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 Mat. Zametki, 2007, Volume 82, Issue 5, Pages 665–677 (Mi mz4081)

On the Properties of a Cauchy-Type Problem for an Abstract Differential Equation with Fractional Derivatives

A. V. Glushak

Belgorod State University

Abstract: We study the relationship between the solutions of abstract differential equations with fractional derivatives and their stability with respect to the perturbation by a bounded operator. Besides, we obtain representations for the solution of an inhomogeneous equation and for an equation containing a fractional power of the generator of a cosine operator function.

Keywords: differential equation with fractional derivatives, Cauchy-type problem, Riemann–Liouville fractional derivative, fractional integral, Mittag-Leffler function

DOI: https://doi.org/10.4213/mzm4081

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English version:
Mathematical Notes, 2007, 82:5, 596–607

Bibliographic databases:

UDC: 517.983
Revised: 10.04.2007

Citation: A. V. Glushak, “On the Properties of a Cauchy-Type Problem for an Abstract Differential Equation with Fractional Derivatives”, Mat. Zametki, 82:5 (2007), 665–677; Math. Notes, 82:5 (2007), 596–607

Citation in format AMSBIB
\Bibitem{Glu07} \by A.~V.~Glushak \paper On the Properties of a Cauchy-Type Problem for an Abstract Differential Equation with Fractional Derivatives \jour Mat. Zametki \yr 2007 \vol 82 \issue 5 \pages 665--677 \mathnet{http://mi.mathnet.ru/mz4081} \crossref{https://doi.org/10.4213/mzm4081} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2399947} \zmath{https://zbmath.org/?q=an:1160.34053} \elib{https://elibrary.ru/item.asp?id=12844045} \transl \jour Math. Notes \yr 2007 \vol 82 \issue 5 \pages 596--607 \crossref{https://doi.org/10.1134/S000143460711003X} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000252128700003} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-38349053288} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Furati Kh.M., “A Cauchy-type problem with a sequential fractional derivative in the space of continuous functions”, Bound. Value Probl., 2012 (2012), 58, 14 pp.
2. Furati Kh.M., “Bounds on the solution of a Cauchy-type problem involving a weighted sequential fractional derivative”, Fract. Calc. Appl. Anal., 16:1 (2013), 171–188
3. Furati Kh.M., “A Cauchy-Type Problem Involving a Weighted Sequential Derivative in the Space of Integrable Functions”, Comput. Math. Appl., 66:5 (2013), 883–891
4. Vijayakumar V., Selvakumar A., Murugesu R., “Controllability For a Class of Fractional Neutral Integro-Differential Equations With Unbounded Delay”, Appl. Math. Comput., 232 (2014), 303–312
5. V. E. Fedorov, M. V. Plekhanova, R. R. Nazhimov, “Degenerate linear evolution equations with the Riemann–Liouville fractional derivative”, Siberian Math. J., 59:1 (2018), 136–146
6. V. E. Fedorov, A. S. Avilovich, “A Cauchy type problem for a degenerate equation with the Riemann–Liouville derivative in the sectorial case”, Siberian Math. J., 60:2 (2019), 359–372
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