Stability and Controllability Analysis of Stochastic Fractional Differential Equations Under Integral Boundary Conditions Driven by Rosenblatt Process with Impulses

1. Introduction

Fractional calculus has acquired significant popularity and importance due to its numerous applications in many mathematical, physical, and engineering disciplines, such as systems of diffusion phenomena [1], optimal control problems [2,3], chaotic synchronization systems [4], stability [5], controllability [6], quantum theory [7], thermoelasticity systems [8], solutions of differential systems [9,10,11], delay problems [12], and impulsive problems [13]. Compared with integral calculus, fractional calculus provides a more accurate and significant description of many real-world problems. New discoveries in several scientific and technological areas have shown that integer-order differential systems have been rapidly replaced by fractional-order differential systems. The primary advantage of fractional calculus is its ability to describe the memory effect and hereditary properties of different materials and processes through the use of fractional derivatives (FDs). Different types of FDs have been used in the analysis of fractional differential systems (DSs), such as Riemann–Liouville, Hadamard, Caputo, Caputo–Hadamard, Riesz, and Grünwald–Letnikov FDs. Recently, Almeida [14] introduced a new

ψ

-Caputo FD in relation to a different function. The

ψ

-Caputo fractional derivative is one such extension, which is particularly useful in the context of fractional stochastic DSs. This allows the model to account for phenomena where the system’s past states influence its future behavior, such as in viscoelastic materials, anomalous diffusion, or systems exhibiting long-range dependence. To find out more about these new

ψ

-Caputo fractional DSs, and fractional calculus, see [15,16,17,18,19]. For their applications, refer to [20,21,22] and the references therein.

Nonlinear analysis has been known to actively research the area of impulsive DSs. Differential equations, including impulse conditions, have been extensively studied in the literature. It has been noted in ecological models, population dynamics, engineering systems, stability analysis, controllability analysis, optimal control theory, electronics, medicine, biology, and biotechnology that impulsive differential systems are suitable formulations for systems facing short-term perturbations. Impulsive effects are often classified into two distinct categories based on their duration. First, an impulse is considered instantaneous if its impulsive effect occurs immediately; an impulse effect that occurs after a period of time is classified as non-instantaneous. Non-instantaneous impulses were introduced by Hernandez and O’Regan [23] while modeling evolution systems. Fixed-point theorems have been used in the literature to investigate the qualitative properties of instantaneous and non-instantaneous DSs. See [23,24,25] and the references therein for more studies.

Considering noise and uncontrollably perturbed systems as common occurrences in both natural and artificial systems, stochastic models require further investigation instead of deterministic ones. In including disturbance, differential system theory has been broadened to include stochastic DSs. Recently, many researchers have given significant attention to stochastic DSs for explaining a variety of occurrences in technological engineering, population motion, biological science, neuroscience, and many other areas of science and technology [26]. Existence, controllability, stability, uniqueness, and other quantitative and qualitative analyses of the solutions for stochastics DSs have attracted many researchers; see [27,28] and the references therein. Studies have indicated that when it comes to analyzing and explaining an event, stochastic frameworks work exceptionally effectively and more precisely, such as for population display, stock prices, and memory-containing materials.

Stochastic DSs describe systems that evolve randomly over time and are frequently used in fields such as finance, physics, and biology. When these systems are driven by the Rosenblatt process, a type of self-similar process with stationary increments that generalizes fractional Brownian motion (fBm), they can model more complex dependencies and long-range interactions. The Rosenblatt process captures memory effects and is widely used in modeling phenomena like turbulent fluid flows and signal processing, where correlations exist over long time scales.

In a presentation at Wisconsin University in 1940, Ulam [29] introduced the concept of stability in functional equations. He put out the following problem: When is it possible that there exists an additive mapping close to an approximation additive mapping? In 1941, Hyers was the first to answer Ulam’s question, particularly in the context of Banach spaces [30]. This form of stability eventually came to be known as Ulam–Hyers stability. By including variables, Rassias presented a significant extension of Ulam–Hyers stability in 1978 [31]. The idea of stability in functional equations appears when an inequality acts as a perturbation of the original equations. Hence, the difference between the solutions of the inequality and the supplied functional equation is the central problem concerning the stability of functional equations. Ulam–Hyers and Ulam–Hyers–Rassias stability in different forms of functional equations has received a significant amount of attention; these topics are discussed in the monographs by [32,33,34,35]. Further insights can be found in [36,37].

Controllability is the ability of a dynamical system to be transferred through a set of controls from its starting state to any desired state, which is a basic and essential concept in control theory. In modern mathematical control theory, controllability is closely related to structural decomposition, quadratic optimization, and other concepts and is crucial to the growth of engineering. In 1960, Kalman was the first mathematician to develop the idea of controllability. Since then, many researchers have extensively used various methodologies to work on the controllability of different DSs. For more details about controllability and its applications, see [38,39,40,41]. See [42,43] for related results.

