*Article* **Analysis of Controllability of Fractional Functional Random Integroevolution Equations with Delay**

**Kinda Abuasbeh 1,\*, Ramsha Shafqat 2,\*, Ammar Alsinai 3,\* and Muath Awadalla <sup>1</sup>**


**Abstract:** Various scholars have lately employed a wide range of strategies to resolve two specific types of symmetrical fractional differential equations. The evolution of a number of real-world systems in the physical and biological sciences exhibits impulsive dynamical features that can be represented via impulsive differential equations. In this paper, we explore some existence and controllability theories for the Caputo order *q* ∈ (1, 2) of delay- and random-effect-affected fractional functional integroevolution equations (FFIEEs). In order to prove that random solutions exist, we must prove a random fixed point theorem using a stochastic domain and the mild solution. Then we demonstrate that our solutions are controllable. At the end, applications and example is illustrated which indicates the applicability of this manuscript.

**Keywords:** random fixed point; state dependent delay; controllability; functional differential equation; mild solution; finite delay; cosine and sine family

**MSC:** 26A33; 34K37

**1. Introduction**

Many different applications have been investigated through the theory of impulsive fractional differential equations (IFDEs) in the accurate mathematical representation of a wide variety of practical problems. It is acknowledged as a crucial area for research, as much as the modelling of impulsive issues in population dynamics, ecology, biotechnology, and other fields. In real-world situations, many processes and phenomena are characterised by rapid shifts in their states. The mentioned quick modifications are called impulsive effects within the system. Instantaneous and noninstantaneous impulses are the two main forms of impulses discussed in the literature to date. In contrast to the length of a whole evolution, such as that of shocks and natural disasters, the period of these fluctuations in instantaneous impulses is insignificant; in the case of noninstantaneous impulses, on the other hand, the duration of the changes exists throughout a finite time period.

Over the past three decades, the field of mathematical analysis has incorporated fractional calculus, FDEs, and integrodifferential equations, and the qualitative theory of these equations on both a theoretical and a practical level. Fundamentally, fractional calculus theory, the qualitative theory of FDEs and fractional integrodifferential equations, numerical simulations, and symmetry analysis are mathematical analytical tools used to study arbitrary-order integrals and derivatives that unify and generalise the conventional ideas of differentiation and integration. Compared to classical formulations, nonlinear operators with a fractional order are more useful. Throughout the development of emerging control theory, the controllability of DEs problems has played a major role. Typically, it means that a dynamical system may be moved from any initial state to the desired terminal

**Citation:** Abuasbeh, K.; Shafqat, R.; Alsinai, A.; Awadalla, M. Analysis of Controllability of Fractional Functional Random Integroevolution Equations with Delay. *Symmetry* **2023**, *15*, 290. https://doi.org/10.3390/ sym15020290

Academic Editor: Dumitru Baleanu

Received: 26 December 2022 Revised: 11 January 2023 Accepted: 17 January 2023 Published: 20 January 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

state using a set of legal controls. Control theory places much emphasis on the qualitative characteristics of control systems. There has been particular focus on the controllability of linear and nonlinear systems in a finite-dimensional space that are described by ordinary DEs; see [1–4] for a list of researchers who have extended the idea to infinite-dimensional systems with bounded operators in Banach spaces (BS). The controllability problem was converted into a fixed-point problem by the authors of [5]. We advise reading [6,7] for additional information. The authors of [8,9] investigated a variety of functional DEs and inclusions, and proposed various controllability findings. A family of integrodifferential evolution equations' controllability was examined by Dilao et al. [10].

It is often advantageous to handle second-order abstract DEs explicitly rather than always reducing them to first-order systems. For the investigation of second-order issues, the theory of strongly continuous cosine families is an invaluable resource. We use some of the core ideas in cosine family theory [11]. Typically, this means that a dynamical system may be moved from any initial state to the desired terminal state using a set of legal controls. Control theory places much emphasis on the qualitative characteristics of control systems. There has been particular focus on the controllability of linear and nonlinear systems in finite-dimensional space that are described by ordinary DEs [12,13].

The reader is recommended to read [14–16] for more information on random differential equations, which are natural generalisations of deterministic DEs and appear in a variety of applications. The accuracy of our knowledge about the system's characteristics determines the nature of a dynamic system. When knowledge about a dynamic system is exact, a deterministic dynamical system emerges. Moreover, many of the available details for identifying and assessing dynamic system characteristics are incorrect, uncertain, or imprecise. To put it another way, determining the parameters of a dynamic system is highly risky. However, when we have probable knowledge and an understanding of statistical characteristics, we can use stochastic DEs in mathematically modelling such systems.

Ji-Huan He [17] studied fractal calculus. Wang et al. [18–20] worked on nondifferentiable exact solutions, the modification of the unsteady model, and diverse exact and explicit solutions. Mehmood et al. [21] worked on a partial DE. Niazi et al. [22], Shafqat et al. [23], Alnahdi [24], and Abuasbeh et al. [25] investigated the existence and uniqueness of FEEs. Inspired by the above studies [26], this paper deals with the controllability of the fractional functional integroevolution equation with delay and random effects:

*c* 0*Dq <sup>ν</sup>U*(*χ*, *<sup>ξ</sup>*) = *<sup>B</sup>*1*U*(*χ*, *<sup>ξ</sup>*) + *<sup>ϕ</sup>*(*χ*, *<sup>U</sup>χ*(., *<sup>ξ</sup>*), *<sup>ξ</sup>*) + *<sup>ν</sup>* <sup>0</sup> *B*<sup>2</sup> *f*(*χ*, *ξ*)*dCv* + *Bx*(*ν*)*Cx*(*ν*)*dν*, *ξ* ∈ Θ := [0, *κ*], *ν* ∈ [0, *T*] *U*(*χ*, *ξ*) + *m*(*U*) = 1(*χ*, *ξ*); *ξ* ∈ (−∞, 0], *U* (*χ*, *ξ*) = 2(*ξ*) (1)

> Knowing that complete probability space (Φ, *F*, ℘) given functions *ϕ* : Θ × *D* × Ψ → Ξ, *σ*<sup>1</sup> ∈ *D* ∈ *D* × Φ, and infinitesimal generator *B*<sup>1</sup> : *D*(*B*1) ⊂ Ξ → Ξ of a strongly continuous cosine family, the phase space is (*Hq*(*χ*))*χ*∈**R<sup>m</sup>** on Ξ, *D*, and a real BS is (Ξ, |.|). Control function <sup>P</sup>(., *<sup>ξ</sup>*) is specified in *<sup>L</sup>*2(Θ, <sup>Ω</sup>), a BS of possible control functions with <sup>Ω</sup> as a BS, and *B*<sup>2</sup> is a bounded linear operator (LO) from Ω into Ξ.

> The component of *D* × Φ determined with *D* × Φ, given by *U<sup>ξ</sup>* (*ι*, *ξ*) = *U*(*ξ* + *ι*, *ξ*), *ι* ∈ (−∞, 0] is denoted by *Uχ*(., *ξ*). Here, the state's existence from the year −∞ to the current day *ξ* is represented by the string *Uχ*(., *ξ*). Eras *Uχ*(., *ξ*) were presumptively part of some abstract phases *D*.

First, we suppose random issue

*c* 0*Dq <sup>ν</sup>U*(*χ*, *<sup>ξ</sup>*) = *<sup>B</sup>*1*U*(*χ*, *<sup>ξ</sup>*) + *<sup>ϕ</sup>*(*χ*, *<sup>U</sup>ϑ*(*χ*,*Uχ*)(., *<sup>ξ</sup>*), *<sup>ξ</sup>*) + *<sup>ν</sup>* <sup>0</sup> *B*<sup>2</sup> *f*(*χ*, *ξ*)*dCv* + *Bx*(*ν*)*Cx*(*ν*)*dν*, *ξ* ∈ Θ := [0, *κ*], *ν* ∈ [0, *T*] *U*(*χ*, *ξ*) + *m*(*U*) = 1(*χ*, *ξ*); *ξ* ∈ (−∞, 0], *U* (*χ*, *ξ*) = 2(*ξ*) (2)

> where *ϕ* : Θ × *D* × Ψ → Ξ, *σ*<sup>1</sup> ∈ *D* ∈ *D* × Φ are given random functions, *B*<sup>1</sup> : *D*(*B*1) ⊂ Ξ → Ξ is as in problem (1), *D* is the phase space, *ψ*; Θ × *D* → (−∞, *κ*], and (Ξ, |.|) is a real

BS. For the key conclusions on Schauder's fixed theorem [27], and random fixed-point theorem paired with the family of cosine operators, we employ our' arguments.

The layout of this article is as follows. Section 2 contains some needed preliminaries and fundamental results. Sections 3 and 4 present our main results in two cases: infinite fixed delay and state-dependent delay, respectively. In Sections 5 and 6, we give applications and an example, respectively. In Section 7, we present the conclusion.

#### *Motivation and Novelties*

The incorporation of fractional-order derivatives in delay DEs provides a range of advantages, including hereditary properties, additional degrees of freedom, and other advantages of fractional modelling. As these equations are primarily used in control theory and robotics, the stability and asymptotics of these equations are of vital importance. However, stability and asymptotic analyses of fractional delay DEs are still in their early stages. Most of the current stability results on autonomous equations of this type are based on the root locus of their corresponding characteristic equations, and do not offer a universal and reliable way of assessing the stability of a given fractional delay DE.

FDEs with a time delay are widely used in natural phenomena, and the fields of science and engineering. To capture the dynamic behavior of travelling wave solutions on the basis of these equations, researchers have created algorithms with high performance for various spatial and time fractional delay DEs. However, there are still challenges to be addressed in the field of fractional delay DEs, such as the stability analysis of numerical time integration schemes and the numerical theory of the numerical scheme. Additionally, there is a need for stability and numerical simulations of travelling wave solutions, critical travelling wave solutions, and the design of compact fourth- and sixth-order schemes for fractional delay DEs with strong nonlinearity.

This paper aims to investigate the existence and controllability of solutions to FDEs with delay and random effects. While the majority of results in the literature have focused on first-order equations, some researchers produced FDE results. In our study, we obtained findings for Caputo derivatives of order (1,2) using a mild solution. Stability is a major area of research in DE theory, and over the past 20 years, stability for FDE has been a major focus of research. In order to illustrate this, we consider the prerequisites for solution stability and FDE asymptotic stability. We also examine delay fractional functional random integroevolution equations.

#### **2. Preliminaries**

We discuss a few of the abbreviations, definitions, and theorems that are used throughout the work in this part. Considering the BS *D*(Ξ) of bounded LOs from Ξ into Ξ, where Θ := [0, *κ*], *κ* > 0,

$$||\aleph||\_{D(\Sigma)} = \sup\_{||\chi||=1} ||\aleph(\mathcal{U})||.$$

Let C := C(*I*, Ξ) be the Banach space of continuous functions *U* : Θ → Ξ with the norm

$$||\mathcal{U}||\_{\mathcal{C}} = \sup\_{\chi \in \Theta} |\mathcal{U}(\chi)|.$$

We follow to the methodology described in [28] and apply the axiomatic description of the phase space *D* given in [29]. Once (*D*, ||.||*D*) is defined as a seminormed linear space of functions translating (−∞, 0] into Ξ, we have

	- (*a*) *U<sup>χ</sup>* ∈ *D*;
	- (*b*) There ∃ a positive constant *ρ*, |*U*(*χ*)| ≤ ||*Uχ*||*D*.

(*c*) There <sup>∃</sup> two functions *<sup>β</sup>*(.), *<sup>ω</sup>*(.) : **<sup>R</sup><sup>m</sup>** <sup>+</sup> <sup>→</sup> **<sup>R</sup><sup>m</sup>** <sup>+</sup> independent of *U* with *β* continuous and bounded and *ω* locally bounded where:

$$||\mathcal{U}\_{\chi}||\_{D} \le \beta(\chi) \sup \{ |\mathcal{U}(\rho)| : 0 \le \rho \le \rho \} + \omega(\chi) ||\mathcal{U}\_0||\_{D}.$$

(*J*2) For function *U* in (*A*1), *U<sup>χ</sup>* is a *D*-valued continuous function on Θ.

(*J*3) The space *D* is complete.

