Topological Properties of Solution Sets for τ-Fractional Non-Instantaneous Impulsive Semi-Linear Differential Inclusions with Infinite Delay

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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.

“…where f : ℑ × E → E. By applying the operator I µ,Φ,w 0,σ on both sides of Equation ( 14) and using (7), we have for σ ∈ (0, b]…”

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

confidence: 99%

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

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“…For more applications, we refer the reader to [11][12][13]. For papers concerning the existence of solutions or mild solutions for different kinds of impulsive differential equations and inclusions, we refer the reader to [14][15][16][17][18][19][20][21][22][23][24]. In the literature, there are two types of problems that contain non-instantaneous impulses.…”

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

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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.

“…In Refs. [10][11][12][13], recent results on different types of impulse differential equations and inclusions are presented.…”

Section: Introductionmentioning

confidence: 99%

Antiperiodic Solutions for Impulsive ω-Weighted ϱ–Hilfer Fractional Differential Inclusions in Banach Spaces

Alsheekhhussain,

Ibrahim,

Al-Sawalha

et al. 2024

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In this article, we construct sufficient conditions that secure the non-emptiness and compactness of the set of antiperiodic solutions of an impulsive fractional differential inclusion involving an ω-weighted ϱ–Hilfer fractional derivative, D0,tσ,v,ϱ,ω, of order σ∈(1,2), in infinite-dimensional Banach spaces. First, we deduce the formula of antiperiodic solutions for the observed problem. Then, we give two theorems regarding the existence of these solutions. In the first, by using a fixed-point theorem for condensing multivalued functions, we show the non-emptiness and compactness of the set of antiperiodic solutions; and in the second, by applying a fixed-point theorem for contraction multivalued functions, we prove the non-emptiness of this set. Because many types of famous fractional differential operators are particular cases from the operator D0,tσ,v,ϱ,ω, our results generalize several recent results. Moreover, there are no previous studies on antiperiodic solutions for this type of fractional differential inclusion, so this work is novel and interesting. We provide two examples to illustrate and support our conclusions.