Recently, many researchers have conducted stability and controllability analyses of various fractional differential equations and systems. In [18], Dhayal and Zhu studied the stability and controllability of impulsive

ψ

-Hilfer fractional DSs; Shen et al. [44] examined the stability and controllability of stochastic fractional differential systems driven by the Rosenblatt process; and Kumar and Djemai [45] established results on the existence, stability, and controllability of piecewise dynamic systems under the influence of impulses. Moreover, Dhayal et al. [46] derived results on the existence, stability, and controllability of non-instantaneous impulsive stochastic DSs, and Kumar and Malik [47] ascertained the existence, Hyers–Ulam stability, and controllability of an impulsive hybrid neutral switched system. In particular, Dhayal et al. [48] recently studied the stability and controllability of

ψ

-Caputo impulsive fractional stochastic DSs driven by the Rosenblatt process. But to the best of our knowledge, no research works concerning the stability and controllability of non-instantaneous

ψ

-Caputo fractional stochastic differential systems driven by Rosenblatt process under integral boundary conditions has been found in the literature as of now.

Motivated by these research works and to fill the gap mentioned above, this research aims to discuss the stability and controllability of non-instantaneous

ψ

-Caputo fractional stochastic differential systems driven by the Rosenblatt process with integral boundary conditions. In this study, first, we investigate the stability of the following problem:

$\{\begin{matrix} C_{D_{ψ}^{μ}} ω (ϱ) = T ω (ϱ) + Φ_{1} (ϱ, ω (ϱ)) + Φ_{2} (ϱ) \frac{d {\hat{E}}^{K} (ϱ)}{d ϱ}, ϱ \in ⋃_{z = 0}^{δ} (m_{z}, ϱ_{z + 1}], \\ ω (ϱ) = F_{z} (ϱ, ω (ϱ_{z}^{-})), ϱ \in ⋃_{z = 0}^{δ} (m_{z}, ϱ_{z + 1}], \\ ω (0) = \int_{0}^{ℏ} \frac{{(ℏ - ς)}^{ϱ}}{Γ (ϱ)} Φ_{3} (ς, ω (ς)) d ς, \end{matrix}$

(1)

Then, we determine the controllability of the following problem:

$\{\begin{matrix} C_{D_{ψ}^{μ}} ω (ϱ) = T ω (ϱ) + ℵ u (ϱ) + Φ_{1} (ϱ, ω (ϱ)) + Φ_{2} (ϱ) \frac{d {\hat{E}}^{K} (ϱ)}{d ϱ}, ϱ \in ⋃_{z = 0}^{δ} (m_{z}, ϱ_{z + 1}], \\ ω (ϱ) = F_{z} (ϱ, ω (ϱ_{z}^{-})), ϱ \in ⋃_{z = 0}^{δ} (m_{z}, ϱ_{z + 1}], \\ ω (0) = \int_{0}^{ℏ} \frac{{(ℏ - ς)}^{ϱ}}{Γ (ϱ)} Φ_{3} (ς, ω (ς)) d ς, \end{matrix}$

(2)

where $C_{D_{ψ}^{μ}}$ is the $ψ$ -Caputo FD of order $\frac{1}{2} < μ < 1$ , and $ω (\cdot)$ takes points in the real separable Hilbert space $M$ . $0 = m_{0} = ϱ_{0} < ϱ_{1} < ϱ_{2} < \dots < ϱ_{δ} < m_{δ} < ϱ_{δ + 1} = d < \infty, J = [0, d]$ , and $T$ is the generator of a $C_{0}$ -semigroup ${S (ϱ)}_{ϱ \geq 0}$ on $M$ . ${\hat{E}}^{K} = {\hat{E}}^{K} (ϱ) : ϱ \geq 0$ is a $Y$ -valued Rosenblatt process with the parameter $K \in (\frac{1}{2}, 1)$ , where $Y$ is a real separable Hilbert space. $F_{z} (ϱ, ω (ϱ_{z}^{-}))$ denotes non-instantaneous impulses on $(ϱ_{z}, m_{z}], z = 1, 2, \dots, δ .$ $u (\cdot) \in L^{2} (J, X)$ is the control function, and $ℵ : X \to M$ is a bounded and linear operator where $X$ is a real separable Hilbert space. The functions $Φ_{1} : J \times M \to M$ , $Φ_{2} : J \to L_{2}^{0} (P^{\frac{1}{2}} Y, M)$ , $Φ_{3} : J \times M \to M$ , and $F_{z} : (ϱ_{z}, m_{z}] \times M \to M, z = 1, 2, \dots, δ$ are satisfied by certain conditions, which will be explained later.

The significant outcomes of this study are presented as follows:

The stability and controllability of non-instantaneous $ψ$ -Caputo fractional stochastic differential systems driven by the Rosenblatt process under integral boundary conditions are rarely available in the literature which is the key inspiration to our research work in this paper.
The existence of a mild solution for system (1) on $J$ is suitably proved by using the Banach fixed-point theorem and Lipschitz criteria.
Unlike conventional exponential stability results, a novel result has been determined on the stability of impulsive stochastic $ψ$ -Caputo fractional DSs driven by the Rosenblatt process.
We examined the controllability results for the proposed system (2) on $J$ by using the Banach fixed-point theorem and a new piecewise control function. Finally, illustrative examples were provided to demonstrate the authenticity and validity of the presented work.