Set

$$\mathfrak{g} = \sup \{ \beta(\chi) : \chi \in \Theta \}, \text{ and } \omega = \sup \{ \omega(\chi) : \chi \in \Theta \}.$$

**Remark 1.** *1. (2) is equivalent to* |1||*<sup>D</sup>* ≤ ||1||*D*∀<sup>1</sup> ∈ *D.*


$$\Xi := \{ \mathcal{U} : ( - \infty, \kappa] : \mathcal{U}|\_{(\infty, 0]} \in D \text{ and } \mathcal{U}|\_{\ominus} \in \mathbb{C} \},$$

*and let* ||*U*||<sup>Ξ</sup> *be the seminorm in* Ξ *given by*

$$||\mathcal{U}||\_{\Xi} = ||\varrho\_1||\_D + ||\mathcal{U}||\_{\mathcal{C}}.$$

**Definition 1.** *Let* {*Hq*(*χ*) : *<sup>χ</sup>* <sup>∈</sup> **<sup>R</sup>m**} *be a family of bounded LOs in the Banach space* <sup>Ψ</sup>*, which is a strongly continuous cosine family if*


*Let* {*Hq*(*χ*) : *<sup>χ</sup>* <sup>∈</sup> **<sup>R</sup>m**} *be a strongly continuous cosine family in* <sup>Ψ</sup>*. Define the sine family* {*Kq*(*χ*) : *<sup>χ</sup>* <sup>∈</sup> **<sup>R</sup>m**} *with*

$$K\_{\eta}(\chi)\eta = \int\_{0}^{\chi} H\_{\eta}(\rho)\eta d\rho, \; \eta \in \Xi, \; \chi \in \mathbf{R}^{\mathbf{m}}.$$

*The infinitesimal generator B*<sup>1</sup> : <sup>Ξ</sup> <sup>→</sup> <sup>Ξ</sup> *of the cosine family* {*S*(*χ*) : *<sup>χ</sup>* <sup>∈</sup> **<sup>R</sup>m**} *is defined by*

$$B\_1 \eta = \frac{d^2}{d\chi^2} H\_\eta(\chi) \eta|\_{\chi=0\prime} \eta \in D(B\_1)\_\prime$$

*where*

$$D(B\_1) = \{ \eta \in \Xi : H\_{\eta}(.)\eta \in C^2(\mathbb{R}^{\mathfrak{m}}, \Xi) \}.$$

**Definition 2.** *Consider the map φ* : Θ × *D* × *ψ* → Ξ *is random Caratheodory if*

(i) *χ* → *φ*(*χ*, *U*, Δ)*, this map measurable* ∀ *U* ∈ *D and for all* Δ ∈ *ψ.*

(ii) *U* → *φ*(*χ*, *U*, Δ) *is measurable* ∀ *U* ∈ *D and for all* Δ ∈ *ψ.*

(iii) Δ → *φ*(*χ*, *U*, Δ) *is measurable* ∀ *U* ∈ *D, and almost χ* ∈ Θ*.*

*Let <sup>D</sup>*<sup>Ξ</sup> *be the Borel <sup>σ</sup>-algebra in separable BS* <sup>Ξ</sup>*. If, for each* <sup>Π</sup> <sup>∈</sup> *<sup>D</sup>*Ξ, *<sup>p</sup>*−1(Π) <sup>∈</sup> *F, then the map p* : *ψ* → Ξ *is a random variable. If G*(., *p*)*, written as G*(Δ, *p*) = *G*(Δ)*p, is measurable for each p* ∈ Ξ*, then G* : *ψ* × Ξ → Ξ *is a random operator.*

**Definition 3** ([30])**.** *Let <sup>G</sup>*´ *be a mapping from <sup>ψ</sup> into* <sup>2</sup>Ξ*. A mapping <sup>G</sup>* : {(Δ, *<sup>p</sup>*) : <sup>Δ</sup> <sup>∈</sup> *<sup>ψ</sup>* <sup>∧</sup> *<sup>p</sup>* <sup>∈</sup> *<sup>G</sup>*´(Δ)} → <sup>Ξ</sup> *is a random operator with stochastic domain <sup>G</sup>*´ *if and only if, for all closed* <sup>Π</sup><sup>1</sup> <sup>⊆</sup> <sup>Ξ</sup>, {<sup>Δ</sup> <sup>∈</sup> *<sup>ψ</sup>* : *<sup>G</sup>*´(Δ) <sup>∩</sup> *<sup>G</sup>*´ <sup>1</sup> = ∅} ∈ *F, and for all open* Π<sup>2</sup> ⊆ Ξ *and all p* ∈ Ξ, {Δ ∈ *ψ* : *p* ∈ *<sup>G</sup>*´(Δ) <sup>∧</sup> *<sup>G</sup>*(Δ, *<sup>p</sup>*) <sup>∈</sup> <sup>Π</sup>2} ∈ *F. <sup>G</sup> is continuous if every <sup>G</sup>*(Δ) *is continuous. A mapping <sup>p</sup>* : *<sup>ψ</sup>* <sup>→</sup> <sup>Ξ</sup> *is a random fixed point of <sup>G</sup> if and only if for all* <sup>Δ</sup> <sup>∈</sup> *<sup>ψ</sup>*, *<sup>p</sup>*(Δ) <sup>∈</sup> *<sup>G</sup>*´(Δ) *and <sup>G</sup>*(Δ)*p*(Δ) = *<sup>p</sup>*(Δ) *and p is measurable if for all open* Π<sup>2</sup> ⊆ Ξ, {Δ ∈ *ψ* : *p*(Δ) ∈ Π2} ∈ *F.*

**Lemma 1** ([30])**.** *Let <sup>G</sup>*´ : *<sup>ψ</sup>* <sup>→</sup> <sup>2</sup><sup>Ξ</sup> *be measurable for every* <sup>Δ</sup> <sup>∈</sup> *<sup>ψ</sup> with <sup>G</sup>*´(Δ) *closed, convex, and solid (i.e., <sup>G</sup>*(Δ) <sup>=</sup> <sup>∅</sup>*). We assumed the existence of a measurable <sup>p</sup>*<sup>0</sup> : *<sup>ψ</sup>* <sup>→</sup> <sup>Ξ</sup> *with <sup>p</sup>*<sup>0</sup> <sup>∈</sup> *<sup>G</sup>*´(Δ) *for all* <sup>Δ</sup> <sup>∈</sup> *<sup>ψ</sup>. Assume that <sup>G</sup> is a continuous random operator with the stochastic domain <sup>G</sup>*´*; as such, <sup>G</sup>*(Δ)*<sup>p</sup>* <sup>=</sup> *<sup>p</sup>* <sup>=</sup> <sup>∅</sup> *for any* <sup>Δ</sup> <sup>∈</sup> *<sup>ψ</sup>*, {*<sup>p</sup>* <sup>∈</sup> *<sup>G</sup>*´(Δ)*. Once this happens, <sup>G</sup> has a stochastic fixed point. If the function p*(*χ*, .) *is measurable for each χ* ∈ Θ*, then the mapping of p of* Θ × *ψ into* Ξ *is stochastic.*

**Definition 4** ([31])**.** *Assume that U is a BS, and φ<sup>U</sup> is the bounded subsets of* Ξ*. The Kuratowski measure of noncompactness is map <sup>μ</sup>* : *<sup>ψ</sup><sup>U</sup>* <sup>→</sup> [0, <sup>∞</sup>) *given by <sup>μ</sup>*(Π) = inf{ <sup>&</sup>gt; <sup>0</sup> : <sup>Π</sup> ⊆ ∪*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> *and diam*(Π*i*) ≤ }*; here* Π ∈ *ψ<sup>U</sup> and verifies the following properties:*


**Lemma 2** ([32])**.** *μ*(*g*(*χ*)) *is continuous on theta if and only if g* ⊂ *C*(Θ, Ξ) *is bounded and equicontinuous:*

$$\mu\left(\left\{\int\_{\Theta} \eta(\rho)d\rho : \eta \in \mathfrak{g}\right\}\right) \leq \int\_{\Theta} \mu(\mathfrak{g}(\rho))d\rho.$$

*where g*(*χ*) = {*η*(*χ*) : *η* ∈ *g*}, *χ* ∈ Θ*.*

**Lemma 3** (Gronwall lemma [28])**.** *Assume <sup>μ</sup>, <sup>y</sup>* ∈ H([0, 1], <sup>R</sup>+) *and let <sup>μ</sup> be increasing. If* <sup>u</sup> <sup>∈</sup> <sup>H</sup>([0, 1], <sup>R</sup>+) *satisfies*

$$\mathfrak{u}(\omega) \lesssim \mathfrak{\mu}(\omega) + \int\_0^\omega \mathfrak{y}(s)\mathfrak{u}(s)ds,\ \omega \in [0,1]\_\prime$$

*then*

$$
\mathfrak{u}(\omega) \lesssim \mu(\omega) \exp \int\_0^\omega y(s)\mathfrak{u}(s)ds, \ \omega \in [0,1].
$$

**Definition 5** ([30])**.** *The fractional Riemann–Liouville (RL) derivative is defined as follows.*

$${}\_{a}D\_{\omega}^{p}\chi(\omega) = \frac{1}{\Gamma(n-p+1)} \left(\frac{d}{d\omega}\right)^{n+1}$$

$$\int\_{a}^{\omega} (\omega - \tau)^{n-p} \chi(\tau) d\tau, \ n \le p \le n+1.$$

**Definition 6** ([30])**.** *Caputo fractional derivatives* C *<sup>a</sup> D<sup>α</sup> ωχ*(*ω*) *of order <sup>α</sup>* <sup>∈</sup> <sup>R</sup><sup>+</sup> *are defined by*

$$\,\_a^{\mathcal{C}}D\_{\omega}^{a}\chi(\omega) = \,\_aD\_{\omega}^{a}(\chi(\omega) - \sum\_{j=0}^{k-1} \frac{\chi^{(j)}(a)}{j!}(\omega - a)^{j})\lambda$$

*in which k* = [*α*] + 1*.*

**Definition 7** ([31])**.** *Wright function ψα is defined by*

$$\begin{aligned} \psi\_{\alpha}(\kappa) &= \sum\_{j=0}^{\infty} \frac{(-\kappa)^{j}}{j! \Gamma(-\alpha j + 1 - \alpha)} \\ &= \frac{1}{\pi} \sum\_{j=1}^{\infty} \frac{(-\kappa)^{j}}{(j-1)!} \Gamma(j\alpha) \sin(j\pi\alpha) \end{aligned}$$

*<sup>α</sup>* <sup>∈</sup> (0, 1), *<sup>κ</sup>* <sup>∈</sup> <sup>C</sup>*.*

#### **3. Results of Controllability for the Steady Delay Case**

**Definition 8.** *Equation* (1) *is controllable on the interval* (−∞, *<sup>κ</sup>*] *if, for all final state <sup>U</sup>*1(*ξ*)*, there* <sup>∃</sup> *a control* <sup>P</sup>(., *<sup>ξ</sup>*) *in L*2(Θ, <sup>Ω</sup>)*, such that the solution U*(*χ*, *<sup>ξ</sup>*) *of* (1) *satisfies U*(*κ*, *<sup>ξ</sup>*) = *<sup>U</sup>*1(*ξ*)*.*

**Definition 9.** *A stochastic process U* : (−∞, *κ*] × Φ → Ξ *is a random mild solution of Problem* (1) *if <sup>U</sup>*(*χ*, *<sup>ξ</sup>*) = 1(*χ*, *<sup>ξ</sup>*); *<sup>χ</sup>* <sup>∈</sup> (−∞, *<sup>χ</sup>*], *<sup>U</sup>*∞(0, *<sup>ξ</sup>*) = 2(*ξ*)*, and the restriction of <sup>U</sup>*(., *<sup>ξ</sup>*) *to the interval* Θ *is continuous and verifies:*

$$\begin{split} \mathcal{U}(\boldsymbol{\chi},\boldsymbol{\xi}) &= \ \mathcal{U}\_{\boldsymbol{\eta}}(\boldsymbol{\chi})(\boldsymbol{\varrho}\_{1}(\boldsymbol{\chi},\boldsymbol{\xi}) - \boldsymbol{m}(\boldsymbol{\mathcal{U}})) + \mathcal{K}\_{\boldsymbol{\eta}}(\boldsymbol{\chi})\boldsymbol{\varrho}\_{2}(\boldsymbol{\chi}) + \int\_{0}^{\boldsymbol{\nu}} (\boldsymbol{\chi} - \boldsymbol{\rho}) \mathcal{P}\_{\boldsymbol{\eta}}(\boldsymbol{\chi} - \boldsymbol{\rho}) \mathcal{B}\_{1} \mathcal{U}(\boldsymbol{\chi},\boldsymbol{\xi}) d\boldsymbol{\rho} + \int\_{0}^{\boldsymbol{\nu}} (\boldsymbol{\chi} - \boldsymbol{\rho}) \mathcal{P}\_{\boldsymbol{\eta}}(\boldsymbol{\chi} - \boldsymbol{\rho}) \mathcal{B}\_{1} \mathcal{U}(\boldsymbol{\chi},\boldsymbol{\xi}) d\boldsymbol{\rho} \\ & \quad [\boldsymbol{\varrho}(\boldsymbol{\chi},\boldsymbol{\mathcal{U}}\_{\boldsymbol{\xi}}(.,\boldsymbol{\xi}),\boldsymbol{\xi})] d\boldsymbol{\rho} + \int\_{0}^{\boldsymbol{\chi}} \Big( (\boldsymbol{\chi} - \boldsymbol{\rho}) \mathcal{P}\_{\boldsymbol{\eta}}(\boldsymbol{\chi} - \boldsymbol{\rho}) \int\_{0}^{\boldsymbol{\nu}} \mathcal{B}\_{2} f(\boldsymbol{\chi},\boldsymbol{\xi}) d\boldsymbol{\mathcal{C}}\_{\boldsymbol{\eta}} + \mathcal{B} \boldsymbol{x}(\boldsymbol{\rho}) \mathcal{C} \boldsymbol{x}(\boldsymbol{\rho}) \Big) d\boldsymbol{\rho} \end{split}$$

*Let*

$$\omega = \sup \{ ||H\_q(\chi)||\_{D(\Xi)} : \chi \ge 0 \} $$

*and*

$$\omega = \sup \{ ||\mathbb{K}\_q(\chi)||\_{D(\Xi)} : \chi \ge 0 \}.$$

The following hypotheses must be introduced:



(*H*4)There <sup>∃</sup> a random function *<sup>Q</sup>* : *<sup>ψ</sup>* <sup>→</sup> **<sup>R</sup><sup>m</sup>** <sup>+</sup> {0} where:

$$\omega(1+\kappa\omega\zeta(||\varrho\_1||\_D+\eta(D\_\prime\Delta||p||\_{L^1})+\kappa\omega\zeta||\eta^1||+\omega^{\prime}(1+\kappa\omega\zeta)|\varrho\_2|\leq Q(\Delta))$$

where

$$D := \zeta Q(\Delta) + \sigma ||\varrho\_1||\_{D'} $$

(*H*5)The linear : *<sup>L</sup>*2(Θ, <sup>Ω</sup>) <sup>→</sup> <sup>Ψ</sup> given by

$$\overline{\mathcal{D}}\mathcal{P} = \int\_0^\kappa H\_\emptyset(\kappa - \rho) B\_2 \mathcal{P}(\rho, \Delta) d\rho$$

has an inverse operator −<sup>1</sup> in *<sup>L</sup>*2(Θ, <sup>Ω</sup>)/ ker, and there <sup>∃</sup> a positive constant *<sup>ζ</sup>*, such that ||*B*2−1|| ≤ *<sup>ζ</sup>*,

(*H*6)for each Δ ∈ *ψ*, (., Δ) is continuous and *χ*, 1(*χ*, .) and Δ ∈ *ψ*, 2(Δ) are measurable.

