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1

On solution of non-instantaneous impulsive Hilfer fractional integro-differential evolution system

Trivedi Gargi J., Shah Vishant, Sharma Jaita, Sanghvi Rajesh

Mathematica Applicanda

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2023

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Vol. 51, no. 1

33--50

EN

In this article, we have considered the non-instantaneous fractional integrodifferential evolution system with Hilfer fractional differential operator in the Banach space and discussed its existence results for the mild solution for the equation with local and non-local conditions. These results are obtained by applying the method of a C0 operator generated by the linear part of the equation combined with the concept of nonlinear functional analysis and the fixed point theorems. We have discussed the examples to highlight the applicability of the results.

PL

Artykuł poświęcony jest ułamkowym, z opóźnieniem, systemom ewolucji całkowo-różniczkowej opisanym ułamkowym operatorem różniczkowym Hilfera w przestrzeni Banacha. Analizowane jest istnienia gładkiego rozwiązania równania z warunkami lokalnymi i nielokalnymi. Wyniki uzyskano stosując do operatora C0 generowanego przez liniową część równania metody nieliniowej analizy funkcjonalnej z twierdzeniami o punkcie stałym. Zamieszczone przykłady podkreślają znaczenie otrzymanych wyników.

2

Controllability of time varying semilinear non-instantaneous impulsive systems with delay, and nonlocal conditions

Cabada Dalia, Garcia Katherine, Guevara Cristi, Leiva Hugo

Archives of Control Sciences

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2022

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Vol. 32, No. 2

335--357

EN

In this paper we prove the exact controllability of a time varying semilinear system considering non-instantaneous impulses, delay, and nonlocal conditions occurring simultaneously. It is done by using the Rothe’s fixed point theorem together with some sub-linear conditions on the nonlinear term, the impulsive functions, and the function describing the nonlocal conditions. Furthermore, a control steering the semilinear system from an initial state to a final state is exhibited.

3

Numerical approach to the controllability of fractional order impulsive differential equations

Kumar Avadhesh, Vats Ramesh K., Kumar Ankit, Chalishajar Dimplekumar N.

Demonstratio Mathematica

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2020

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Vol. 53, nr 1

193--207

EN

In this manuscript, a numerical approach for the stronger concept of exact controllability (total controllability) is provided. The proposed control problem is a nonlinear fractional differential equation of order α ∈ (1, 2] with non-instantaneous impulses in finite-dimensional spaces. Furthermore, the numerical controllability of an integro-differential equation is briefly discussed. The tool for studying includes the Laplace transform, the Mittag-Leffler matrix function and the iterative scheme. Finally, a few numerical illustrations are provided through MATLAB graphs.

4

Existence results of noninstantaneous impulsive fractional integro-differential equation

Kataria Haribhai R., Patel Prakashkumar H., Shah Vishant

Demonstratio Mathematica

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2020

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Vol. 53, nr 1

373--384

EN

Existence of mild solution for noninstantaneous impulsive fractional order integro-differential equations with local and nonlocal conditions in Banach space is established in this paper. Existence results with local and nonlocal conditions are obtained through operator semigroup theory using generalized Banach contraction theorem and Krasnoselskii’s fixed point theorem, respectively. Finally, illustrations are added to validate derived results.

5

Differential equations with random Gamma distributed moments of non-instantaneous impulses and p-moment exponential stability

Agarwal R., Hristova S., O’Regan D., Kopanov P.

Demonstratio Mathematica

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2018

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Vol. 51, nr 1

151--170

EN

Nonlinear differential equations with impulses occurring at random time and acting noninstantaneously on finite intervals are studied. We consider the case when the time where the impulses occur is Gamma distributed. Lyapunov functions are applied to obtain sufficient conditions for the p-moment exponential stability of the trivial solution of the given system.