This article is organized in the following manner. A few important preliminary materials that will be utilized subsequently are provided in Section 2. In Section 3, we investigate the existence of solutions and new stability results for the proposed system (1). Section 4 investigates the findings on the controllability for the suggested system (2). In the last part, Section 5, two examples are presented to demonstrate the certainty of the obtained theory.

2. Preliminaries

Let $L (Y, M)$ denote the space of all bounded linear operators and $∥ \cdot ∥$ represent the norms of the vectors belonging to the spaces $X$ , $Y$ , $M$ , and $L (Y, M)$ . Assume $L^{2} (G_{d}, M)$ denotes the Banach space of all $G_{d}$ -measurable, square integrable, random variables on $M$ . Let $(Θ, G, Q)$ be a complete probability space, and for $ϱ \geq 0$ , $G_{ϱ}$ represents the $σ$ -field generated by ${\hat{E}}^{\hat{K}} (p) : p \in [0, ϱ]$ and the Q-null sets.

Assume

L_{2}^{0} = L_{2}^{0} (P^{\frac{1}{2}} Y, M)

denotes the separable Hilbert space of all Hilbert–Schmidt operators from

P^{\frac{1}{2}} Y

into

M

, with

${∥ ϕ ∥}_{L_{2}^{0}}^{2} = {∥ ϕ P^{\frac{1}{2}} ∥}^{2} = T r (ϕ P ϕ^{*})$

Let

{\hat{E}}^{K} (ϱ) : ϱ \geq 0

denote the one-dimensional Rosenblatt process with the parameter

K \in (\frac{1}{2}, 1)

and written as [49]

${\hat{E}}^{K} (ϱ) = q (K) \int_{0}^{ϱ} \int_{0}^{ϱ} [\int_{y_{1} \lor y_{2}}^{ϱ} \frac{\partial H^{\overset{´}{K}}}{\partial t} (t, y_{1}) \frac{\partial H^{\overset{´}{K}}}{\partial t} (t, y_{2})] d \hat{E} (y_{1}) d \hat{E} (y_{2}),$

where $\hat{E}$ is a Wiener process, and $H^{K} (q, p)$ is defined as

$H^{K} (q, p) = a_{K} p^{(\frac{1}{2} - K)} \int_{p}^{ϱ} {(t - p)}^{K - (\frac{3}{2})} t^{K - (\frac{1}{2})} d t,$

ϱ \geq p

and

H^{K} (ϱ, p) = 0

when

ϱ \leq p

. Here,

a_{K} = \sqrt{\frac{K (2 K - 1)}{Γ (2 - 2 K, K - \frac{1}{2})}}

\overset{´}{K} = \frac{(1 + K)}{2}

q (K) = (\frac{1}{(1 + K)}) \sqrt{\frac{K}{(2 K - 1)}}

is a normalizing constant, and

Γ (\cdot, \cdot)

represents the gamma function. The Rosenblatt process covariance satisfies

$E ({\hat{E}}^{K} (ϱ), {\hat{E}}^{K} (ϱ)) = \frac{1}{2} (p^{2 K} + ϱ^{2 K} - {| ϱ - p |}^{2 K}) .$

Let

{\hat{E}}^{P} (ϱ)

be a

Y

-valued process given by

${\hat{E}}^{P} (ϱ) = \sum_{i = 1}^{\infty} q_{i} (ϱ) P^{\frac{1}{2}} t_{i}, ϱ \geq 0,$

where $q_{i} (ϱ)$ represents a series of mutually independent, two-sided, one-dimensional Rosenblatt processes on $(Θ, G, Q)$ . Assume $P \in L (Y, Y)$ is an operator defined by $P t_{i} = λ_{i} t_{i}$ with the finite-trace $Tr (P) = \sum_{i = 1}^{\infty} λ_{i} < \infty$ , where $λ_{i} \geq 0 : i = 1, 2, \dots$ are real numbers and $t_{i} : i = 1, 2, \dots$ is a complete orthonormal basis in $Y$ . Let ${\hat{E}}^{P} (ϱ)$ be a $Y$ -valued Rosenblatt process with the covariance P as

${\hat{E}}^{P} (ϱ) = \sum_{i = 1}^{\infty} \sqrt{λ_{i}} q_{i} (ϱ) t_{i} .$

Definition 1

([49]). Assume

ϕ : J \to L_{2}^{0} (P^{\frac{1}{2}} Y, M)

such that

\sum_{i = 1}^{\infty} {∥ H_{K}^{*} (ϕ P^{\frac{1}{2}} t_{i}) ∥}_{L^{2} (J, M)} \leq \infty .

