Existence of Mild Solutions for Hilfer Fractional Neutral Integro-Differential Inclusions via Almost Sectorial Operators

Bose, Chandrabose Sindhu Varun; Udhayakumar, R.

doi:10.3390/fractalfract6090532

Cited by 19 publications

(8 citation statements)

References 41 publications

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“…In [41], Shu and Shi presented the correct form of mild solutions to a linear fractional impulsive evolution equation. Varun Bose and Udhayakumar [42] utilized the fractional calculus theory, the semigroup of operator method, and the Martelli fixed point theorem to investigate the existence of a mild solution fractional neutral integrodifferential inclusion with almost sectorial operator with the Hilfer derivative. Varun Bose et al studied the approximate controllability of neutral Volterra integrodifferential inclusions via the Hilfer fractional derivative and with almost sectorial operators.…”

Section: Journal Of Function Spacesmentioning

confidence: 99%

Existence of Mild Solutions for Nonlocal Evolution Equations with the Hilfer Derivatives

Elbukhari

Fan

2023

Journal of Function Spaces

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The existence of mild solutions for Hilfer fractional evolution equations with nonlocal conditions in a Banach space is investigated in this manuscript. No assumptions about the compactness of a function or the Lipschitz continuity of a nonlinear function are imposed on the nonlocal item and the nonlinear function, respectively. However, we assumed that the nonlocal item is continuous, the nonlinear term is continuous and satisfies some specified assumptions, and the associated semigroup is compact. Our theorems are proved by means of approximate techniques, semigroup methods, and fixed point theorem. These methods are useful for fixing the noncompactness of operators caused by some specified given assumptions on this paper. The results obtained here improve some known results. Finlay, two examples are presented for illustration of our main results.

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Section: Journal Of Function Spacesmentioning

confidence: 99%

Existence of Mild Solutions for Nonlocal Evolution Equations with the Hilfer Derivatives

Elbukhari

Fan

2023

Journal of Function Spaces

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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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“…Over the last decade, a significant number of academics have produced neutral fractional differential systems with or without delays, utilizing a variety of fixed-point procedures, mild solutions, noncompactness measures, and nonlocal conditions. For more details, we may refer to [16][17][18][19][20]. Many researchers have extensively studied the existence of mild solutions for neutral stochastic differential systems in [21][22][23][24].…”

Section: Introductionmentioning

confidence: 99%

Hilfer Fractional Neutral Stochastic Differential Inclusions with Clarke’s Subdifferential Type and fBm: Approximate Boundary Controllability

Nandhaprasadh,

Udhayakumar

2024

Contemp. Math.

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In this paper, the approximate boundary controllability of Hilfer fractional neutral stochastic differential inclusions with fractional Brownian motion (fBm) and Clarke’s subdifferential in Hilbert space is discussed. The existence of a mild solution of Hilfer fractional neutral stochastic differential inclusions with fractional Brownian motion and Clarke’s subdifferential is proved by using fractional calculus, compact semigroups, the fixed point theorem, stochastic analysis, and multivalued maps. The required conditions for the approximate boundary controllability of this system are defined according to a corresponding linear system that is approximately controllable. To demonstrate how our primary findings may be used, a final example is provided.

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Existence of Mild Solutions for Hilfer Fractional Neutral Integro-Differential Inclusions via Almost Sectorial Operators

Cited by 19 publications

References 41 publications

Existence of Mild Solutions for Nonlocal Evolution Equations with the Hilfer Derivatives

Existence of Mild Solutions for Nonlocal Evolution Equations with the Hilfer Derivatives

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

Hilfer Fractional Neutral Stochastic Differential Inclusions with Clarke’s Subdifferential Type and fBm: Approximate Boundary Controllability

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