**Theorem 1.** *Assume that* (*H*1)*–*(*H*2) *are met; then Problem* (1) *is controllable on* Θ*.*

**Proof.** Define the control:

$$\begin{split} \mathcal{P}(\boldsymbol{\chi},\boldsymbol{\Delta}) &= \ \ & \ -\nabla^{-1} \Big( p^{1}(\Theta) - H\_{\boldsymbol{\eta}}(\boldsymbol{\chi})(\boldsymbol{\varrho}\_{1}(\boldsymbol{\chi},\boldsymbol{\xi}) - \boldsymbol{m}(\boldsymbol{\mathcal{U}})) - \boldsymbol{K}\_{\boldsymbol{\eta}}(\boldsymbol{\chi})\boldsymbol{\varrho}\_{2}(\boldsymbol{\chi}) - \int\_{0}^{\boldsymbol{\upsilon}} (\boldsymbol{\chi} - \rho) P\_{\boldsymbol{\eta}}(\boldsymbol{\chi} - \rho) \boldsymbol{B}\_{1} \boldsymbol{\mathcal{U}}(\boldsymbol{\chi},\boldsymbol{\xi}) d\rho \\ & - \int\_{0}^{\boldsymbol{\upsilon}} (\boldsymbol{\chi} - \rho) P\_{\boldsymbol{\eta}}(\boldsymbol{\chi} - \rho) [\boldsymbol{\varrho}(\boldsymbol{\chi},\boldsymbol{\mathcal{U}}\_{\boldsymbol{\chi}}(., \boldsymbol{\xi}), \boldsymbol{\xi})] d\rho \Big) .\end{split}$$

The operator *I* : *ψ* × Ξ → Ξ defined by (*I*(*ξ*)*p*)(*χ*) = 1(*χ*, *ξ*), if *χ* ∈ (−∞, 0], and for *χ* ∈ Θ:

*<sup>U</sup>*(*χ*, *<sup>ξ</sup>*) = *Hq*(*χ*)(1(*χ*, *<sup>ξ</sup>*) <sup>−</sup> *<sup>m</sup>*(*U*)) + *Kq*(*χ*)2(*χ*) + *<sup>ν</sup>* 0 (*χ* − *ρ*)*Pq*(*χ* − *ρ*)*B*1*U*(*χ*, *ξ*)*dρ* + *ν* 0 (*χ* − *ρ*)*Pq*(*χ* − *ρ*) [*ϕ*(*χ*, *Uχ*(., *ξ*), *ξ*)]*dρ* + *χ* 0 (*<sup>χ</sup>* <sup>−</sup> *<sup>ρ</sup>*)*Pq*(*<sup>χ</sup>* <sup>−</sup> *<sup>ρ</sup>*)*B*−<sup>1</sup> *<sup>U</sup>*1(Θ) <sup>−</sup> *Hq*(*χ*)(1(*χ*, *<sup>ξ</sup>*) <sup>−</sup> *<sup>m</sup>*(*U*)) −*Kq*(*χ*)2(*χ*) − *ν* 0 (*χ* − *ρ*)*Pq*(*χ* − *s*)*B*1*U*(*χ*, *ξ*)*dρ* − *ν* 0 (*χ* − *ρ*)*Pq*(*χ* − *ρ*) [*ϕ*(*χ*, *Uχ*(., *ξ*), *ξ*)]*dC<sup>ρ</sup>* + *Bx*(*ρ*)*Cx*(*ρ*) *dρ*. (3)

We use (*H*5) to show that *I* has a fixed point *U*(*χ*, *ξ*) that is a mild solution of (1). This suggests that Issue (1) is manageable on Θ. Additionally, we establish that *I* is a random operator. To prove this, we show that *I*(.)(*U*) : *ψ* → Ξ is a random variable for any *U* ∈ Ξ. The measurement of *I*(.)(*U*) : *ψ* → Ξ is then shown. Because of the assumptions (*H*2) and (*H*6), the mapping *<sup>ϕ</sup>*(*χ*, *<sup>U</sup>*, .), *<sup>χ</sup>* <sup>∈</sup> <sup>Θ</sup>, *<sup>U</sup>* <sup>∈</sup> <sup>Ξ</sup> is measurable. Assume that *<sup>D</sup>* : *<sup>ψ</sup>* <sup>→</sup> <sup>2</sup><sup>Ξ</sup> is provided by:

$$D(\xi) = \{ \mathcal{U} \in \Xi : \|\mathcal{U}\|\_{\Xi} \le Q(\xi) \}.$$

*D*(*χ*) is bounded, convex, closed, and solid for all *ξ* ∈ *ψ*. So, *D* is measurable. Suppose *ξ* ∈ *ψ* is fixed; then, *U* ∈ *D*(*ξ*) and by (*A*1), we obtain:

$$\begin{array}{rcl} \|\mathcal{U}\_{\boldsymbol{\rho}}\|\_{D} & \leq & \beta(\boldsymbol{\rho})|\mathcal{U}(\boldsymbol{\rho})|\mathcal{W} + \omega(\boldsymbol{\rho})||\mathcal{U}\_{0}||\_{D} \\ & \leq & \mathcal{Z}\_{\kappa}|\mathcal{U}(\boldsymbol{\rho})| + \omega\_{\kappa}||\boldsymbol{\varrho}\_{1}||\_{D} \end{array}$$

and via (*H*3) and (*H*4), we have


Then, we have

$$\left| \left( I(\xi) \mathcal{U}(\chi) \right) \right| \leq \left. \omega (1 + \kappa \omega \zeta) \right( \left| \varrho\_1 \vert\_{D} + \mathcal{U}(D\_{\mathbb{K}} \zeta) \int\_{0}^{\chi} p(\rho\_{\prime} \xi) d\rho \right) \kappa \omega \zeta \| p^{1}(\xi) \| + \omega^{\prime} |\varrho\_2| (1 + \kappa \omega \zeta) .$$

Thus

$$\begin{aligned} \|I(\boldsymbol{\xi})U\|\_{\boldsymbol{\Xi}} &\leq \ \omega (1 + \kappa \omega \boldsymbol{\zeta}) (\|\boldsymbol{\varrho}\_{1}\|\_{D} + \mathcal{U}(\boldsymbol{D\_{\kappa}}\omega) \|\boldsymbol{\varrho}\|\_{L}^{1}) \kappa \omega \boldsymbol{\zeta} |\mathcal{U}^{1}(\boldsymbol{\xi})| + \omega \boldsymbol{\zeta} (1 + \kappa \omega \boldsymbol{\zeta}) |\boldsymbol{\varrho}\_{2}| \\ &\leq \ \boldsymbol{\varrho}(\omega). \end{aligned}$$

Thus, we deduce that, with stochastic domain *D*, *I* is a random operator and *I*(*ξ*) : *D*(*ξ*) → *D*(*ξ*) for each *ξ* ∈ *ψ*.

**Claim 1:** *I* is continuous.

Assume that *<sup>U</sup><sup>n</sup>* is a sequence where *<sup>U</sup><sup>n</sup>* <sup>→</sup> *<sup>U</sup>* in *<sup>Y</sup>*. Then,

$$\begin{split} |(I(\boldsymbol{\xi})\mathcal{U}^{\boldsymbol{n}})(\boldsymbol{\chi}) - (I(\boldsymbol{\xi})\mathcal{U}(\boldsymbol{\chi}))| &\leq \quad \omega \int\_{0}^{\chi} |\boldsymbol{\varrho}(\rho, \mathcal{U}^{\boldsymbol{n}}\_{\rho}, \boldsymbol{\xi}) - \boldsymbol{\varrho}(\rho, \mathcal{U}\_{\boldsymbol{\theta}}, \boldsymbol{\xi})| d\boldsymbol{\varepsilon} d\rho + \boldsymbol{\xi}\omega \int\_{0}^{\chi} \int\_{0}^{\chi} \|H\_{\boldsymbol{\theta}}(\mathbf{x} - \boldsymbol{\varepsilon})\| \\ &|\boldsymbol{\varrho}(\boldsymbol{\varepsilon}, \mathcal{U}^{\boldsymbol{n}}\_{\varepsilon}(., \boldsymbol{\xi}) - \boldsymbol{\varrho}(\boldsymbol{\varepsilon}, \mathcal{U}\_{\boldsymbol{\varepsilon}}, \boldsymbol{\xi})) d\boldsymbol{\varepsilon} d\rho \\ &\leq \quad \omega \int\_{0}^{\chi} |\boldsymbol{\varrho}(\rho, \mathcal{U}^{\boldsymbol{n}}\_{\rho}, \boldsymbol{\xi}) - \boldsymbol{\varrho}(\rho, \mathcal{U}\_{\boldsymbol{\theta}}, \boldsymbol{\xi})| d\boldsymbol{\varepsilon} d\rho + \kappa\omega^{2}\zeta \int\_{0}^{\chi} |\boldsymbol{\varrho}(\boldsymbol{\varepsilon}, \mathcal{U}^{\boldsymbol{n}}\_{\varepsilon}(., \boldsymbol{\xi}) - \boldsymbol{\varrho}(\boldsymbol{\varepsilon}, \mathcal{U}\_{\boldsymbol{\varepsilon}}, \boldsymbol{\xi})| d\boldsymbol{\varepsilon} d\rho \\ &\leq \quad \omega(1 + \kappa\omega\zeta) \int\_{0}^{\chi} |\boldsymbol{\varrho}(\boldsymbol{\varepsilon}, \mathcal{U}^{\boldsymbol{n}}\_{\varepsilon}(., \boldsymbol{\xi}) - \boldsymbol{\varrho}(\boldsymbol{\varepsilon}, \mathcal{U}\_{\boldsymbol{\varepsilon}}, \boldsymbol{\xi})| d\boldsymbol{\varepsilon} \end{split}$$

As *ϕ*(*χ*, ., *ξ*) is continuous, we obtain

$$||\varrho(.,\mathcal{U}^n,\xi) - \varrho(.,\mathcal{U},\xi)||\_{L^1} \to 0 \text{ as } n \to +\infty.$$

*I* is continuous.

**Claim 2:** we show that *<sup>ξ</sup>* <sup>∈</sup> *<sup>ψ</sup>*, {*<sup>U</sup>* <sup>∈</sup> *<sup>D</sup>*(*ξ*) : *<sup>I</sup>*(*ξ*)*<sup>U</sup>* <sup>=</sup> *<sup>U</sup>*} <sup>=</sup> <sup>∅</sup> by applying Schauder's theorem.

(*a*) *I* maps bounded sets into equicontinuous sets in *D*(*ξ*). Assume that 1, <sup>2</sup> ∈ [0, *κ*] with <sup>2</sup> > 1, *D*(*ξ*) are a bounded set, as in Claim 2, and *U* ∈ *D*(*ξ*). Now,

$$\begin{split} \left| \left( I(\xi) \mathcal{U} \right) (\varepsilon\_{2}) - \left( I(\xi) \mathcal{U} \right) (\varepsilon\_{1}) \right| &\leq \quad \left\| H\_{q} (\varepsilon\_{2}) - H\_{q} (\varepsilon\_{1}) \right\| \left\| \mathcal{U}\_{\mathbb{P}} \right\| \left\| \mathcal{U}\_{\mathbb{P}} \right\| + \left\| K\_{q} (\varepsilon\_{2}) - K\_{q} (\varepsilon\_{1}) \right\| \left\| \mathcal{U}\_{\mathbb{P}} \right\| \left\| \mathcal{U}\_{\mathbb{P}} \right\| \left\| \mathcal{U}\_{\mathbb{P}} \right\| \left\| \mathcal{U}\_{\mathbb{P}} \right\| \left\| \mathcal{U}\_{\mathbb{P}} \right\| \left\| \mathcal{U}\_{\mathbb{P}} \right\| \left\| \mathcal{U}\_{\mathbb{P}} \right\| \left\| \mathcal{U}\_{\mathbb{P}} \right\| \left\| \mathcal{U}\_{\mathbb{P}} \right\| \left\| \mathcal{U}\_{\mathbb{P}} \right\| \left\| \mathcal{U}\_{\mathbb{P}} \right\| \left\| \mathcal{U}\_{\mathbb{P}} \right\| \left\| \mathcal{U}\_{\mathbb{P}} \right\| \left\| \mathcal{U}\_{\mathbb{P}} \right\| \left\| \mathcal{U}\_{\mathbb{P}} \right\| \left\| \mathcal{U}\_{\mathbb{P}} \right\| \left\| \mathcal{U}\_{\mathbb{P}} \right\| \left\| \mathcal{U}\_{\mathbb{P}} \right\| \left\| \mathcal{U}\_{\mathbb{P}} \right\| \left\| \mathcal{U}\_{\mathbb{P}} \right\| \left\| \mathcal{U}\_{\mathbb{P}} \right\| \left\| \mathcal{U}\_{\mathbb{P}} \right\| \left\| \mathcal{U}\_{\mathbb{P}} \right\| \left$$

*ϕ*(, *U*(., *ξ*), *ξ*)|*ddρ* + *ζ* 1 *C*(<sup>2</sup> <sup>−</sup> *<sup>ρ</sup>*)*D*(Ψ)[|*U*1(*ξ*)<sup>|</sup> <sup>+</sup> *Hq*(*κ*)*D*(Ψ)1*<sup>D</sup>* <sup>+</sup> *Hq*(*κ*)*D*(Ψ)|2|]*d<sup>ρ</sup>* +*ζ* <sup>2</sup> 1 *C*(<sup>2</sup> − *ρ*)*D*(Ψ) *κ* 0 *Hq*(*κ* − )*D*(Ψ)|*ϕ*(, *U*(., *ξ*)*ξ*)|*ddρ* ≤ *Hq*( − *ρ*) − *Hq*(<sup>1</sup> − *ρ*)*D*(Ψ)1*<sup>D</sup>* + *Kq*(2) − *Kq*(1)*D*(Ψ)|2|*U*(*Dκ*, *ξ*) <sup>1</sup> 0 *Hq*(<sup>2</sup> − *ρ*) −*Hq*(<sup>1</sup> − *ρ*)*D*(Ψ)*U*(*ρ*, *ξ*)*dρ* + *ωx*(*Dκ*, *ξ*) <sup>2</sup> 1 *p*(*ρ*, *ξ*)*dρ* + *ζ* <sup>1</sup> 0 *Hq*(<sup>2</sup> − *ρ*) − *Hq*(<sup>1</sup> − *ρ*)*D*(Ψ) <sup>×</sup>[|*U*1(*ξ*)<sup>|</sup> <sup>+</sup> *Hq*(*κ*)*D*(Ψ)1*<sup>D</sup>* <sup>+</sup> *Kq*(*κ*)*D*(Ψ)|2|]*d<sup>ρ</sup>* <sup>+</sup> *ζωU*(*Dκ*, *<sup>ξ</sup>*) <sup>1</sup> 0 *Hq*(<sup>2</sup> − *ρ*) −*Hq*(<sup>1</sup> − *ρD*(Ψ) *κ* 0 *<sup>U</sup>*(, *<sup>ξ</sup>*)*ddρζω* <sup>2</sup> 1 (|*U*1(*ξ*)<sup>|</sup> <sup>+</sup> *Hq*(*κ*)*D*(Ψ)1*<sup>D</sup>* <sup>+</sup> *Kq*(*κ*)*D*(Ψ)|2<sup>|</sup> +*ωU*(*Dκ*, *ξ*) *κ* 0 *U*(, *ξ*)*ddρ*.