Then, for

ϱ \geq 0

, with respect to the Rosenblatt process, its stochastic integral is defined as

$\begin{matrix} \int_{0}^{ϱ} ϕ (p) d {\hat{E}}^{P} (p) & = & \sum_{i = 1}^{\infty} \int_{0}^{ϱ} ϕ (p) P^{\frac{1}{2}} t_{i} d q_{i} (p) \\ = & \sum_{i = 1}^{\infty} \int_{0}^{ϱ} \int_{0}^{ϱ} H_{K}^{*} (ϕ P^{\frac{1}{2}} t_{i}) (y_{1}, y_{2}) d \hat{E} (y_{1}) d \hat{E} (y_{2}) . \end{matrix}$

Lemma 1

([50]). For

ϕ : J \to L_{2}^{0} (P^{\frac{1}{2}} t_{i} Y, M)

such that

\sum_{i = 1}^{\infty} {∥ ϕ P^{\frac{1}{2}} t_{i} ∥}_{L^{\frac{1}{K}}} (J, M) < \infty

holds, and for any

c, d \in J

with

d > c

, we have

$E [∥ \int_{c}^{d} ϕ (p) d {\hat{E}}^{P} (p) ∥^{2}] \leq a_{K} {(d - c)}^{2 K - 1} \sum_{i = 1}^{\infty} \int_{c}^{d} {∥ ϕ (p) P^{\frac{1}{2}} t_{i} ∥}^{2} d p .$

If $\sum_{i = 1}^{\infty} ∥ ϕ P^{\frac{1}{2}} t_{i} ∥$ is uniformly convergent for $ϱ \in J$ , then it satisfies

$E [∥ \int_{c}^{d} ϕ (p) d {\hat{E}}^{P} (p) ∥^{2}] \leq a_{K} {(d - c)}^{2 K - 1} \int_{c}^{d} {∥ ϕ (p) ∥}_{L_{2}^{0}}^{2} d p .$

Let $J_{2} = [c, d]$ and $ψ \in C^{n} (J_{2}, R)$ be an increasing function such that $ψ^{'} (q) \neq 0$ for all $q \in J_{2}$ .

Definition 2

([51]). The ψ-Caputo FD of order

μ (h - 1 < μ < h, h \in N)

for the function

W

is defined as

$(C_{D_{ψ}^{μ}} W) (ϱ) = (c J_{ψ}^{h - μ} W^{[h]}) = \frac{1}{Γ (h - μ)} \int_{c}^{ϱ} {(ψ (ϱ) - ψ (p))}^{h - μ - 1} W^{[h]} (p) ψ^{'} (p) d p,$

where $h = [μ] + 1$ and $W^{[h]} (ϱ) = {(\frac{1}{ψ^{'} (p)} \frac{d}{d ϱ})}^{h} W (ϱ) .$

Lemma 2

([51]). Let

W \in C^{h} ([c, d])

and

μ > 0

. Then, we have

$c^{J_{ψ}^{μ}} C_{D_{ψ}^{μ}} W (ϱ) = W (ϱ) - \sum_{m = 0}^{h - 1} \frac{W^{[m]} (c^{+})}{m!} {(ψ (ϱ) - ψ (p))}^{m} .$

Particularly, for $μ \in (0, 1)$ , we obtain

$c^{J_{ψ}^{μ}} C_{D_{ψ}^{μ}} W (ϱ) = W (ϱ) - W (c) .$

Lemma 3

([52]). Let

μ > 0

and

Π > 0

; then,

i: $J_{ψ}^{μ} {(ψ (ϱ) - ψ (c))}^{Π - 1} (ϱ) = \frac{Γ (Π)}{Γ (Π + μ)} {(ψ (ϱ) - ψ (p))}^{Π + μ - 1}$ .
ii: $C_{J_{ψ}^{μ}} {(ψ (ϱ) - ψ (c))}^{Π - 1} (ϱ) = \frac{Γ (Π)}{Γ (Π - μ)} {(ψ (ϱ) - ψ (p))}^{Π - μ - 1}$ .

Lemma 4.

The ψ-Caputo fractional-order Cauchy system has a solution:

$\{\begin{matrix} C_{D_{ψ}^{μ}} ω (ϱ) = T ω (ϱ) + ℜ (ϱ), ϱ \in (0, d], \\ ω (0) = \int_{0}^{ℏ} \frac{{(ℏ - ς)}^{ϱ}}{Γ (ϱ)} Φ_{3} (ς, ω (ς)) d ς . \end{matrix}$

(3)

given by

$\{\begin{matrix} ω (ϱ) = & T_{ψ}^{μ} (ϱ, 0) \int_{0}^{ℏ} \frac{{(ℏ - ς)}^{ϱ}}{Γ (ϱ)} Φ_{3} (ς, ω (ς)) d ς \\ + \int_{c}^{ϱ} {(ψ (ϱ) - ψ (p))}^{μ - 1} S_{ψ}^{μ} (q, p) ℜ (p) ψ^{'} (p) d p, \end{matrix}$

(4)

where

$T_{ψ}^{μ} (ϱ, p) ω = \int_{0}^{\infty} φ_{μ} (ϑ) S ({(ψ (ϱ) - ψ (p))}^{μ} ϑ) ω d ϑ,$

$S_{ψ}^{μ} (ϱ, p) ω = μ \int_{0}^{\infty} ϑ φ_{μ} (ϑ) S ({(ψ (ϱ) - ψ (p))}^{μ} ϑ) ω d ϑ, 0 \leq p \leq ϱ \leq d,$

and $φ_{μ} (ϑ) = \frac{1}{μ} ϑ^{(- 1 - (\frac{1}{μ}))} ς_{μ} (ϑ^{- \frac{1}{μ}})$ is the probability density function defined on $(0, - \infty)$ , i.e., $φ_{μ} (ϑ) \geq 0,$ $ϑ \in (0, \infty)$ and $\int_{0}^{\infty} φ_{μ} (ϑ) d ϑ = 1$ .

