On Ψ-Hilfer Fractional Integro-Differential Equations with Non-Instantaneous Impulsive Conditions

Arul, Ramasamy; Karthikeyan, P.; Karthikeyan, Kulandhaivel; Geetha, Palanisamy; Alruwaily, Ymnah; Almaghamsi, Lamya; El‐hady, El‐sayed

doi:10.3390/fractalfract6120732

Cited by 5 publications

(4 citation statements)

References 39 publications

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“…Then, Equation ( 37) is yielded from (39) and (41). Now, by (35), (37) and assumption (A), we obtain…”

Section: Non-emptiness and Compactness Of The Mild Solution Set For (1)mentioning

confidence: 99%

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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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“…Then, Equation ( 37) is yielded from (39) and (41). Now, by (35), (37) and assumption (A), we obtain…”

Section: Non-emptiness and Compactness Of The Mild Solution Set For (1)mentioning

confidence: 99%

“…In [39][40][41][42][43][44][45][46], there are studies on the existence of mild solutions of differential equations and inclusions involving the w -weighted Φ-Hilfer fractional derivative of order µ ∈ (0, 1) and of type v ∈ [0, 1] in the special case w(σ) = 1; ∀σ ∈ ℑ.…”

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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“…Differential equations concerning the ψ-Hilfer fractional derivative D σ,v,ψ 0,ϱ , σ ∈ (0, 1), υ ∈ [0, 1], without impulses are studied in [37][38][39][40], with instantaneous impulses in [41] and with non-instantaneous impulses in [41][42][43][44][45]. Also, Sitho [46] studied implicit fractional integro-differential equations with the ψ-Hilfer fractional derivative D σ,v,ψ 0,ϱ without impulses and σ ∈ (1, 2),υ ∈ [0, 1].…”

Section: Introductionmentioning

confidence: 99%

The Existence of Solutions for w-Weighted ψ-Hilfer Fractional Differential Inclusions of Order μ ∈ (1, 2) with Non-Instantaneous Impulses in Banach Spaces

Alsheekhhussain,

Ibrahim,

Al-Sawalha

et al. 2024

Fractal Fract

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In this research, we obtain the sufficient conditions that guarantee that the set of solutions for an impulsive fractional differential inclusion involving a w-weighted ψ-Hilfer fractional derivative, D0,tσ,v,ψ,w,of order μ∈(1,2), in infinite dimensional Banach spaces that are not empty and compact. We demonstrate the exact relation between a differential equation involving D0,tσ,v,ψ,w of order μ ∈(1,2) in the presence of non-instantaneous impulses and its corresponding fractional integral equation. Then, we derive the formula for the solution for the considered problem. The desired results are achieved using the properties of the w-weighted ψ-Hilfer fractional derivative and appropriate fixed-point theorems for multivalued functions. Since the operator D0,tσ,v,ψ,w includes many types of well-known fractional differential operators, our results generalize several results recently published in the literature. We give an example that illustrates and supports our theoretical results.

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“…It was first proposed by Steffenson as poweriods in 1941 and later developed by Dattoli [5,6]. As demonstrated by works such as [7][8][9][10][11][12], these operational approaches are useful and successful research tools. The study and use of hybrid special polynomials in numerous areas of mathematics have greatly benefited from the adoption of these ideas.…”

Section: Introductionmentioning

confidence: 99%

Certain Properties and Characterizations of Multivariable Hermite-Based Appell Polynomials via Factorization Method

2023

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This paper introduces a new type of polynomials generated through the convolution of generalized multivariable Hermite polynomials and Appell polynomials. The paper explores several properties of these polynomials, including recurrence relations, explicit formulas using shift operators, and differential equations. Further, integrodifferential and partial differential equations for these polynomials are also derived. Additionally, the study showcases the practical applications of these findings by applying them to well-known polynomials, such as generalized multivariable Hermite-based Bernoulli and Euler polynomials. Thus, this research contributes to advancing the understanding and utilization of these hybrid polynomials in various mathematical contexts.

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On Ψ-Hilfer Fractional Integro-Differential Equations with Non-Instantaneous Impulsive Conditions

Cited by 5 publications

References 39 publications

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

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

The Existence of Solutions for w-Weighted ψ-Hilfer Fractional Differential Inclusions of Order μ ∈ (1, 2) with Non-Instantaneous Impulses in Banach Spaces

Certain Properties and Characterizations of Multivariable Hermite-Based Appell Polynomials via Factorization Method

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