In the above inequality, right-hand side tends to zero as <sup>2</sup> − <sup>1</sup> → 0, since *Hq*(*χ*), *Kq*(*χ*) are compact for *χ* > 0 and strongly continuous; then, we obtain the continuity in the uniform operator topology [12,33].

(*b*) Assume that *χ* ∈ [0, *κ*] is, fixed and *U* ∈ *D*(*ξ*): by assumption (*H*3),(*H*5); since *Hq*(*χ*) is compact, the set

$$\left\{ \int\_{0}^{\chi} H\_{\eta}(\chi-\rho) \, \uprho (\rho, \mathsf{U}\_{\rho}(.,\mathfrak{f}),\mathfrak{f}) d\rho \int\_{0}^{\chi} H\_{\eta}(\chi-\rho) B\_{2} \mathfrak{p}(\chi,\mathfrak{f}) d\rho \right\}$$

is precompact in Ψ; then, the set

$$\begin{cases} \mathcal{H}\_{\boldsymbol{\eta}}(\boldsymbol{\chi})(\boldsymbol{\varrho}\_{1}(\boldsymbol{\chi},\boldsymbol{\xi})-\boldsymbol{m}(\boldsymbol{\mathcal{U}}))+\mathcal{K}\_{\boldsymbol{\eta}}(\boldsymbol{\chi})\boldsymbol{\varrho}\_{2}(\boldsymbol{\chi})+\int\_{0}^{\boldsymbol{\chi}}(\boldsymbol{\chi}-\rho)P\_{\boldsymbol{\eta}}(\boldsymbol{\chi}-\boldsymbol{s})\mathcal{B}\_{1}\mathcal{U}(\boldsymbol{\chi},\boldsymbol{\xi})d\rho+\int\_{0}^{\boldsymbol{\chi}}(\boldsymbol{\chi}-\rho)P\_{\boldsymbol{\eta}}(\boldsymbol{\chi}-\boldsymbol{s})\mathcal{U}(\boldsymbol{\chi})d\boldsymbol{\chi}+\int\_{0}^{\boldsymbol{\chi}}(\boldsymbol{\chi}-\rho)P\_{\boldsymbol{\eta}}(\boldsymbol{\chi}-\boldsymbol{s})\mathcal{U}(\boldsymbol{\chi})d\rho\\ \mathcal{B}\_{1}(\boldsymbol{\chi},\boldsymbol{\mathcal{U}}\_{2}(\boldsymbol{\chi},\boldsymbol{\xi}),\boldsymbol{\xi})]d\rho+\int\_{0}^{\boldsymbol{\chi}}\left((\boldsymbol{\chi}-\rho)P\_{\boldsymbol{\eta}}(\boldsymbol{\chi}-\rho)\int\_{0}^{\boldsymbol{\chi}}B\_{2}f(\boldsymbol{\chi},\boldsymbol{\xi})d\mathbf{C}\_{\boldsymbol{\eta}}+B\mathbf{x}(\boldsymbol{\rho})\mathbf{C}\mathbf{x}(\boldsymbol{\rho})\right)d\rho\end{cases}$$

is precompact in Ψ. Thus, *I*(*ξ*) : *D*(*ξ*) → *D*(*ξ*) is continuous. Through compact Schauder's theorem, we obtain that *I*(*ξ*) has a fixed point *U*(*ξ*) in *D*(*ξ*). Since <sup>∩</sup>*ξ*∈*ψD*(*ξ*) <sup>=</sup> <sup>∅</sup>, and a measurable selector of *<sup>D</sup>* exists, then via Lemma 4, *<sup>I</sup>* has a stochastic fixed point *U*∗(*ξ*), which is a random mild solution of (1).

#### **4. Results for State-Dependent Delay Case Controllability**

**Definition 10.** *A stochastic process U* : (−∞, *κ*] × *ψ* → Ψ *is a random mild solution of Problem* (2) *if U*(*χ*, *ξ*) = (*χ*, *ξ*); *χ* ∈ (−∞, 0], *U* (0, *ξ*) = 2(*ξ*)*, and the restriction of U*(., *ξ*) *to the interval* Θ *is continuous and verifies the following equation:*

$$\begin{split} \mathcal{U}(\boldsymbol{\chi},\boldsymbol{\xi}) &= \ &H\_{\boldsymbol{\eta}}(\boldsymbol{\chi})(\boldsymbol{\varrho}\_{1}(\boldsymbol{\chi},\boldsymbol{\xi})-\boldsymbol{m}(\boldsymbol{\mathsf{U}}))+K\_{\boldsymbol{\eta}}(\boldsymbol{\chi})\boldsymbol{\varrho}\_{2}(\boldsymbol{\chi}) + \int\_{0}^{\boldsymbol{\chi}} (\boldsymbol{\chi}-\rho)P\_{\boldsymbol{\eta}}(\boldsymbol{\chi}-\boldsymbol{s})B\_{1}\mathcal{U}(\boldsymbol{\chi},\boldsymbol{\xi})d\rho + \int\_{0}^{\boldsymbol{\chi}} (\boldsymbol{\chi}-\rho) \\ &P\_{\boldsymbol{\eta}}(\boldsymbol{\chi}-\rho)[\boldsymbol{\varrho}(\boldsymbol{\chi},\boldsymbol{\mathcal{U}}\_{2}(\boldsymbol{\chi},\boldsymbol{\xi}),\boldsymbol{\xi})]d\rho + \int\_{0}^{\boldsymbol{\chi}} \Big{(}(\boldsymbol{\chi}-\rho)P\_{\boldsymbol{\eta}}(\boldsymbol{\chi}-\rho)\int\_{0}^{\boldsymbol{\chi}}B\_{2}f(\boldsymbol{\chi},\boldsymbol{\xi})d\mathbf{C}\_{\sigma} + \mathcal{B}\mathcal{U}(\boldsymbol{\rho})\mathcal{C}\mathcal{U}(\boldsymbol{\rho})\Big{)}d\rho \end{split}$$

*Set*

$$Q(\theta^{-1}) = \{ \theta(\rho\_\prime \varrho\_2) : (\rho\_\prime \varrho\_2) \in \Theta \times D, \theta(\rho\_\prime \varrho\_2) \le 0 \}.$$

*Suppose that θ* : Θ → (−∞, *κ*] *is continuous.* (*H*<sup>1</sup> ) *the function χ* → 1*<sup>χ</sup> is continuous from <sup>Q</sup>*(*θ*−1) *into D, and there exists a continuous and bounded function <sup>β</sup>*<sup>1</sup> : *<sup>Q</sup>*(*θ*−) <sup>→</sup> (0, <sup>∞</sup>) *where <sup>β</sup>*<sup>1</sup> (*χ*)||1||*<sup>D</sup> for every <sup>χ</sup>* <sup>∈</sup> *<sup>Q</sup>*(*θ*−)*.*

**Remark 2** ([28])**.** *Hypothesis H*<sup>1</sup> *is satisfied through continuous and bounded functions.*

**Lemma 4** ([34])**.** *If U* : (−∞, *κ*] → Ψ *is a function, such that U*<sup>0</sup> = 1*, then*

$$\|\|U\_{\varrho}\|\|\_{D} \le (\omega\_{\mathfrak{k}} + \beta^{\varrho\_1}) \|\varrho\_1\|\_{D} + \zeta\_{\mathfrak{k}} \sup \{ |\mathcal{U}(i)| \colon I \in [0, \max\{0, \rho\}] \}, \varrho \in \mathcal{Q}(\theta^-) \bigcup \Theta.$$

*where <sup>β</sup>*<sup>1</sup> <sup>=</sup> sup*χ*∈*Q*(*θ*−1) *<sup>β</sup>*<sup>1</sup> (*χ*)*.*

#### **The hypotheses**


$$|\phi(\chi, \mathcal{P}, \Delta)| \le p(\chi, \Delta)\eta((|\mathcal{P}||\_{\mathcal{D}}, \Delta) \text{ for a.e.} \chi \in \Theta \text{ and each } \mathcal{P} \in \mathcal{D}\_{\Delta}$$

(*H* <sup>4</sup>) There <sup>∃</sup> a random function *<sup>α</sup>* : <sup>Θ</sup> <sup>×</sup> *<sup>ψ</sup>* <sup>→</sup> **<sup>R</sup><sup>m</sup>** <sup>+</sup> with *<sup>α</sup>*(., *<sup>χ</sup>*) <sup>∈</sup> *<sup>L</sup>*1(Θ, **<sup>R</sup><sup>m</sup>** <sup>+</sup> ) for each *ξ* ∈ *ψ* such that for any bounded *B* ⊆ Ψ.

$$
\mu(\varrho(\chi\_\nu B\_\prime \chi)) \le \alpha(\chi\_\nu \mathfrak{E})\mu(B).
$$

(*H* <sup>5</sup>) There <sup>∃</sup> a random function *<sup>Q</sup>* : *<sup>ψ</sup>* <sup>→</sup> **<sup>R</sup><sup>m</sup>** <sup>+</sup> {0} where:

$$\omega(1+\kappa\omega\lambda)\left(\|\varrho\_1\|\_{D}+\eta(\omega\_{\mathfrak{k}}+\beta^{\varrho\_1})\|\varrho\_1\|\_{D}+\zeta\_{\mathfrak{k}}Q(\chi),\chi)\int\_{0}^{\mathfrak{x}}p(\rho,\chi)d\rho\right)+\kappa\omega\lambda\left|\|\mathcal{U}^{1}(\chi)\|+\omega'(1+\kappa\omega\lambda)\|\varrho\_2\|\leq Q(\mathfrak{f}).$$

(*H* <sup>6</sup>) The linear LO : *<sup>L</sup>*2(Θ, <sup>Ω</sup>) <sup>→</sup> <sup>Ψ</sup> defined by:

$$
\Delta U = \int\_0^\kappa H\_\emptyset(\kappa - \rho) B\_2 \mathcal{U}(\rho, \mathfrak{f}) d\rho
$$

has an inverse operator −<sup>1</sup> that takes values in *<sup>L</sup>*2(Θ, <sup>Ω</sup>)/*ker*, and there <sup>∃</sup> <sup>a</sup> positive constant *<sup>λ</sup>*, such that *B*2−1 ≤ *<sup>λ</sup>*.

(*H* <sup>7</sup>) For each Δ ∈ *ψ*, (., Δ) is continuous and, for each *χ*, 1(*χ*, .), is measurable, and, for each Δ ∈ *ψ*, 2(Δ), is measurable.

**Theorem 2.** *Suppose that* (*H* 1)*–*(*H* <sup>7</sup>) *and* (*H*<sup>1</sup> ) *hold. If*

$$
\omega(1+\omega\lambda\kappa)\int\_0^\kappa \alpha(\rho)\xi(\rho)d\rho < 1. \tag{4}
$$

*Therefore, Theta can be used to control Random Problem* (2)*.*

**Proof.** Using (*H*6), the control is

$$\mathcal{U}(\boldsymbol{\chi},\boldsymbol{\xi}) = \mathsf{T}^{-1}(\boldsymbol{U}^{1}(\boldsymbol{\xi}) - H\_{\boldsymbol{\xi}}(\boldsymbol{\kappa})\boldsymbol{\varrho}\_{1}(0,\boldsymbol{\xi}) - K\_{\boldsymbol{\xi}}(\boldsymbol{\kappa})\boldsymbol{\varrho}\_{2}(\boldsymbol{\xi}) - \int\_{0}^{\kappa} H\_{\boldsymbol{\xi}}(\boldsymbol{\kappa} - \boldsymbol{\rho})B\_{2}\mathcal{U}(\boldsymbol{\chi},\boldsymbol{\xi})d\boldsymbol{\rho} - \int\_{0}^{\kappa} H\_{\boldsymbol{\xi}}(\boldsymbol{\kappa} - \boldsymbol{\rho})\boldsymbol{\varrho}(\boldsymbol{\rho},\boldsymbol{U}\_{\boldsymbol{\theta}(\boldsymbol{\varrho},\boldsymbol{\ell}\boldsymbol{\xi})}(.,\boldsymbol{\xi}),\boldsymbol{\xi})d\boldsymbol{\rho}\Big).$$