Proof.

We can rewrite Equation (3) in integral equation form as

$\begin{matrix} ω (ϱ) = & \int_{0}^{ℏ} \frac{{(ℏ - ς)}^{ϱ}}{Γ (ϱ)} Φ_{3} (ς, ω (ς)) d ς \\ + \frac{1}{Γ (μ)} + \int_{c}^{ϱ} {(ψ (ϱ) - ψ (p))}^{μ - 1} [T ω (p) + ℜ (p)] ψ^{'} (p) d p, \end{matrix}$

(5)

given that Equation (5) exists. Assume that $Π \geq 0$ . With the help of a Laplace transform, one can obtain

$\hat{M} (Π) = \frac{\int_{0}^{ℏ} \frac{{(ℏ - ς)}^{ϱ}}{Γ (ϱ)} Φ_{3} (ς, ω (ς)) d ς}{Π} + \frac{1}{Π^{μ}} (T \hat{M} (Π) + \hat{ℜ} (Π)),$

where

$\hat{M} (Π) = \int_{0}^{\infty} e^{- Π (ψ (ζ) - ψ (0))} ω (ζ) ψ^{'} (ζ) d ζ,$

and

$\hat{ℜ} (Π) = \int_{0}^{\infty} e^{- Π (ψ (ζ) - ψ (0))} ℜ (ζ) ψ^{'} (ζ) d ζ .$

It follows that

$\begin{matrix} \hat{M} (Π) & = & Π^{μ - 1} {(Π^{μ} J - T)}^{- 1} \int_{0}^{ℏ} \frac{{(ℏ - ς)}^{ϱ}}{Γ (ϱ)} Φ_{3} (ς, ω (ς)) d ς + {(Π^{μ} J - T)}^{- 1} \hat{ℜ} (Π) \\ = & Π^{μ - 1} \int_{0}^{\infty} e^{- Π^{μ} p} S (p) \int_{0}^{ℏ} \frac{{(ℏ - ς)}^{ϱ}}{Γ (ϱ)} Φ_{3} (ς, ω (ς)) d ς d p \\ + \int_{0}^{\infty} e^{- Π^{μ} p} S (p) \hat{ℜ} (Π) d p . \end{matrix}$

Taking $p = \hat{ϱ^{μ}}$ , we obtain

$\begin{matrix} \hat{M} (Π) & = & μ \int_{0}^{\infty} {(Π \hat{ϱ})}^{μ - 1} {e^{- (Π \hat{ϱ})}}^{μ} S ({\hat{ϱ}}^{μ}) \int_{0}^{ℏ} \frac{{(ℏ - ς)}^{ϱ}}{Γ (ϱ)} Φ_{3} (ς, ω (ς)) d ς d \hat{ϱ} \\ + μ \int_{0}^{\infty} {(\hat{ϱ})}^{μ - 1} {e^{- (Π \hat{ϱ})}}^{μ} S ({\hat{ϱ}}^{μ}) \hat{ℜ} (Π) d \hat{ϱ} \\ = & J_{1} + J_{2}, \end{matrix}$

where

$J_{1} = μ \int_{0}^{\infty} {(Π \hat{ϱ})}^{μ - 1} {e^{- (Π \hat{ϱ})}}^{μ} S ({\hat{ϱ}}^{μ}) \int_{0}^{ℏ} \frac{{(ℏ - ς)}^{ϱ}}{Γ (ϱ)} Φ_{3} (ς, ω (ς)) d ς d \hat{ϱ}$

$J_{2} = μ \int_{0}^{\infty} {(\hat{ϱ})}^{μ - 1} {e^{- (Π \hat{ϱ})}}^{μ} S (\hat{ϱ μ}) \hat{ℜ} (Π) d \hat{ϱ}$

Taking

\hat{ϱ} = ψ (ϱ) - ψ (0)

