Hadamard Itô-Doob Stochastic Fractional Order Systems

Mchiri, Lassaad; Arfaoui, Hassen; Dhahri, Slim; El‐hady, El‐sayed; Cherif, Bahri

doi:10.3934/dcdss.2022184

Cited by 10 publications

(8 citation statements)

References 21 publications

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“…Remark Ben Makhlouf and Mchiri [15] investigated the averaging principle of Hadamard Itô–Doob Stochastic Fractional Order Systems without delay.…”

Section: Averaging Principlementioning

confidence: 99%

“…Abouagwa et al [12] studied the existence, uniqueness, and mean square stability of solutions of FIDSE with the non-Lipschitz condition. The notion of the averaging principle is an important way to approximate the solutions of the initial system by the corresponding solutions to the standard system, in mean square and probability senses (see previous works [13,[15][16][17][18][19][20]).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Existence, uniqueness, and averaging principle for Hadamard Itô–Doob stochastic delay fractional integral equations

Mchiri

Mtiri

2023

Math Methods in App Sciences

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“…Remark Ben Makhlouf and Mchiri [15] investigated the averaging principle of Hadamard Itô–Doob Stochastic Fractional Order Systems without delay.…”

Section: Averaging Principlementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Existence, uniqueness, and averaging principle for Hadamard Itô–Doob stochastic delay fractional integral equations

Mchiri

Mtiri

2023

Math Methods in App Sciences

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“…Xia and Yan et al [11] derived an averaging theorem for a new kind of FSDE. Considering the different types of fractional derivatives, Makhlouf et al [12] showed an averaging principle for Hadarmard Itô-Doob FSDEs. Liu et al [13] established an averaging result for Caputo-Hadamard FSDEs.…”

Section: Introductionmentioning

confidence: 99%

A Note on Averaging Principles for Fractional Stochastic Differential Equations

Liu,

Zhang,

Wang

et al. 2024

Fractal Fract

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Over the past few years, many scholars began to study averaging principles for fractional stochastic differential equations since they can provide an approximate analytical method to reduce such systems. However, in the most previous studies, there is a misunderstanding of the standard form of fractional stochastic differential equations, which consequently causes the wrong estimation of the convergence rate. In this note, we take fractional stochastic differential equations with Lévy noise as an example to clarify these two issues. The corrections herein have no effect on the main proofs except the two points mentioned above. The innovation of this paper lies in three aspects: (i) the standard form of the fractional stochastic differential equations is derived under natural time scale; (ii) it is first proved that the convergence interval and rate are related to the fractional order; and (iii) the presented results contain and improve some well known research achievements.

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“…More than thirty years ago, the study of the existence of a mild solution to semi-linear differential Equations and semi-linear differential inclusions containing a fractional differential operator became of interest. Some of these equations contained the Caputo fractional derivative [10][11][12], some involved the Riemann-Liouville fractional differential operator [13,14], some contained the Caputo-Hadamard fractional differential operator [15,16], some included the Hilfer fractional differential operator of order α ∈ (0, 1) in [17][18][19][20][21][22][23][24][25][26], some contained the Katugampola fractional differential operator [27], some contained the Hilfer-Katugampola fractional differential operator of order α ∈ (0, 1) [28][29][30][31][32] and others involved the Hilfer fractional differential operator of order λ ∈ (1, 2) [33].…”

Section: Introductionmentioning

confidence: 99%

Mild Solutions for w-Weighted, Φ-Hilfer, Non-Instantaneous, Impulsive, w-Weighted, Fractional, Semilinear Differential Inclusions of Order μ ∈ (1, 2) in Banach Spaces

Alsheekhhussain,

Ibrahim,

Al-Sawalha

et al. 2024

Fractal Fract

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The aim of this work is to obtain novel and interesting results for mild solutions to a semilinear differential inclusion involving a w-weighted, Φ-Hilfer, fractional derivative of order μ∈(1,2) with non-instantaneous impulses in Banach spaces with infinite dimensions when the linear term is the infinitesimal generator of a strongly continuous cosine family and the nonlinear term is a multi-valued function. First, we determine the formula of the mild solution function for the considered semilinear differential inclusion. Then, we give sufficient conditions to ensure that the mild solution set is not empty or compact. The desired results are achieved by using the properties of both the w-weighted Φ-Laplace transform, w-weighted ψ-convolution and the measure of non-compactness. Since the operator, the w-weighted Φ-Hilfer, includes well-known types of fractional differential operators, our results generalize several recent results in the literature. Moreover, our results are novel because no one has previously studied these types of semilinear differential inclusions. Finally, we give an illustrative example that supports our theoretical results.

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Hadamard Itô-Doob Stochastic Fractional Order Systems

Cited by 10 publications

References 21 publications

Existence, uniqueness, and averaging principle for Hadamard Itô–Doob stochastic delay fractional integral equations

Existence, uniqueness, and averaging principle for Hadamard Itô–Doob stochastic delay fractional integral equations

A Note on Averaging Principles for Fractional Stochastic Differential Equations

Mild Solutions for w-Weighted, Φ-Hilfer, Non-Instantaneous, Impulsive, w-Weighted, Fractional, Semilinear Differential Inclusions of Order μ ∈ (1, 2) in Banach Spaces

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