The operator *I* : *ψ* × Ξ → Ξ given by: (*I*(*ξ*)*U*)(*χ*) = 1(*χ*, *ξ*), if *χ* ∈ (−∞, 0], and for *χ* ∈ Θ:

$$\begin{split} \mathcal{U}(\boldsymbol{\chi},\boldsymbol{\xi}) &= \ & \mathcal{U}\_{\boldsymbol{\xi}}(\boldsymbol{\chi}) [\boldsymbol{\varrho}\_{1}(\boldsymbol{\chi},\boldsymbol{\xi}) - \boldsymbol{m}(\boldsymbol{\mathsf{U}})] + \mathcal{K}\_{\boldsymbol{\eta}}(\boldsymbol{\chi}) \boldsymbol{\varrho}\_{2}(\boldsymbol{\chi}) + \int\_{0}^{\boldsymbol{\chi}} (\boldsymbol{\chi} - \rho) P\_{\boldsymbol{\eta}}(\boldsymbol{\chi} - \boldsymbol{s}) B\_{1} \mathcal{U}(\boldsymbol{\chi},\boldsymbol{\xi}) d\rho + \int\_{0}^{\boldsymbol{\chi}} (\boldsymbol{\chi} - \rho) P\_{\boldsymbol{\eta}}(\boldsymbol{\chi} - \rho) \\ & \quad [\boldsymbol{\varrho}(\boldsymbol{\chi},\boldsymbol{\mathcal{U}}\_{2}(.,\boldsymbol{\xi}),\boldsymbol{\xi})] d\rho + \int\_{0}^{\boldsymbol{\chi}} \Big{(}(\boldsymbol{\chi} - \rho) P\_{\boldsymbol{\eta}}(\boldsymbol{\chi} - \rho) B\_{\boldsymbol{\omega}}^{-1} \Big{(} \boldsymbol{p}^{1}(\boldsymbol{\Theta}) - H\_{\boldsymbol{\eta}}(\boldsymbol{\chi})) \boldsymbol{\varrho}\_{1}(\boldsymbol{\chi},\boldsymbol{\xi}) - m(\boldsymbol{\mathcal{U}}) \\ & \qquad - \mathcal{K}\_{\boldsymbol{\eta}}(\boldsymbol{\chi}) \boldsymbol{\varrho}\_{2}(\boldsymbol{\chi}) - \int\_{0}^{\boldsymbol{\chi}} (\boldsymbol{\chi} - \rho) P\_{\boldsymbol{\eta}}(\boldsymbol{\chi} - \boldsymbol{s}) [\boldsymbol{\varrho}(\boldsymbol{\chi},\boldsymbol{\mathcal{U}}\_{2}(.,\boldsymbol{\xi}),\boldsymbol{\xi})] d\mathbf{\mathcal{C}}\_{\boldsymbol{\rho}} \Big{) + \mathcal{B}\boldsymbol{\ell}(\boldsymbol{\rho}) \boldsymbol{\c} \mathcal{U}(\boldsymbol{\rho}) \Big{)} d\rho \end{split} \tag{5}$$

This proves that *I* has a fixed point *U*(*χ*, *ξ*), and that (2) is controllable. Moreover, we demonstrate that *I* is a random operator by showing that, for any *U* ∈ Ξ, *I*(.)(*U*) : *ψ* → Ξ is a random variable. We also show that *I*(.)(*U*) : *ψ* → Ξ is measurable, as a mapping *ϕ*(*χ*, *U*, .), *χ* ∈ Θ, *U* ∈ Ξ is measurable through assumptions (*H* <sup>2</sup>) and (*H* <sup>6</sup>). Assume that *<sup>D</sup>* : *<sup>ψ</sup>* <sup>→</sup> <sup>2</sup><sup>Ξ</sup> is given by:

$$D(\mathfrak{f}) = \{ \mathcal{U} \in \Xi : \|\mathcal{U}\|\_{\Xi} \le \mathcal{Q}(\mathfrak{f}) \}.$$

*D*(*χ*) is bounded, convex, closed and solid for all *ξ* ∈ *ψ*. Then, *D* is measurable. Let *ξ* ∈ *ψ* be fixed; if *p* ∈ *D*(*ξ*), then

$$\|\|\mathcal{U}\_{\mathfrak{q}(\chi,\mathcal{U}\_{\mathfrak{K}})}\|\|\_{D\_{-}} = \|\left(\omega\_{\mathfrak{k}} + L^{\varrho\_{1}}\right)\|\varrho\_{1}\|\|\_{D} + \zeta\_{\mathfrak{k}}Q(\mathfrak{f})\_{\mathfrak{k}}$$

For each *U* ∈ *D*(*ξ*), (*H* <sup>3</sup>), and (*H* <sup>4</sup>), for each *χ* ∈ Θ, we have


Thus, with stochastic domain *D*, *I* is a random operator and *I*(*ξ*) : *D*(*ξ*) → *D*(*ξ*) for each *ξ* ∈ *ψ*.

#### **Claim 1:** *I* is continuous.

Suppose that *<sup>U</sup><sup>n</sup>* is a sequence where *<sup>U</sup><sup>n</sup>* <sup>→</sup> *<sup>U</sup>* in <sup>Ξ</sup>. Then,

<sup>|</sup>(*I*(*ξ*)*Un*)(*χ*) <sup>−</sup> (*I*(*ξ*)*U*(*χ*)| ≤ *<sup>ω</sup> χ* 0 <sup>|</sup>*ϕ*(*ρ*, *<sup>U</sup>ϑ*(*χ*, *<sup>U</sup><sup>n</sup> <sup>χ</sup>*)*n*, *<sup>ξ</sup>*) <sup>−</sup> *<sup>ϕ</sup>*(*ρ*, *<sup>U</sup>ϑ*(*χ*,*Uχ*), *<sup>ξ</sup>*)|*dd<sup>ρ</sup>* <sup>+</sup>*ζω <sup>χ</sup>* 0 *κ* 0 *Hq*(*<sup>κ</sup>* <sup>−</sup> )|*ϕ*(, *<sup>p</sup><sup>n</sup>* (., *ξ*) − *ϕ*(, *p*, *ξ*))|*ddρ* ≤ *ω χ* 0 <sup>|</sup>*ϕ*(*ρ*, *<sup>U</sup>ϑ*(*χ*, *<sup>U</sup><sup>n</sup> <sup>χ</sup>*), *<sup>ξ</sup>*)*n*) <sup>−</sup> *<sup>ϕ</sup>*(*ρ*, *<sup>U</sup>ϑ*(*χ*, *<sup>U</sup>χ*), *<sup>ξ</sup>*))|*dd<sup>ρ</sup> κω*2*ζ κ* 0 <sup>|</sup>*ϕ*(, *<sup>U</sup>ϑ*(*χ*, *<sup>U</sup><sup>n</sup> <sup>χ</sup>*)*n*(., *<sup>ξ</sup>*)) <sup>−</sup> *<sup>ϕ</sup>*(*Uϑ*(*χ*, *<sup>U</sup>χ*), *<sup>ξ</sup>*)|*d* ≤ *ω*(1 + *κωζ*) *κ* 0 <sup>|</sup>*ϕ*(, *<sup>U</sup><sup>n</sup> ϑ*(*χ*,*U<sup>n</sup> χ*)(., *<sup>ξ</sup>*) − *<sup>ϕ</sup>*(*Uϑ*(*χ*, *<sup>U</sup>χ*), *<sup>ξ</sup>*)|*d*

As *ϕ*(*χ*, ., *ξ*) is continuous, we have

$$||\varrho(.,\mathcal{U}^n,\xi) - \varrho(.,\mathcal{U},\xi)||\_{\Xi} \to 0 \text{ as } n \to +\infty.$$

*I* is continuous.

**Claim 2:** We show that *<sup>ξ</sup>* <sup>∈</sup> *<sup>ψ</sup>*, {*<sup>U</sup>* <sup>∈</sup> *<sup>D</sup>*(*ξ*) : *<sup>I</sup>*(*ξ*)*<sup>U</sup>* <sup>=</sup> *<sup>U</sup>*} <sup>=</sup> <sup>∅</sup>. We apply Mönch fixed point theorem [35,36].

(*a*) In *D*(*ξ*), *I* transforms bounded sets into equicontinuous sets. Let 1, <sup>2</sup> ∈ [0, *κ*] with <sup>2</sup> > 1, *D*(*ξ*) be a bounded set as in Claim 2, and *U* ∈ *D*(*ξ*). Then,


Thus,


Hence,


In the previous inequality, the right-hand side went to zero as <sup>2</sup> − <sup>1</sup> → 0, *Hq*(*χ*), *Kq*(*χ*) are a strongly continuous operator, and *Hq*(*χ*) and *Kq*(*χ*) for *χ* > 0 are compact, which implies that uniform operator topology is continuous. Suppose that *ξ* ∈ *ψ* is fixed.

(*b*) Suppose that Λ is a subset of *D*(*ξ*) where Λ ⊂ *conv*(*I*(Λ) {0}). <sup>Λ</sup> is bounded and equicontinuous, and function *χ* → *v*(*χ*) = *ς*(Λ(*χ*)) is continuous on (−∞, *κ*]. Via (*H*2), and by considering the characteristics of the measure Λ, we have *χ* ∈ (−∞, *κ*]:

*v* ≤ *ς*(*I*(Λ))(*χ*) {0}) ≤ *ς*(*I*(Λ)(*χ*)) ≤ *ς*(*Hq*(*χ*)1(0, *ξ*)) + *ς*(*Kq*(*χ*)2(*ξ*)) + *ς χ* 0 *Hq*(*χ* − *ρ*)*ϕ*(, *Uϑ*(*χ*,*Uχ*)(., *ξ*)*dρ* <sup>+</sup> *ωλ <sup>χ</sup>* 0 *ς*(*U*1(*ξ*) −*Hq*(*κ*)1(0, *ξ*) − *Kq*(*κ*)2(*ξ*)) + *ς κ* 0 *Hq*(*κ* − )*ϕ*(, *Uϑ*(*χ*,*Uχ*)(., *ξ*), *ξ dρ* ≤ *ω χ* 0 *<sup>ς</sup>*(*ϕ*(*ρ*, *<sup>U</sup>ϑ*(*χ*,*Uχ*)(., *<sup>ξ</sup>*), *<sup>ξ</sup>*))*dρωλ <sup>χ</sup>* 0 *κ* 0 *ς*(*Hq*(*κ* − )*ϕ*(, *Uϑ*(*χ*,*Uχ*)(., *ξ*), *ξ*)*ddρ* ≤ *ω χ* 0 *<sup>α</sup>*(*ρ*)*ς*({*Uϑ*(*χ*,*pχ*) : *<sup>p</sup>* <sup>∈</sup> <sup>Λ</sup>})*dρωλ <sup>χ</sup>* 0 *κ* 0 *ς*(*Hq*(*κ* − )*ϕ*(, *Uϑ*(*χ*,*Uχ*)(., *ξ*), *ξ*)*ddρ* ≤ *ω χ* 0 *γ*(*ρ*)*ζ*(*ρ*) sup 0≤≤*ρ ς*(Λ())*ρ* + *ω*2*λ χ* 0 *κ* 0 *ς*(*ϕ*(, *Uϑ*(*χ*,*Uχ*), *ξ*)*ddρ* ≤ *ω χ* 0 *<sup>γ</sup>*(*ρ*)*ζ*(*ρ*)*ς*(Λ(*ρ*))*d<sup>ρ</sup>* <sup>+</sup> *<sup>ω</sup>*2*λκ <sup>κ</sup>* 0 *α*()*ς*(*ϕ*({*Uϑ*(*χ*,*Uχ*) : *U* ∈ Λ)*d* ≤ *ω χ* 0 *<sup>v</sup>*(*ρ*)*α*(*ρ*)*ζ*(*ρ*)*d<sup>ρ</sup>* <sup>+</sup> *<sup>ω</sup>*2*λκ <sup>κ</sup>* 0 *α*()*ζ*()*ς*(Λ())*d* = *ω χ* 0 *<sup>α</sup>*(*ρ*)*ζ*(*ρ*)*v*(*ρ*)*d<sup>ρ</sup>* <sup>+</sup> *<sup>ω</sup>*2*λκ <sup>κ</sup>* 0 *α*()*ζ*()*v*())*d* ≤ *ω χ* 0 *<sup>α</sup>*(*ρ*)*ζ*(*ρ*)*v*(*ρ*)*d<sup>ρ</sup>* <sup>+</sup> *<sup>ω</sup>*2*λκ <sup>κ</sup>* 0 *α*()*ζ*()*v*())*d* ≤ *ω*(1 + *ωλκ*) *κ* 0 *α*(*ρ*)*ζ*(*ρ*)*v*(*ρ*))*dρ* ≤ *ω*(1 + *ωλκ*) *κ* 0 *α*(*ρ*)*ζ*(*ρ*) sup 0≤≤*ρ v*())*dρ* ≤ *ω*(1 + *ωλκ*)*v*<sup>∞</sup> *κ* 0 *α*(*ρ*)*ζ*(*ρ*)*dρ*.

Thus,

$$||v||\_{\infty} \le \omega (1 + \omega \lambda \kappa) ||v||\_{\infty} \int\_0^\kappa \alpha(\rho) \zeta(\rho) d\rho$$

Then,

$$||v||\_{\infty} \left(1 - \omega(1 + \omega \lambda \kappa) \int\_0^\kappa \alpha(\rho) \zeta(\rho) d\rho \right) \le 0.$$

Hereby, *v*<sup>∞</sup> = 0; thus, *v*(*χ*) = 0 for each *χ* ∈ Θ, this implies Λ(*χ*) is relatively compact in Ψ. Through the result of Ascoli-Arzel *a*` theorem, Λ is relatively compact in *D*(*ξ*). Via *Monch* ¨ fixed-point theorem, we show that I has a fixed point *U*(*ξ*) ∈ *D*(*ξ*). As 6 *<sup>ξ</sup>*∈*<sup>ϕ</sup> <sup>D</sup>*(*ξ*) <sup>=</sup> <sup>∅</sup>; moreover, a measurable selector of *<sup>D</sup>* exists. Lemma implies that I has a stochastic fixed point *U*∗(*ξ*), which is a mild solution of (2).

#### **5. Applications**

The qualitative theory of FDEs, fractional integrodifferential equations, and fractionalorder operators can be applied to a wide range of scientific fields, including fluid mechanics, viscoelasticity, physics, biology, chemistry, dynamical systems, signal processing, and entropy theory. Due to this, academics from all over the world have become interested in the applications of the theory of fractional calculus and the qualitative theory of the aforementioned equations, and many researchers have included them into their most recent research.

For a very long time, DEs driven by a Brownian motion (or Wiener process) have been the focus of study on the qualitative characteristics of stochastic DEs and their applications. Furthermore, applications from a variety of domains, including storage, queueing, economic, and neurophysiological systems, can be found frequently in stochastic DEs driven by a Poisson process. Additionally, stochastic DEs with Poisson jumps have gained much traction in modelling phenomena from a variety of disciplines, especially economics, where jump processes are frequently used to describe asset and commodity price dynamics. These factors are sufficient for the existence and uniqueness of non-Lipschitz stochastic neutral delay DEEs driven by Poisson jumps.