, we obtain

$\begin{matrix} J_{1} & = & μ \int_{0}^{\infty} {(Π^{μ - 1} (ψ (ϱ) - ψ (0)))}^{μ - 1} {e^{- (Π (ψ (ϱ) - ψ (0)))}}^{μ} S {((ψ (ϱ) - ψ (0)))}^{μ} \\ \times \int_{0}^{ℏ} \frac{{(ℏ - ς)}^{ϱ}}{Γ (ϱ)} Φ_{3} (ς, ω (ς)) d ς ψ^{'} (ϱ) d ϱ \\ = & \int_{0}^{\infty} \frac{- 1}{Π} \frac{d}{d ϱ} ({e^{- (Π (ψ (ϱ) - ψ (0)))}}^{μ}) S {((ψ (ϱ) - ψ (0)))}^{μ} \\ \times \int_{0}^{ℏ} \frac{{(ℏ - ς)}^{ϱ}}{Γ (ϱ)} Φ_{3} (ς, ω (ς)) d ς ψ^{'} (ϱ) d ϱ . \end{matrix}$

and

$\begin{matrix} J_{2} & = & μ \int_{0}^{\infty} {((ψ (ϱ) - ψ (0)))}^{μ - 1} {e^{- (Π (ψ (ϱ) - ψ (0)))}}^{μ} S {((ψ (ϱ) - ψ (0)))}^{μ} \hat{ℜ} (Π) ψ^{'} (ϱ) d ϱ \\ = & \int_{0}^{\infty} \int_{0}^{\infty} μ {((ψ (ϱ) - ψ (0)))}^{μ - 1} {e^{- (Π (ψ (ϱ) - ψ (0)))}}^{μ} S {((ψ (ϱ) - ψ (0)))}^{μ} \\ \times e^{- (Π (ψ (ϱ) - ψ (0)))} ℜ ψ^{'} (p) ψ^{'} (ϱ) d p d ϱ . \end{matrix}$

Now, we take the below one-sided stable probability density:

$ς_{μ} (ϑ) = \frac{1}{π} \sum_{j = 1}^{\infty} {(- 1)}^{j - 1} ϑ^{- μ j - 1} \frac{Γ (μ j + 1)}{j!} sin (j π μ), ϑ \in (0, \infty),$

whose integration is defined as follows:

$\int_{0}^{\infty} e^{- Π ϑ} ς_{μ} (ϑ) d ϑ = e^{- Π^{μ}}, μ \in (0, 1) .$

(6)

Using Equation (6), we obtain

$\begin{matrix} J_{1} & = & \int_{0}^{\infty} \frac{- 1}{Π} \frac{d}{d ϱ} (\int_{0}^{\infty} {e^{- (Π (ψ (ϱ) - ψ (0)))}}^{ϑ} ς_{μ} (ϑ) d ϑ) S {((ψ (ϱ) - ψ (0)))}^{μ} \\ \times \int_{0}^{ℏ} \frac{{(ℏ - ς)}^{ϱ}}{Γ (ϱ)} Φ_{3} (ς, ω (ς)) d ς d ϱ \\ = & \int_{0}^{\infty} \int_{0}^{\infty} ϑ ς_{μ} (ϑ) {e^{- (Π (ψ (ϱ) - ψ (0)))}}^{ϑ} S {((ψ (ϱ) - ψ (0)))}^{μ} \\ \times \int_{0}^{ℏ} \frac{{(ℏ - ς)}^{ϱ}}{Γ (ϱ)} Φ_{3} (ς, ω (ς)) d ς ψ^{'} (ϱ) d ϑ d ϱ \\ = & \int_{0}^{\infty} e^{- (Π (ψ (ϱ) - ψ (0)))} (\int_{0}^{\infty} ς_{μ} (ϑ) S (\frac{(ψ (ϱ) - ψ (0)}{ϑ^{μ}}) d ϑ) \\ \times \int_{0}^{ℏ} \frac{{(ℏ - ς)}^{ϱ}}{Γ (ϱ)} Φ_{3} (ς, ω (ς)) d ς ψ^{'} (ϱ) d ϱ . \end{matrix}$

and

$\begin{matrix} J_{2} & = & \int_{0}^{\infty} \int_{0}^{\infty} \int_{0}^{\infty} μ {((ψ (ϱ) - ψ (0)))}^{μ - 1} ς_{μ} (ϑ) e^{- (Π (ψ (ϱ) - ψ (0))) ϑ} S ({((ψ (ϱ) - ψ (0)))}^{μ}) \\ \times e^{- (Π (ψ (ϱ) - ψ (0)))} ℜ (p) ψ^{'} (p) ψ^{'} (ϱ) d ϑ d p d ϱ \\ = & \int_{0}^{\infty} \int_{0}^{\infty} \int_{0}^{\infty} μ e^{- (Π (ψ (ϱ) + ψ (p) - 2 ψ (0)))} \frac{{((ψ (ϱ) - ψ (0)))}^{μ - 1}}{ϑ^{μ}} \\ \times (\frac{{((ψ (ϱ) - ψ (0)))}^{μ - 1}}{ϑ^{μ}}) ℜ (p) ψ^{'} (p) ψ^{'} (ϱ) d ϑ d p d ϱ \\ = & \int_{0}^{\infty} \int_{0}^{\infty} \int_{0}^{\infty} μ e^{- (Π (ψ (ϱ) - ψ (0)))} ς_{μ} (ϑ) \frac{{((ψ (ϱ) - ψ (0)))}^{μ - 1}}{ϑ^{μ}} \\ \times S (\frac{{((ψ (ϱ) - ψ (0)))}^{μ}}{ϑ^{μ}}) ℜ (ψ^{- 1} (ψ (ζ) - ψ (ϱ) + ψ (0))) ψ^{'} (p) ψ^{'} (ϱ) d ϑ d ζ d ϱ \\ = & \int_{0}^{\infty} e^{- (Π (ψ (ϱ) - ψ (0)))} (\int_{o}^{ζ} \int_{0}^{\infty} μ ς_{μ} (ϑ) \frac{{((ψ (ϱ) - ψ (0)))}^{μ - 1}}{ϑ^{μ}} \\ \times S (\frac{{((ψ (ζ) - ψ (p)))}^{μ}}{ϑ^{μ}}) ℜ (p) ψ^{'} (p) d ϑ d p) ψ^{'} (ζ) d ζ . \end{matrix}$

