(s\right)d{\Re }_s^{H_3},s\in U,\hfill \\ \omega \left(0\right)=\kappa \left(\omega ,\upsilon ,\theta \right),\upsilon \left(0\right)=\tau \left(\omega ,\upsilon ,\theta \right)and\theta \left(0\right)=\mu \left(\omega ,\upsilon ,\theta \right).\hfill \end{array}\right.$$

Now, we consider the continuous functions

$$\begin{array}{ccc}\hfill {¯h}_1\left(s,\omega ,\upsilon ,\theta \right)& =& \frac{s{\varpi }_1}{1+{\varpi }_1^2+{\varpi }_2^2+{\varpi }_3^2},\hfill \\ \hfill {¯h}_2\left(s,\omega ,\upsilon ,\theta \right)& =& \frac{s{\varpi }_2}{1+{\varpi }_1^2+{\varpi }_2^2+{\varpi }_3^2},\hfill \\ \hfill and{¯h}_3\left(s,\omega ,\upsilon ,\theta \right)& =& \frac{s{\varpi }_3}{1+{\varpi }_1^2+{\varpi }_2^2+{\varpi }_3^2},\hfill \end{array}$$

with

$${\left|{¯h}_1\left(s,\omega ,\upsilon ,\theta \right)\right|}^2\le c{\left|{\varpi }_1\right|}^2,{\left|{¯h}_2\left(s,\omega ,\upsilon ,\theta \right)\right|}^2\le c{\left|{\varpi }_2\right|}^2,{\left|{¯h}_1\left(s,\omega ,\upsilon ,\theta \right)\right|}^2\le c{\left|{\varpi }_3\right|}^2.$$

Moreover, if we take

$$\begin{array}{ccc}\hfill \kappa \left({\varpi }_1,{\varpi }_2,{\varpi }_3\right)& =& {\int }_0^{\pi }r\left(\zeta ,{\varpi }_2\right){\varpi }_1\left(s,{\varpi }_2\right)d{\varpi }_2,\hfill \\ \hfill \tau \left({\varpi }_1,{\varpi }_2,{\varpi }_3\right)& =& {\int }_0^{\pi }r\left(\zeta ,{\varpi }_3\right){\varpi }_2\left(s,{\varpi }_3\right)d{\varpi }_3,\hfill \\ \hfill and\mu \left({\varpi }_1,{\varpi }_2,{\varpi }_3\right)& =& {\int }_0^{\pi }r\left(\zeta ,{\varpi }_1\right){\varpi }_3\left(s,{\varpi }_1\right)d{\varpi }_1,\hfill \end{array}$$

we find that the hypotheses $\left(A_3\right)-\left(A_6\right)$ are satisfied. Therefore, by Theorem 5, there is at least one solution to the Problem 5 on $U\times [0,\pi ].$

5. Conclusions and Future Work

Understanding nonlinear fractional stochastic systems is essential for systematizing real-world processes with memory and randomness. Researchers continue to explore new mathematical tools and techniques to address the complexities posed by these equations. So, in this manuscript, we investigate the existence and uniqueness of solutions for a novel system of differential equations. This system combines fractional operators and stochastic variables. Our study is conducted under nonlocal functional boundary conditions. To analyze the existence of a solution, we employ the fixed-point method, specifically utilizing the theorems of Leray–Schauder and Perov in generalized metric spaces. Additionally, we explore an illustrative example to bolster our findings. There are many exciting avenues to explore within the field of nonlinear fractional stochastic differential systems. Researchers can build upon existing results, develop new mathematical techniques, and address practical applications in various scientific domains.