Levy procedures are becoming increasingly significant in the world of banking. While Levy processes are often employed in newer models to accommodate jumps (which can be regarded as external shocks) and achieve a better fit to empirical data, Brownian motion is still frequently used in older models as a source of randomness. As a result, Levy process applications in finance are simple to locate. There have been numerous applications of the theory of impulsive DEs of an integer order in accurate mathematical modelling. It has recently become a crucial subject of research due to the large range of practical problems. This is because many evolutionary systems' states are frequently exposed to rapid disturbances and undergo abrupt shifts from time to time. These changes have a very brief and insignificant length when compared to the lifespan of the process under consideration, and can be viewed as impulses. Due to the lack of effective methods, the control analysis of problems, including the impulse effect, fractional calculus, and white noise, is challenging.

#### **6. Example**

Consider

$$\begin{cases} \, \_0^\eta D\_\upsilon^\eta \mathcal{U}(\chi, \mathfrak{f}, \mathfrak{c}) = \varrho(\chi, \mathcal{U}(\chi, \mathfrak{f}, \mathfrak{c}), \mathfrak{c}) + \int\_0^\upsilon B\_2 f(\chi, \mathfrak{c}) d\mathcal{C}\_{\upsilon \iota} \, \mathfrak{f} \in \Theta := [0, \kappa], \, \upsilon \in [0, T] \\ \mathcal{U}(\chi, \pi, \mathfrak{c}) + m(\mathcal{U}) = \mathcal{U}\_1(\chi, 2\pi, \mathfrak{c}); \, \mathfrak{f} \in [0, \kappa], \\ \mathcal{U}'(\chi, \mathfrak{f}, \mathfrak{c}) = \mathcal{U}\_2(\mathfrak{f}), \end{cases} \tag{6}$$

where <sup>Φ</sup> : <sup>Θ</sup> <sup>×</sup> *<sup>R</sup>* <sup>×</sup> *<sup>ζ</sup>* <sup>→</sup> **<sup>R</sup><sup>m</sup>** is a given function. If <sup>Ξ</sup> <sup>=</sup> *<sup>L</sup>*2[*π*, 2*π*], and *<sup>B</sup>*<sup>1</sup> : <sup>Ξ</sup> <sup>→</sup> <sup>Ξ</sup> given by *B*1*U* = *U* with domain *D*(*B*1) = {*U* ∈ Φ; *U*, *U are absolutely continuous*, *U* ∈ Ξ, *U*(*π*) = *U*(2*π*) = <sup>0</sup>}. Let the strongly continuous cosine function (*Hq*(*χ*))*χ*∈**R<sup>m</sup>** on Φ be infinitesimally generated by the operator *B*1. Furthermore, *B*<sup>1</sup> has a discrete spectrum, and the eigenvalues are <sup>−</sup>*n*2, *<sup>n</sup>* <sup>∈</sup> *IN* with corresponding normalized eigenvectors

$$\mathcal{U}\_n(\varepsilon) := \left(\frac{2}{2\pi}\right)^{\frac{1}{2}} \cos(n\varepsilon).$$

and


(iii) For *<sup>x</sup>* <sup>∈</sup> <sup>Φ</sup>, *Hq*(*ϑ*)*<sup>x</sup>* <sup>=</sup> <sup>∑</sup><sup>∞</sup> *<sup>n</sup>*=<sup>1</sup> sin(*nt*)*x*, *UnUn*, and the associated cosine family is

$$K\_{\emptyset}(\vartheta)\mathfrak{x} = \sum\_{n=1}^{\infty} \frac{\cos(nt)}{n} \langle \mathfrak{x}, \mathsf{U}\_{n} \rangle \mathsf{U}\_{n}.$$

Consequently, *Kq*(*χ*) is compact for all *χ* > 0 and

$$\|H\_{\emptyset}(\theta)\| = \|K\_{\emptyset}(\chi)\| \le 1, \forall \chi \ge 0.$$

(iv) Let the group of translation be denoted by Φ:

$$
\overline{\psi}(\chi)\mathfrak{x}(\mathcal{U},\emptyset) = \widetilde{\mathfrak{x}}(\mathcal{U} + \chi\_{\mathcal{U}}\emptyset).
$$

where *<sup>x</sup>*<sup>1</sup> is the extension of *<sup>x</sup>* with period 4*π*. Then,

$$H\_{\mathfrak{q}}(\chi) = \frac{1}{2}(\overline{\psi} + \psi(-\chi)) ; \mathcal{U}\_1 = D\_\prime$$

where D is the infinitesimal generator of the group on

$$X = \{ \mathfrak{x}(.,\mathfrak{g}) \in H^1(\mathfrak{n}, 2\pi) : \mathfrak{x}(\pi, \mathfrak{g}) = \mathfrak{x}(2\pi, \mathfrak{g}) = 0 \}. $$

Suppose that *<sup>B</sup>*<sup>2</sup> is a bounded LO from <sup>Ω</sup> into <sup>Ξ</sup> and the linear operator *<sup>K</sup>* : *<sup>L</sup>*2(Θ, <sup>Ω</sup>) <sup>→</sup> Ξ given by:

$$Kf = \int\_0^k H\_\emptyset(k-\varrho)B\_2f(\varrho,\varrho)d\rho\_\varkappa$$

has an inverse operator *K*−<sup>1</sup> in *L*2(Θ, Ω)/ ker *K*. We deduce that Equation (1) is an abstract formulation of Equation (6) if *H*<sup>1</sup> to *H*<sup>6</sup> are met. Via Theorem 1, we conclude that Equation (6) is controllable.

#### **7. Conclusions**

Existence and controllability results were presented for a couple of classes of secondorder fractional functional differential equations. A stochastic random fixed-point theorem established the basis for our claims. Then, we demonstrated that our problems were controllable. Some of the findings in this area form the basis of our future research plans. New results can be obtained by either changing or generalising the conditions and the functional spaces, or even by involving some fractional differential problems.

**Author Contributions:** Conceptualization, R.S.; Methodology, R.S.; Software, A.A.; Formal analysis, M.A.; Investigation, R.S. and A.A.; Resources, A.A. and M.A.; Writing—original draft, R.S.; Writing review and editing, R.S.; Supervision, A.A.; Project administration, K.A.; Funding acquisition, K.A. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia (grant no. 2384).

**Data Availability Statement:** The data are original, and references are given where required.

**Conflicts of Interest:** the authors declare that they have no known competing financial interest or personal relationships that could have appeared to influence the work reported in this paper.

#### **References**


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**Laila F. Seddek 1,2,\*, Essam R. El-Zahar 1,3 and Abdelhalim Ebaid <sup>4</sup>**


**Abstract:** This paper considers the classes of the first-order fractional differential systems containing a finite number *n* of sinusoidal terms. The fractional derivative employs the Riemann–Liouville fractional definition. As a method of solution, the Laplace transform is an efficient tool to solve linear fractional differential equations. However, this method requires to express the initial conditions in certain fractional forms which have no physical meaning currently. This issue formulated a challenge to solve fractional systems under real/physical conditions when applying the Riemann– Liouville fractional definition. The principal incentive of this work is to overcome such difficulties via presenting a simple but effective approach. The proposed approach is successfully applied in this paper to solve linear fractional systems of an oscillatory nature. The exact solutions of the present fractional systems under physical initial conditions are derived in a straightforward manner. In addition, the obtained solutions are given in terms of the entire exponential and periodic functions with arguments of a fractional order. The symmetric/asymmetric behaviors/properties of the obtained solutions are illustrated. Moreover, the exact solutions of the classical/ordinary versions of the undertaken fractional systems are determined smoothly. In addition, the properties and the behaviors of the present solutions are discussed and interpreted.

**Keywords:** Riemann–Liouville fractional derivative; fractional differential equation; sinusoidal; exact solution

#### **1. Introduction**

Unlike the classical calculus (CC) with integer derivatives, the fractional calculus (FC) implements the derivatives of an arbitrary order (non-integer) [1–3]. So, the FC is considered as a generalization of the CC. During the past decades, numerous physical, engineering, and biological problems have been investigated by means of the FC ([4–9]). There are several definitions for the derivatives of an arbitrary order, such as the Caputo fractional derivative (CFD) [10–22], the Riemann–Liouville fractional derivative (RLFD) [23–25], and the conformable derivative [26–29]. However, some difficulties arise when applying the RLFD to solve fractional models under real physical conditions. The present paper is an attempt to face such an issue by considering the following class of first-order fractional ordinary equations (FODEs):

$$\begin{aligned} \label{eq:SDAR-} & \sum\_{j=1}^{RL} D\_{1}^{\underline{n}} y(t) + \omega^{2} y(t) &=& b\_{1} \sin(\Omega\_{1} t) + b\_{2} \sin(\Omega\_{2} t) + \dots + b\_{n} \sin(\Omega\_{n} t), \\ & =& \sum\_{j=1}^{n} b\_{j} \sin(\Omega\_{j} t), \qquad y(0) = A, \quad a \in (0, 1], \end{aligned} \tag{1}$$

**Citation:** Seddek, L.F.; El-Zahar, E.R.; Ebaid, A. The Exact Solutions of Fractional Differential Systems with *n* Sinusoidal Terms under Physical Conditions. *Symmetry* **2022**, *14*, 2539. https://doi.org/10.3390/ sym14122539

Academic Editors: Francisco Martínez González and Mohammed K. A. Kaabar

Received: 4 November 2022 Accepted: 15 November 2022 Published: 1 December 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

where *α* is the non-integer order of the RLFD. The constant *A* is real while *ω*, *bj*, and Ω*<sup>j</sup>* may be real or complex ∀ *j* = 1, 2, 3, . . . , *n*.

The applications of the class (1) may arise in oscillatory models in engineering when the FC is incorporated. This class splits to other physical classes. As examples, for complex *ω*, i.e., *ω* = *iμ* (*μ* is real), where *i* is the imaginary number, the model (1) becomes

$$D\_{-\\\\\infty}^{RL}D\_t^{\alpha}y(t) - \mu^2y(t) = \sum\_{j=1}^n b\_j \sin(\Omega\_j t), \ y(0) = A, \ \alpha \in (0,1]. \tag{2}$$

In addition, if Ω*<sup>j</sup>* = *iσ<sup>j</sup>* and *bj* = −*idj*, the classes (1) and (2) take the form:

$$D\_{-\\\infty}^{RL}D\_t^{\alpha}y(t) + \omega^2 y(t) = \sum\_{j=1}^n d\_j \sinh(\sigma\_j t), \ y(0) = A, \ a \in (0,1]. \tag{3}$$

and

$$\,\_{-\infty}^{RL}D\_t^{\alpha}y(t) - \mu^2 y(t) = \sum\_{j=1}^n d\_j \sinh(\sigma\_{\bar{\jmath}}t)\_{\prime} \; \; y(0) = A, \; \; a \in (0, 1], \tag{4}$$

in terms of hyperbolic functions, respectively.

In Refs. [1–3], the RLFD of order *<sup>α</sup>* <sup>∈</sup> <sup>R</sup><sup>+</sup> <sup>0</sup> of function *<sup>f</sup>* : [*c*, *<sup>d</sup>*] <sup>→</sup> <sup>R</sup> (−<sup>∞</sup> <sup>&</sup>lt; *<sup>c</sup>* <sup>&</sup>lt; *<sup>d</sup>* <sup>&</sup>lt; <sup>∞</sup>) is defined as

$$\, \_c^{RL}D\_t^n f(t) = \frac{1}{\Gamma(n-a)} \frac{d^n}{dt^n} \left( \int\_c^t \frac{f(\tau)}{(t-\tau)^{a-n+1}} d\tau \right) \quad n = [a] + 1, \ t > c,\tag{5}$$

where [*α*] is the integral part of *α*. If 0 < *α* ≤ 1 and *c* → −∞, then

$${}^{RL}\_{-\\\\\infty}D\_t^af(t) = \frac{1}{\Gamma(1-\alpha)}\frac{d}{dt}\left(\int\_{-\\\infty}^t \frac{f(\tau)}{(t-\tau)^a}d\tau\right). \tag{6}$$

It is important to refer to the initial condition (IC) *y*(0) = *A* being physical, unlike the nonphysical condition *Dα*−<sup>1</sup> *<sup>t</sup> y*(0) = *A* that has been considered by the authors [30]. In fact, the IC in the last fractional form is required when solving an FODE via the Laplace transform (LT). This is, simply, because the LT of the RLFD as *<sup>c</sup>* <sup>→</sup> 0, i.e., *RL* <sup>0</sup> *<sup>D</sup><sup>α</sup> <sup>t</sup>* , is [1–3,23,30]

$$\mathcal{L}\left[{}^{RL}\_{0}D\_{t}^{a}y(t)\right] = s^{a}\mathcal{Y}(s) - D\_{t}^{a-1}y(0),\tag{7}$$

which is given in terms of *Dα*−<sup>1</sup> *<sup>t</sup> <sup>y</sup>*(0). Really, the main difference between *RL* −∞*D<sup>α</sup> <sup>t</sup>* and *RL* <sup>0</sup> *<sup>D</sup><sup>α</sup> t* lies in the nature of the considered IC of the problem. In the literature, one can see that the obtained solutions of the physical models depend on both the nature of the given classical/fractional ICs along with the implemented method of solution.

In this regard, Ebaid and Al-Jeaid [30] applied the RLFDs *RL* −∞*D<sup>α</sup> <sup>t</sup>* and *RL* <sup>0</sup> *<sup>D</sup><sup>α</sup> <sup>t</sup>* to obtain a dual solution for a similar model under the nonphysical IC *Dα*−<sup>1</sup> *<sup>t</sup> y*(0) using the LT. Although the LT was shown as an effective tool to exactly investigate several models [31–37], it may not be appropriate to deal with the class (1) under the physical IC *y*(0) = *A* by means of the RLFD operator *RL* <sup>0</sup> *<sup>D</sup><sup>α</sup> <sup>t</sup>* . However, the solution is still available under this physical condition via the RLFD operator *RL* −∞*D<sup>α</sup> <sup>t</sup>* along with avoiding the LT, as will be shown through this paper.