Hence, we obtain

$\begin{matrix} \hat{M} (Π) & = & \int_{0}^{\infty} e^{- (Π (ψ (ϱ) - ψ (0)))} (\int_{o}^{ζ} ς_{μ} (ϑ) S (\frac{{((ψ (ζ) - ψ (p)))}^{μ}}{ϑ^{μ}}) \\ \times \int_{0}^{ℏ} \frac{{(ℏ - ς)}^{ϱ}}{Γ (ϱ)} Φ_{3} (ς, ω (ς)) d ς d ϑ) ψ^{'} (ϱ) d ϱ \\ + \int_{0}^{\infty} e^{- (Π (ψ (ϱ) - ψ (0)))} (\int_{o}^{ζ} \\ \times \int_{0}^{\infty} μ ς_{μ} (ϑ) \frac{{((ψ (ϱ) - ψ (0)))}^{μ - 1}}{ϑ^{μ}} \\ \times S (\frac{{((ψ (ζ) - ψ (p)))}^{μ}}{ϑ^{μ}}) ℜ (p) ψ^{'} (p) d ϑ d p) ψ^{'} (ζ) d ζ . \end{matrix}$

By using an inverse Laplace transform, we obtain

$\begin{matrix} ω (ϱ) & = & \int_{o}^{\infty} ς_{μ} (ϑ) S (\frac{{((ψ (ζ) - ψ (p)))}^{μ}}{ϑ^{μ}}) \int_{0}^{ℏ} \frac{{(ℏ - ς)}^{ϱ}}{Γ (ϱ)} Φ_{3} (ς, ω (ς)) d ς d ϑ \\ + \int_{0}^{ϱ} \int_{0}^{\infty} μ ς_{μ} (ϑ) \frac{{((ψ (ϱ) - ψ (p)))}^{μ - 1}}{ϑ^{μ}} S (\frac{{((ψ (ζ) - ψ (p)))}^{μ}}{ϑ^{μ}}) ℜ (p) ψ^{'} (p) d ϑ d p . \end{matrix}$

Thus, we obtain

$\begin{matrix} ω (ϱ) & = & \int_{o}^{\infty} φ_{μ} (ϑ) S ({((ψ (ζ) - ψ (p)))}^{μ} ϑ) \\ + μ \int_{0}^{ϱ} \int_{0}^{\infty} ϑ φ_{μ} (ϑ) {((ψ (ϱ) - ψ (p)))}^{μ - 1} S {((ψ (ζ) - ψ (p)))}^{μ} ϑ) ℜ (p) ψ^{'} (p) d ϑ d p, \end{matrix}$

where $φ_{μ} (ϑ) = \frac{1}{μ} ϑ^{- 1 - \frac{1}{μ}} ς_{μ} (ϑ^{- \frac{1}{μ}})$ . For any $ω \in M$ , the operators $T_{ψ}^{μ} (ϱ, p) ω$ and $S_{ψ}^{μ} (ϱ, p) ω$ are defined as

$T_{ψ}^{μ} (ϱ, p) ω = \int_{0}^{\infty} φ_{μ} (ϑ) S ({(ψ (q) - ψ (p))}^{μ} ϑ) ω d ϑ,$

and

$S_{ψ}^{μ} (ϱ, p) ω = μ \int_{0}^{\infty} ϑ φ_{μ} (ϑ) S ({(ψ (q) - ψ (p))}^{μ} ϑ) ω d ϑ,) \leq p \leq ϱ \leq d .$

Hence, we obtain

$\begin{matrix} ω (ϱ) & = & T_{ψ}^{μ} (ϱ, 0) \int_{0}^{ℏ} \frac{{(ℏ - ς)}^{ϱ}}{Γ (ϱ)} Φ_{3} (ς, ω (ς)) d ς \\ + \int_{c}^{ϱ} {(ψ (ϱ) - ψ (p))}^{μ - 1} S_{ψ}^{μ} (q, p) ℜ (p) ψ^{'} (p) d p . \end{matrix}$

□

Lemma 5

([51]). For any fixed

ϱ \geq p \geq o

T_{ψ}^{μ} (ϱ, p)

and

S_{ψ}^{μ} (ϱ, p)

are linear bounded operators, and

$\begin{matrix} ∥ T_{ψ}^{μ} (ϱ, p) (ω) ∥ & \leq & \hat{W} ∥ ω ∥, \\ ∥ S_{ψ}^{μ} (ϱ, p) (ω) ∥ & \leq & \frac{μ \hat{W}}{Γ (1 + μ)} ∥ ω ∥ = \frac{\hat{W}}{Γ (μ)} ∥ ω ∥ . \end{matrix}$

Definition 3.

