Impulsive Hilfer fractional differential equations

Ahmed, Hamdy M.; El-Borai, Mahmoud M.; El-Owaidy, H. M.; Ghanem, Ahmed S.

doi:10.1186/s13662-018-1679-7

Cited by 36 publications

(31 citation statements)

References 27 publications

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“…Ahmed and Okasha discussed the existence of mild solutions for a class of Hilfer fractional neutral integrodifferential equations involving nonlocal initial conditions. Ahmed et al studied the nonlinear Hilfer FDEs with impulsive. Sousa et al investigated the mild solutions for Hilfer FDEs with noninstantaneous impulses.…”

Section: Introductionmentioning

confidence: 99%

Approximate controllability of Hilfer fractional differential equations

Yang

2019

Math Methods in App Sciences

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In this paper, the approximate controllability for a class of Hilfer fractional differential equations (FDEs) of order 1 < < 2 and type 0 ≤ ≤ 1 is considered.The existence and uniqueness of mild solutions for these equations are established by applying the Banach contraction principle. Further, we obtain a set of sufficient conditions for the approximate controllability of these equations.Finally, an example is presented to illustrate the obtained results. KEYWORDSapproximate controllability, fractional differential equations, Hilfer fractional derivative MSC CLASSIFICATION26A33; 34A08; 35R11; 93B05 k=0 (−z) k k!Γ(1− − k) is defined only when ∈ (0, 1). There are only few papers that deal with the FDEs of order 1 < < 2. By using sectorial operator and -resolvent family, Shu and Wang 16 investigated the existence of mild solutions for Caputo FDEs of order ∈ (1, 2). Li et al. 17 concerned the Cauchy problems for Riemann-Liouville FDEs of order ∈ (1, 2). Li et al. 18 researched the Caputo FDEs of order ∈ (1, 2). To the best of our knowledge, there is no results about Hilfer FDEs of order 1 < < 2 and type 0 ≤ ≤ 1.Math Meth Appl Sci. 2020;43:242-254. wileyonlinelibrary.com/journal/mma

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Section: Introductionmentioning

confidence: 99%

Approximate controllability of Hilfer fractional differential equations

Yang

2019

Math Methods in App Sciences

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“…13,15,16,[27][28][29][30][31] Further, looking towards the multidimensional utilization of impulsive FDES, 1,2 it is important to consider the new class implicit FDEs with impulse condition that incorporates a wide class of impulsive FDEs as particular cases. [16][17][18][19][20][21][22][23][24][25] The paper is organized as follows. If we take Ψ(t) = t and = 1, then the problem (1)-(4) reduces to implicit impulsive FDEs with the Caputo fractional derivative.…”

Section: Introductionmentioning

confidence: 99%

“…Then again, there are many fascinating research papers involving Hilfer fractional derivative, which incorporates the Riemann-Liouville and Caputo fractional derivative as special cases. 20,21 For more recent advancement in the theory of FDEs involving the Hilfer fractional derivative, one can see Ahmed and El-Borai [22][23][24] and the references referred to in that.…”

Section: Introductionmentioning

confidence: 99%

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On the impulsive implicit Ψ‐Hilfer fractional differential equations with delay

Kharade

Kucche

2019

Math Methods in App Sciences

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In this paper, we investigate the existence and uniqueness of solutions and derive the Ulam-Hyers-Mittag-Leffler stability results for impulsive implicit Ψ-Hilfer fractional differential equations with time delay. It is demonstrated that the Ulam-Hyers and generalized Ulam-Hyers stability are the specific cases of Ulam-Hyers-Mittag-Leffler stability. Extended version of the Gronwall inequality, abstract Gronwall lemma, and Picard operator theory are the primary devices in our investigation. We provide an example to illustrate the obtained results.

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“…Recently, fractional differential equations have been considered greatly by research community in various aspects due to its salient features for real world problems (see [1][2][3][4][5][6][7]). Controllability problems for different kinds of dynamical systems have been studied by several authors (see [8][9][10][11][12][13][14][15]) and references therein.…”

Section: Introductionmentioning

confidence: 99%

Existence Solution and Controllability of Sobolev Type Delay Nonlinear Fractional Integro-Differential System

et al. 2019

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Fractional integro-differential equations arise in the mathematical modeling of various physical phenomena like heat conduction in materials with memory, diffusion processes, etc. In this manuscript, we prove the existence of mild solution for Sobolev type nonlinear impulsive delay integro-differential system with fractional order 1 < q < 2. We establish the sufficient conditions for the approximate controllability of Sobolev type nonlinear impulsive delay integro-differential system with fractional order 1 < q < 2. In addition, we prove the exact null controllability of Sobolev type nonlinear impulsive delay integro-differential system with fractional order 1 < q < 2. Finally, an example is given to illustrate the obtained results.

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Impulsive Hilfer fractional differential equations

Cited by 36 publications

References 27 publications

Approximate controllability of Hilfer fractional differential equations

Approximate controllability of Hilfer fractional differential equations

On the impulsive implicit Ψ‐Hilfer fractional differential equations with delay

Existence Solution and Controllability of Sobolev Type Delay Nonlinear Fractional Integro-Differential System

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