Therefore, the main incentive of the present work is to introduce a new approach to obtain the real solution of the current model under the physical IC *y*(0) = *A* through the following properties (see Refs. [30,38]):

$${}^{RL}\_{-\infty}D\_t^{\alpha}e^{i\omega t} = (i\omega)^{\alpha}e^{i\omega t},\tag{8}$$

$$\, \_{-\infty}^{RL} D\_t^{\mathfrak{a}} \cos(\omega t) = \omega^{\mathfrak{a}} \cos \left( \omega t + \frac{\mathfrak{a} \cdot \mathfrak{T}}{2} \right) , \tag{9}$$

$$D\_{-\\\infty}^{RL}D\_t^{\mu}\sin(\omega t) = \omega^{\mu}\sin\left(\omega t + \frac{\alpha\pi}{2}\right). \tag{10}$$

By using the above properties, it will be shown that the real solution of class (1) exists at specific values of the fractional-order *α*. The symmetric/asymmetric behaviors/properties of the obtained solutions will be demonstrated. Furthermore, it will be declared that the solution of the class (2) is real at any arbitrary value *α*. In addition, the solutions of the corresponding classes with the classical/ordinary derivative, i.e., as *α* → 1, will be evaluated.

A brief description of the structure of this paper is as follows. In Section 2, an analysis of the complementary and particular solutions is presented. Section 3 is devoted to obtaining the exact solutions for the fractional classes. In Section 4, the exact solutions for the ordinary classes are obtained. The behaviors/properties of the solution are introduced in Section 5. The paper is concluded in Section 6.

#### **2. Analysis**

The complementary solution *yc*(*t*) of Equation (1) can be obtained in the form, see [30]:

$$y\_{\varepsilon}(t) = c \, e^{i\delta t}, \quad \delta = -i \left(-\omega^2\right)^{1/\alpha}, \tag{11}$$

which satisfies the homogeneous equation:

$$D\_{-\\\\\infty}^{RL}D\_t^{\alpha}y(t) + \omega^2 y(t) = 0. \tag{12}$$

In order to evaluate the constant *c*, the given IC will be applied on the general solution *y*(*t*) = *yc*(*t*) + *yp*(*t*) in a subsequent section where *yp*(*t*) is a particular solution of the non-homogeneous Equation (1). A simple method to calculate *yp*(*t*) is explained through the following theorem.

**Theorem 1.** *The yp*(*t*) *of the class (1) is in the form:*

$$y\_p(t) = \sum\_{j=1}^n b\_j \left( \frac{\omega^2 \sin\left(\Omega\_j t\right) + \Omega\_j^a \sin\left(\Omega\_j t - \frac{\pi a}{2}\right)}{\omega^4 + \Omega\_j^{2a} + 2\omega^2 \Omega\_j^a \cos\left(\frac{\pi a}{2}\right)} \right), \tag{13}$$

**Proof.** Let us assume that

$$y\_p(t) = \sum\_{j=1}^{n} (\rho\_{1j} \cos(\Omega\_j t) + \rho\_{2j} \sin(\Omega\_j t)). \tag{14}$$

Using the preceding properties of the RLFD operator *RL* −∞*D<sup>α</sup> <sup>t</sup>* , we have

*RL* −∞*D<sup>α</sup> <sup>t</sup> yp* = *n* ∑ *j*=1 *ρ*1*j RL* −∞*D<sup>α</sup> <sup>t</sup>* cos(Ω*jt*) + *ρ*2*j*(*α*) *RL* −∞*D<sup>α</sup> <sup>t</sup>* sin(Ω*jt*) , = *n* ∑ *j*=1 Ω*<sup>α</sup> <sup>j</sup>* cos- Ω*jt <sup>ρ</sup>*1*<sup>j</sup>* cos*πα* 2 <sup>+</sup> *<sup>ρ</sup>*2*<sup>j</sup>* sin*πα* 2 <sup>+</sup> *n* ∑ *j*=1 Ω*<sup>α</sup> <sup>j</sup>* sin- Ω*jt <sup>ρ</sup>*2*<sup>j</sup>* cos*πα* 2 <sup>−</sup> *<sup>ρ</sup>*1*<sup>j</sup>* sin*πα* 2 . (15)

Thus,

$$\begin{aligned} \, \_{-\infty}^{RL}D\_t^a y\_p + \omega^2 y\_p &= \quad \sum\_{j=1}^n \left[ \left( \Omega\_j^a \cos \left( \frac{\pi a}{2} \right) + \omega^2 \right) \rho\_{1j} + \Omega\_j^a \sin \left( \frac{\pi a}{2} \right) \rho\_{2j} \right] \cos \left( \Omega\_j t \right) + \\ &\quad \sum\_{j=1}^n \left[ \left( \Omega\_j^a \cos \left( \frac{\pi a}{2} \right) + \omega^2 \right) \rho\_{2j} - \Omega\_j^a \sin \left( \frac{\pi a}{2} \right) \rho\_{1j} \right] \sin \left( \Omega\_j t \right) . \end{aligned} \tag{16}$$

Inserting the last result into Equation (1) yields

$$\begin{aligned} \rho \left( \Omega\_j^a \cos\left(\frac{\pi \alpha}{2}\right) + \omega^2 \right) \rho\_{1j} + \Omega\_j^a \sin\left(\frac{\pi \alpha}{2}\right) \rho\_{2i} &= 0, \\ \left( \Omega\_j^a \cos\left(\frac{\pi \alpha}{2}\right) + \omega^2 \right) \rho\_{2j} - \Omega\_j^a \sin\left(\frac{\pi \alpha}{2}\right) \rho\_{1j} &= b\_{j\prime} \end{aligned} \tag{17}$$

which can be easily solved to obtain *ρ*1*<sup>j</sup>* and *ρ*2*<sup>j</sup>* in the forms:

$$\rho\_{1j} = -\frac{\Omega^a b\_j \sin\left(\frac{\pi a}{2}\right)}{\omega^4 + \Omega\_j^{2a} + 2\omega^2 \Omega\_j^a \cos\left(\frac{\pi a}{2}\right)}, \quad \rho\_{2j} = \frac{b\_j \omega^2 + \Omega\_j^a b\_j \cos\left(\frac{\pi a}{2}\right)}{\omega^4 + \Omega\_j^{2a} + 2\omega^2 \Omega\_j^a \cos\left(\frac{\pi a}{2}\right)}.\tag{18}$$

Employing (18) into (14), we find

$$y\_p(t) = \sum\_{j=1}^n b\_j \left( \frac{\omega^2 \sin\left(\Omega\_j t\right) + \Omega\_j^a \sin\left(\Omega\_j t - \frac{\pi a}{2}\right)}{\omega^4 + \Omega\_j^{2a} + 2\omega^2 \Omega\_j^a \cos\left(\frac{\pi a}{2}\right)}\right),\tag{19}$$

which completes the proof.

#### **3. Solution of the Fractional Models:** *α ∈* **(0, 1)**

**Lemma 1.** *The solution of the fractional class (1) is*

$$y(t) = \left(A + \sum\_{j=1}^{n} \frac{\Omega\_{\uparrow}^{a} b\_{\uparrow} \sin\left(\frac{\pi a}{2}\right)}{\omega^{4} + \Omega\_{\downarrow}^{2a} + 2\omega^{2}\Omega\_{\uparrow}^{a} \cos\left(\frac{\pi a}{2}\right)}\right) e^{\left(-\omega^{2}\right)^{\frac{1}{2}}t} + \sum\_{j=1}^{n} b\_{j} \left(\frac{\omega^{2}\sin\left(\Omega\_{\uparrow}t\right) + \Omega\_{\downarrow}^{a} \sin\left(\Omega\_{\uparrow}t - \frac{\pi a}{2}\right)}{\omega^{4} + \Omega\_{\downarrow}^{2a} + 2\omega^{2}\Omega\_{\uparrow}^{a} \cos\left(\frac{\pi a}{2}\right)}\right). \tag{20}$$

**Proof.** The preceding analysis reveals that the general solution of the class (1) is in the form: 

$$y(t) = c \, e^{i\delta t} + \sum\_{j=1}^{n} b\_j \left( \frac{\omega^2 \sin\left(\Omega\_j t\right) + \Omega\_j^a \sin\left(\Omega\_j t - \frac{\pi a}{2}\right)}{\omega^4 + \Omega\_j^{2a} + 2\omega^2 \Omega\_j^a \cos\left(\frac{\pi a}{2}\right)} \right). \tag{21}$$

From this equation, at *t* = 0, we obtain

$$y(0) = c - \sum\_{j=1}^{n} \frac{\Omega\_j^a b\_j \sin\left(\frac{\pi a}{2}\right)}{\omega^4 + \Omega\_j^{2a} + 2\omega^2 \Omega\_j^a \cos\left(\frac{\pi a}{2}\right)},\tag{22}$$

and hence the IC can be applied to give

$$c = A + \sum\_{j=1}^{n} \frac{\Omega\_j^a b\_j \sin\left(\frac{\pi a}{2}\right)}{\omega^4 + \Omega\_j^{2a} + 2\omega^2 \Omega\_j^a \cos\left(\frac{\pi a}{2}\right)}.\tag{23}$$

Substituting (23) into (21), the solution reads

$$y(t) = \left(A + \sum\_{j=1}^{u} \frac{\Omega\_j^a b\_j \sin\left(\frac{\pi a}{2}\right)}{\omega^4 + \Omega\_j^{2a} + 2\omega^2 \Omega\_j^a \cos\left(\frac{\pi a}{2}\right)}\right) e^{\left(-\omega^2\right)^{\frac{1}{2}}t} + \sum\_{j=1}^{u} b\_j \left(\frac{\omega^2 \sin\left(\Omega\_j t\right) + \Omega\_j^a \sin\left(\Omega\_j t - \frac{\pi a}{2}\right)}{\omega^4 + \Omega\_j^{2a} + 2\omega^2 \Omega\_j^a \cos\left(\frac{\pi a}{2}\right)}\right). \tag{24}$$

It can be seen that the above solution satisfies the IC. In addition, the solution (24) is real at specific values of *α*; this point will be discussed later.

**Lemma 2.** *The solution of the fractional class (2) is*

$$y(t) = \left(A + \sum\_{j=1}^{n} \frac{\Omega\_j^{\underline{u}} b\_j \sin\left(\frac{\pi u}{2}\right)}{\mu^4 + \Omega\_j^{2\underline{u}} - 2\mu^2 \Omega\_j^{\underline{u}} \cos\left(\frac{\pi u}{2}\right)}\right) e^{\mu^{\frac{2}{3}}t} - \sum\_{j=1}^{n} b\_j \left(\frac{\mu^2 \sin\left(\Omega\_j t\right) - \Omega\_j^{\underline{u}} \sin\left(\Omega\_j t - \frac{\pi u}{2}\right)}{\mu^4 + \Omega\_j^{2\underline{u}} - 2\mu^2 \Omega\_j^{\underline{u}} \cos\left(\frac{\pi u}{2}\right)}\right). \tag{25}$$

**Proof.** As mentioned in Section 1, the class (2) is a transformed version of the class (1) when *ω* = *iμ*. Hence, the solution of the class (2) can be directly obtained from the solution of the class (1), given in lemma 1, with the aide of the substitution *ω* = *iμ*, which yields

$$y(t) = \left(A + \sum\_{j=1}^{n} \frac{\Omega\_j^a b\_j \sin\left(\frac{\pi a}{2}\right)}{\mu^4 + \Omega\_j^{2a} - 2\mu^2 \Omega\_j^a \cos\left(\frac{\pi a}{2}\right)}\right) e^{\mu^2 t} + \sum\_{j=1}^{n} b\_j \left(\frac{-\mu^2 \sin\left(\Omega\_j t\right) + \Omega\_j^a \sin\left(\Omega\_j t - \frac{\pi a}{2}\right)}{\mu^4 + \Omega\_j^{2a} - 2\mu^2 \Omega\_j^a \cos\left(\frac{\pi a}{2}\right)}\right), \tag{26}$$

$$y(t) = \left(A + \sum\_{j=1}^{n} \frac{\Omega\_j^{\underline{u}} b\_j \sin\left(\frac{\pi u}{2}\right)}{\mu^4 + \Omega\_j^{2\underline{u}} - 2\mu^2 \Omega\_j^{\underline{u}} \cos\left(\frac{\pi u}{2}\right)}\right) e^{\mu^{\frac{2}{n}}t} - \sum\_{j=1}^{n} b\_j \left(\frac{\mu^2 \sin\left(\Omega\_j t\right) - \Omega\_j^{\underline{u}} \sin\left(\Omega\_j t - \frac{\pi u}{2}\right)}{\mu^4 + \Omega\_j^{2\underline{u}} - 2\mu^2 \Omega\_j^{\underline{u}} \cos\left(\frac{\pi u}{2}\right)}\right),\tag{27}$$

which completes the proof.