G_{ϱ}

-adapted stochastic process

ω : J \to M

is said to be a mild solution of the system (1) if for any

ϱ \in J, ω (ϱ)

satisfies

ω (0) = \int_{0}^{ℏ} \frac{{(ℏ - ς)}^{ϱ}}{Γ (ϱ)} Φ_{3} (ς, ω (ς)) d ς

, and

ω (ϱ) = F_{z} (ϱ, ω (ϱ_{z}^{-}))

ϱ \in (ϱ_{z}, m_{z}],

z = 1, 2, \dots, δ

, and

$\begin{matrix} ω (ϱ) & = & T_{ψ}^{μ} (ϱ, 0) \int_{0}^{ℏ} \frac{{(ℏ - ς)}^{ϱ}}{Γ (ϱ)} Φ_{3} (ς, ω (ς)) d ς \\ + \int_{0}^{ϱ} {(ψ (ϱ) - ψ (p))}^{μ - 1} S_{ψ}^{μ} (q, p) Φ_{1} (p, ω (p)) ψ^{'} (p) d p \\ + \int_{0}^{ϱ} {(ψ (ϱ) - ψ (p))}^{μ - 1} S_{ψ}^{μ} (q, p) Φ_{2} (p) ψ^{'} (p) d {\hat{E}}^{H} (p), \end{matrix}$

for all $ϱ \in [0, ϱ_{1}]$ , $z = 0$ , and

$\begin{matrix} ω (ϱ) & = & T_{ψ}^{μ} (ϱ, m_{z}) F_{z} (m_{z}, ω (ϱ_{z}^{-})) \\ + \int_{m_{z}}^{ϱ} {(ψ (ϱ) - ψ (p))}^{μ - 1} S_{ψ}^{μ} (q, p) Φ_{1} (p, ω (p)) ψ^{'} (p) d p \\ + \int_{0}^{ϱ} {(ψ (ϱ) - ψ (p))}^{μ - 1} S_{ψ}^{μ} (q, p) Φ_{2} (p) ψ^{'} (p) d {\hat{E}}^{H} (p), \end{matrix}$

for all $ϱ \in (m_{z}, ϱ_{z + 1}]$ , $z = 1, 2, \dots, δ$ .

Definition 4

([53]). System (1) assures a stable mild solution ω if for any

ϵ > 0

℧ > 0

exists such that

E ∥ ω (ϱ) - \hat{ω} (ϱ) ∥ < ϵ

, whenever

E ∥ ω (0) - {\hat{ω}}_{0} ∥ < ℧

, where

\hat{ω}

is the mild solution of (1) with initial conditions

\hat{ω} (0) = {\hat{ω}}_{0}

, and the impulsive conditions

ω (ϱ) = f_{z} (ϱ, \hat{ω} (ϱ_{z}^{-})),

ϱ \in (ϱ_{z}, m_{z}]

z = 1, 2, \dots, δ

Let

Q C (M)

be the space formed by all

G_{ϱ}

-adapted measurable,

M

-valued stochastic process

ω (ϱ) : ϱ \in J

such that

ω

is continuous at

ϱ \neq ϱ_{z}

ω (ϱ_{z}^{-}) = ω (ϱ_{z}^{+})

and

ω (ϱ_{z}^{+})

exist for every

z = 1, 2, \dots, δ

with the norm described as

${∥ ω ∥}_{Q C} = {(sup_{0 \leq ϱ \leq d} E {∥ ω (ϱ) ∥}^{2})}^{\frac{1}{2}} .$

Then $(Q C (M), ∥ . ∥_{Q C})$ represents a Banach Space.

Lemma 6

([54]). Assume

V \in Q C (J, R)

satisfies the inequality

$V (ϱ) \leq U (ϱ) + f (ϱ) \int_{0}^{ϱ} {(ψ (ϱ) - ψ (t))}^{μ - 1} V (t) ψ^{'} (t) d t + \sum_{0 < ϱ_{z} < ϱ} β_{z} V (ϱ_{z}^{-}), ϱ > 0,$

where f is an increasing function, $U \in Q C (J, R)$ and $U \geq 0$ , $β_{z} > 0$ for $z = 1, 2, \dots, δ$ , then

$\begin{matrix} V (ϱ) & \leq & U (ϱ) [\prod_{j = 1}^{z} 1 + β_{j} Z_{μ} {(f (ϱ) Γ (μ) (ψ (ϱ_{z}) - ψ (0)))}^{μ})] \\ \times Z_{μ} {(f (ϱ) Γ (μ) (ψ (ϱ_{z}) - ψ (0)))}^{μ}), ϱ \in (m_{z}, ϱ_{z + 1}] . \end{matrix}$

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Mohamed S. Algolam www.mdpi.com

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