Nonexistence of global solutions for a Hilfer–Katugampola fractional differential problem

Bhairat, Sandeep P.; Samei, Mohammad Esmael

doi:10.1016/j.padiff.2023.100495

Cited by 9 publications

(4 citation statements)

References 11 publications

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“…The technique used in this article can be used as a generalization in the area of solutions of nonlinear fractional and q-fractional differential equations via the bpp theory. The results of this research can establish more capabilities in the articles, such as [18,[38][39][40][41][42][43],…”

Section: Discussionmentioning

confidence: 69%

Optimum solution of $(k,\gimel )$-Hilfer FDEs by $\mathcal{A}$-condensing operators and the incorporated measure of noncompactness

Khokhar,

Patel,

Patle

et al. 2024

J Inequal Appl

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The notion of $\mathcal{A}$ A -condensing operators via the measure of noncompactness is proposed, which retains the existing classes of condensing operators. Results concerning the existence of the best proximity point (pair) of cyclic (noncyclic) $\mathcal{A}$ A -condensing operators along with the coupled best proximity-point theorem for cyclic $\mathcal{A}$ A -condensing operators have been formulated. An application to a $(k, \gimel )$ ( k , ℷ ) -Hilfer fractional differential equation of order $2< p<3$ 2 < p < 3 , type $q\in [0,1]$ q ∈ [ 0 , 1 ] satistfying some boundary conditions is presented. The paper is the first to investigate the optimum solution of such a generalized fractional differential equation. The hypothesis involved in the investigation is independent of the incorporated measure of noncompactness, thereby making our result better in application than that present in the literature. Moreover, added numerical examples validate the theoretical conclusions.

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Section: Discussionmentioning

confidence: 69%

Optimum solution of $(k,\gimel )$-Hilfer FDEs by $\mathcal{A}$-condensing operators and the incorporated measure of noncompactness

Khokhar,

Patel,

Patle

et al. 2024

J Inequal Appl

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“…Differential equations containing Hilfer-Katugampola fractional derivatives are considered in [31][32][33][34][35].…”

Section: Introductionmentioning

confidence: 99%

“…II-Study the topological properties of the set of solutions to Problem (1). III-Extend the recent work conducted in [21,[25][26][27][28][29][30][31][32][33][34][35][36][37] when the considered fractional differential operator is replaced by D σ,v,ψ,w s i ,ϱ and the dimension of the space is infinite. IV-Generalize this work to the case where the right-hand side contains the infinitesimal generator of a strongly continuous cosine family and the nonlinear part.…”

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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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“…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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Nonexistence of global solutions for a Hilfer–Katugampola fractional differential problem

Cited by 9 publications

References 11 publications

Optimum solution of $(k,\gimel )$-Hilfer FDEs by $\mathcal{A}$-condensing operators and the incorporated measure of noncompactness

Optimum solution of $(k,\gimel )$-Hilfer FDEs by $\mathcal{A}$-condensing operators and the incorporated measure of noncompactness

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

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