**Remark 1.** *The analytic method used to obtain the exact solutions of the fractional classes (1) and (2) is shown in this section. The other fractional classes (3) and (4) can also be obtained similarly. It can be seen from the solution (20) of the fractional class (1) that it is not always a real solution for <sup>α</sup>* <sup>∈</sup> (0, 1)*. This is simply because* (−*ω*2)1/*<sup>α</sup>* <sup>∈</sup> <sup>R</sup> <sup>∀</sup> *<sup>α</sup>* <sup>∈</sup> (0, 1)*, but there are certain values of the fractional-order <sup>α</sup> at which the solution (20) is real, <sup>y</sup>*(*t*) <sup>∈</sup> <sup>R</sup>*. Such values of <sup>α</sup> will be addressed in a subsequent section.*

*However, the solution (25) of the fractional class (2) is always a real solution* ∀ *α* ∈ (0, 1) *where <sup>μ</sup>*2/*<sup>α</sup>* <sup>∈</sup> <sup>R</sup> *for <sup>μ</sup>* <sup>∈</sup> <sup>R</sup>*. In the case of the ordinary/classical derivative, i.e., as <sup>α</sup>* <sup>→</sup> <sup>1</sup>*, then the solutions (20) and (25) are real. The solution of the fractional classes (3) and (4) can be obtained via substituting* Ω*<sup>j</sup>* = *iσ<sup>j</sup> and bj* = −*idj into the solutions (20) and (25), respectively. Although, the resulting solutions of fractional classes (3) and (4) are not real at any value of α. In fact, the solutions of classes (3) and (4) are only real when α* → 1*. The solutions of the four classes (1)–(4), as α* → 1*, are determined in the next section.*

#### **4. Solution of the Classical/Ordinary Models:** *α →* **1**

This section focuses on obtaining the exact solutions of the classical/ordinary versions of the classes (1)–(4) when *α* → 1,

#### *4.1. Class (1)*

As *α* → 1, the class (1) is transformed to the following class of ODEs:

$$y'(t) + \omega^2 y(t) = \sum\_{j=1}^{n} b\_j \sin(\Omega\_j t), \ y(0) = A. \tag{28}$$

The solution of this class can be derived from Equation (20) by letting *α* → 1, and accordingly, we have

$$y(t) = \left(A + \sum\_{j=1}^{n} \frac{\Omega\_j b\_j}{\omega^4 + \Omega\_j^2} \right) e^{-\omega^2 t} + \sum\_{j=1}^{n} b\_j \left(\frac{\omega^2 \sin\left(\Omega\_j t\right) + \Omega\_j \sin\left(\Omega\_j t - \frac{\pi}{2}\right)}{\omega^4 + \Omega\_j^2} \right), \tag{29}$$

which is equivalent to

$$y(t) = \left(A + \sum\_{j=1}^{n} \frac{\Omega\_j b\_j}{\omega^4 + \Omega\_j^2} \right) e^{-\omega^2 t} + \sum\_{j=1}^{n} b\_j \left(\frac{\omega^2 \sin(\Omega\_j t) - \Omega\_j \cos(\Omega\_j t)}{\omega^4 + \Omega\_j^2} \right). \tag{30}$$

The validity of the solution (30) can be easily verified by direct substitution into (28). Moreover, this solution satisfies the given IC.

#### *4.2. Class (2)*

The class (2), as *α* → 1, reduces to ODEs:

$$y'(t) - \mu^2 y(t) = \sum\_{j=1}^{n} b\_j \sin(\Omega\_j t), \ y(0) = A. \tag{31}$$

From Equation (24), we obtain as *α* → 1 that

$$y(t) = \left(A + \sum\_{j=1}^{n} \frac{\Omega\_j b\_j}{\mu^4 + \Omega\_j^2} \right) e^{\mu^2 t} - \sum\_{j=1}^{n} b\_j \left(\frac{\mu^2 \sin\left(\Omega\_j t\right) - \Omega\_j \sin\left(\Omega\_j t - \frac{\pi}{2}\right)}{\mu^4 + \Omega\_j^2}\right), \tag{32}$$

or

$$y(t) = \left(A + \sum\_{j=1}^{n} \frac{\Omega\_j b\_j}{\mu^4 + \Omega\_j^2} \right) e^{\mu^2 t} - \sum\_{j=1}^{n} b\_j \left(\frac{\mu^2 \sin(\Omega\_j t) + \Omega\_j \cos(\Omega\_j t)}{\mu^4 + \Omega\_j^2} \right). \tag{33}$$

*4.3. Class (3)*

The class (3) as *α* → 1 becomes

$$y'(t) + \omega^2 y(t) = \sum\_{j=1}^{n} d\_j \sinh(\sigma\_j t), \ y(0) = A. \tag{34}$$

Because this class is transformed from the class (1) when Ω*<sup>j</sup>* = *iσj*, and *bj* = −*idj*, then the solution of the current class is determined from Equation (30) as

$$y(t) = \left(A + \sum\_{j=1}^{n} \frac{\sigma\_{j} d\_{j}}{\omega^{4} - \sigma\_{j}^{2}}\right) e^{-\omega^{2}t} - \sum\_{j=1}^{n} i d\_{j} \left(\frac{\omega^{2}\sin\left(i\sigma\_{j}t\right) - i\sigma\_{j}\cos\left(i\sigma\_{j}t\right)}{\omega^{4} - \sigma\_{j}^{2}}\right),\tag{35}$$

i.e.,

$$y(t) = \left(A + \sum\_{j=1}^{n} \frac{\sigma\_{\dot{j}} d\_{\dot{j}}}{\omega^4 - \sigma\_{\dot{j}}^2} \right) e^{-\omega^2 t} + \sum\_{j=1}^{n} d\_j \left(\frac{\omega^2 \sinh(\sigma\_{\dot{j}} t) - \sigma\_{\dot{j}} \cosh(\sigma\_{\dot{j}} t)}{\omega^4 - \sigma\_{\dot{j}}^2} \right). \tag{36}$$

*4.4. Class (4)*

If *ω* = *iμ*, Ω*<sup>j</sup>* = *iσj*, and *bj* = −*idj*, then the class (1) as *α* → 1 is equivalent to the following class of ODEs:

$$y'(t) - \mu^2 y(t) = \sum\_{j=1}^{n} d\_j \sinh(\sigma\_j t), \ y(0) = A. \tag{37}$$

In this case, we have three possible ways to obtain the solution of the current class. The first way is to substitute *ω* = *iμ*, Ω*<sup>j</sup>* = *iσj*, and *bj* = −*idj* into Equation (30). The second is to substitute Ω*<sup>j</sup>* = *iσ<sup>j</sup>* and *bj* = −*idj* into Equation (33). The third way is the simplest one, by substituting only *ω* = *iμ* into Equation (36). Following the third option, one can obtain the exact solution:

$$y(t) = \left(A + \sum\_{j=1}^{n} \frac{\sigma\_j d\_j}{\mu^4 - \sigma\_j^2} \right) e^{\mu^2 t} + \sum\_{j=1}^{n} d\_j \left(\frac{-\mu^2 \sinh(\sigma\_j t) - \sigma\_j \cosh(\sigma\_j t)}{\mu^4 - \sigma\_j^2} \right), \tag{38}$$

or

$$y(t) = \left(A + \sum\_{j=1}^{n} \frac{\sigma\_{j} d\_{j}}{\mu^{4} - \sigma\_{j}^{2}}\right) e^{\mu^{2}t} - \sum\_{j=1}^{n} d\_{j} \left(\frac{\mu^{2}\sinh\left(\sigma\_{j}t\right) + \sigma\_{j}\cosh\left(\sigma\_{j}t\right)}{\mu^{4} - \sigma\_{j}^{2}}\right),\tag{39}$$

for the present class of ODEs.

**Remark 2.** *The obtained exact solutions for the four classes of ODEs satisfy the condition y*(0) = *A. On the other hand, the validity of the obtained solutions can be easily checked through direct substitutions into the governing ODEs of these classes. We can say that the FC is of great importance and benefits. This is because the FC not only gives the solutions of fractional models but also helps in deriving the solutions of corresponding classical/ordinary models.*

#### **5. Behavior of Solution**

It is seen from the previous sections that the fractional systems (1) and (2) have the exact solutions given by Equation (20) and Equation (24), respectively. The main observation is that the solution (20) of the class (1) is real if the quantity (−*ω*2)1/*<sup>α</sup>* is real. For real *<sup>ω</sup>*, we note that (−*ω*2)1/*<sup>α</sup>* <sup>=</sup> *νω*2/*<sup>α</sup>* where *<sup>ν</sup>* = (−1)1/*α*. So, the solution (20) is real when *<sup>ν</sup>* is real. The authors [31] were able to specify the *<sup>α</sup>*-values such that *<sup>ν</sup>* = (−1)1/*<sup>α</sup>* is real and this occurs that the *α*-values follow the next theorem [30].

**Theorem 2.** *For <sup>n</sup>*, *<sup>k</sup>* <sup>∈</sup> <sup>N</sup>+*, the solution (20) is real when <sup>α</sup>* <sup>=</sup> <sup>2</sup>*n*−<sup>1</sup> <sup>2</sup>(*k*+*n*−1) *(<sup>ν</sup>* <sup>=</sup> <sup>1</sup>*) and <sup>α</sup>* <sup>=</sup> 2*n*−1 <sup>2</sup>(*k*+*n*)−<sup>1</sup> *(<sup>ν</sup>* <sup>=</sup> <sup>−</sup>1*).*

Based on the above theorem, the solution (20) for the fractional class (1) is plotted in Figure 1 for *α* = <sup>1</sup> <sup>2</sup> at different numbers of the sinusoidal terms. Figure 2 shows the variation in the solution (20) for the fractional class (1) with two sinusoidal terms at different values of the initial condition *A*. In addition, Figure 3 indicates the behavior of the solution at various values of the fractional-order *α* when ten sinusoidal terms are incorporated in the fractional class (1). Furthermore, the solution is depicted in Figure 4 at some selected values *α* close to unity. This figure declares that the fractional solution becomes identical to the ordinary/classical solution as *α* → 1 which validates the present results.

For the fractional class (2), the solution (25) is displayed in Figure 5 when *α* = <sup>1</sup> <sup>2</sup> at different numbers of the sinusoidal terms. The behavior of the solution of this class is similar to Figure 1 but with a slightly higher magnitude of the oscillations for the same numbers of the sinusoidal terms. Figure 6 gives us a picture of the solution profile as the fractional-order *α* varies regarding the fractional class (2). Moreover, Figure 7 displays the profile of the solution (25) at various values of the parameter *μ*. The current results reveal the oscillatory nature of the obtained solutions for the fractional systems (1) and (2). Finally, the present analysis may be extended to effectively analyze higher-order fractional systems containing a finite number of sinusoidal terms.

**Figure 1.** Plots of the solution for the fractional class (1) when *α* = <sup>1</sup> <sup>2</sup> , *<sup>A</sup>* <sup>=</sup> 0, *<sup>ω</sup>* <sup>=</sup> <sup>1</sup> <sup>2</sup> , *bj* = *j*, and Ω*<sup>j</sup>* = *jπ*/2 at different values of *n* (number of sinusoidal terms).

**Figure 2.** Plots of the solution for the fractional class (1) when *α* = <sup>1</sup> <sup>2</sup> , *<sup>ω</sup>* <sup>=</sup> <sup>1</sup> <sup>2</sup> , *bj* = *j*, and Ω*<sup>j</sup>* = *jπ*/2 at different values of *A* = −2, −1, 0, 1, 2 for two sinusoidal terms (*n* = 2).

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**Figure 3.** Plots of the solution for the fractional class (1) when *α* = <sup>1</sup> <sup>2</sup> , *<sup>A</sup>* <sup>=</sup> 0, *<sup>ω</sup>* <sup>=</sup> <sup>1</sup> <sup>5</sup> , *bj* = *j*, and Ω*<sup>j</sup>* = *jπ*/10 at different values of *α* = <sup>1</sup> 4 , 1 2 , 3 4 , 7 <sup>8</sup> for ten sinusoidal terms (*n* = 10).

**Figure 4.** Plots of the solution for the fractional class (1) when *A* = 0, *ω* = <sup>1</sup> <sup>5</sup> , *bj* = *j*, and Ω*<sup>j</sup>* = *jπ*/10 at different values of *α* = <sup>27</sup> <sup>29</sup> , <sup>45</sup> <sup>47</sup> , <sup>61</sup> <sup>63</sup> , <sup>81</sup> <sup>83</sup> , 1 for ten sinusoidal terms (*n* = 10).

**Figure 5.** Plots of the solution for the fractional class (2) when *α* = <sup>1</sup> <sup>2</sup> , *<sup>A</sup>* <sup>=</sup> 0, *<sup>μ</sup>* <sup>=</sup> <sup>1</sup> <sup>2</sup> , *bj* = *j*, and Ω*<sup>j</sup>* = *jπ*/2 at different values of *n* (number of sinusoidal terms).

**Figure 6.** Plots of the solution for the fractional class (2) when *μ* = <sup>1</sup> <sup>2</sup> , *A* = 0, *bj* = *j*, and Ω*<sup>j</sup>* = *jπ*/2 at different values of *α* for five sinusoidal terms (*n* = 5).

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**Figure 7.** Plots of the solution for the fractional class (2) when *α* = <sup>1</sup> <sup>2</sup> , *A* = 0, *bj* = *j*, and Ω*<sup>j</sup>* = *jπ*/2 at different values of *μ*.

#### **6. Conclusions**

In this paper, a class of first-order fractional differential systems containing a finite number *n* of sinusoidal terms was analyzed by means of the Riemann–Liouville fractional definition. The difficulties in solving fractional systems under real/physical initial conditions using the Riemann–Liouville fractional definition are overcome in this paper. This task was achieved via a straightforward method. The suggested method was successfully applied to extract the exact solutions of the considered fractional systems. In addition, the corresponding exact solutions of the classical/ordinary versions were determined. The obtained results reveal the oscillatory nature of the present fractional systems. Moreover, the properties/behaviors of the obtained solutions were investigated graphically and hence interpreted. Accordingly, the current approach may deserve a further extension to include fractional systems of a higher order when the sinusoidal terms of a finite number are incorporated. Finally, the current approach may be applied to include other ideas [39–47].

**Author Contributions:** Conceptualization, L.F.S., E.R.E.-Z. and A.E.; methodology, L.F.S., E.R.E.-Z. and A.E.; software, L.F.S., E.R.E.-Z. and A.E.; validation, L.F.S., E.R.E.-Z. and A.E.; formal analysis, L.F.S., E.R.E.-Z. and A.E.; investigation, L.F.S., E.R.E.-Z. and A.E.; resources, L.F.S., E.R.E.-Z. and A.E.; data curation, L.F.S., E.R.E.-Z. and A.E.; writing—original draft preparation, L.F.S., E.R.E.-Z. and A.E.; writing—review and editing, L.F.S., E.R.E.-Z. and A.E.; visualization, L.F.S., E.R.E.-Z. and A.E.; supervision, L.F.S. and E.R.E.-Z.; project administration, L.F.S. and E.R.E.-Z.; funding acquisition, L.F.S. and E.R.E.-Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** The authors extend their appreciation to the Deputyship for Research and Innovation, the Ministry of Education in Saudi Arabia, for funding this research work through the project number (IF2/PSAU/2022/01/22726)

. **Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors extend their appreciation to the Deputyship for Research and Innovation, the Ministry of Education in Saudi Arabia, for funding this research work through the project number (IF2/PSAU/2022/01/22726). Moreover, they would like to thank the referees for their valuable comments and suggestions which helped to improve the manuscript.

**Conflicts of Interest:** The authors have no competing interest regarding the publication of this paper.

#### **References**

