Geodesic
On the Properties of a Cauchy-Type Problem for an Abstract Differential Equation with Fractional Derivatives
Matematičeskie zametki, Tome 82 (2007) no. 5, pp. 665-677.

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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. 
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A. V. Glushak. On the Properties of a Cauchy-Type Problem for an Abstract Differential Equation with Fractional Derivatives. Matematičeskie zametki, Tome 82 (2007) no. 5, pp. 665-677. http://geodesic.mathdoc.fr/item/MZM_2007_82_5_a2/

[1] S. G. Samko, A. A. Kilbas, O. I. Marichev, Integraly i proizvodnye drobnogo poryadka i nekotorye ikh prilozheniya, Nauka i tekhnika, Minsk, 1987 | MR | Zbl

[2] A. V. Glushak, “Zadacha tipa Koshi dlya abstraktnogo differentsialnogo uravneniya s drobnymi proizvodnymi”, Matem. zametki, 77:1 (2005), 28–41 | MR | Zbl

[3] A. V. Glushak, “O zadache tipa Koshi dlya abstraktnogo differentsialnogo uravneniya s drobnoi proizvodnoi”, Vestn. VGU. Ser. fiz., matem., 2001, no. 2, 74–77 | Zbl

[4] A. V. Glushak, “O zadache tipa Koshi dlya neodnorodnogo abstraktnogo differentsialnogo uravneniya s drobnoi proizvodnoi”, Vestn. VGU. Ser. fiz., matem., 2002, no. 1, 121–123 | Zbl

[5] A. V. Glushak, “O svyazi reshenii abstraktnykh differentsialnykh uravnenii, soderzhaschikh drobnye proizvodnye”, Vestn. VGU. Ser. fiz., matem., 2002, no. 2, 61–63

[6] A. V. Glushak, “O periodicheskikh resheniyakh abstraktnykh differentsialnykh uravnenii s drobnoi proizvodnoi”, Vestn. VGU. Ser. fiz., matem., 2003, no. 1, 96–98

[7] V. V. Vasilev, S. G. Krein, S. I. Piskarev, “Polugruppy operatorov, kosinus operator-funktsii i lineinye differentsialnye uravneniya”, Itogi nauki i tekhniki. Matem. analiz., 28, VINITI, M., 1990, 87–202 | MR | Zbl

[8] A. N. Kochubei, “Zadacha Koshi dlya evolyutsionnykh uravnenii drobnogo poryadka”, Differents. uravneniya, 25:8 (1989), 1359–1368 | MR | Zbl

[9] V. A. Kostin, “K zadache Koshi dlya abstraktnykh differentsialnykh uravnenii s drobnymi proizvodnymi”, Dokl. RAN, 326:4 (1992), 597–600 | MR | Zbl

[10] Ph. Clement, G. Gripenberg, S.-O. Loden, “Regularity properties of solution of fractional evolution equation”, Evolution Equations and Their Applicationsin Physical and Life Sciences (Bad Herrenalb, 1998), Lecture Notes in Pure and Appl. Math., 215, Dekker, New York, 2001, 235–246 | MR | Zbl

[11] G. Beitmen, A. Erdeii, Vysshie transtsendentnye funktsii: Ellipticheskie i avtomorfnye funktsii. Funktsii Lame i Mate, Nauka, M., 1967 | MR | Zbl

[12] S. A. Tersenov, Vvedenie v teoriyu uravnenii, vyrozhdayuschikhsya na granitse, Novosibirskii GU, Novosibirsk, 1973 | MR

[13] A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integraly i ryady. Spetsialnye funktsii, Nauka, M., 1983 | MR | Zbl

[14] A. G. Zemanyan, Integralnye preobrazovaniya obobschennykh funktsii, Nauka, M., 1974 | MR | Zbl

[15] G. Beitmen, A. Erdeii, Vysshie transtsendentnye funktsii: Gipergeometricheskaya funktsiya. Funktsii Lezhandra, Nauka, M., 1965 | MR | Zbl

[16] A. G. Sveshnikov, A. N. Tikhonov, Teoriya funktsii kompleksnoi peremennoi, Kurs vysshei matematiki i matematicheskoi fiziki, 4, Nauka, M., 1967 | MR | Zbl

[17] C. C. Travis, G. F. Webb, “Perturbation of strongly continuous cosine family generators”, Colloq. Math., 45:2 (1981), 277–285 | MR | Zbl

[18] Dzh. Goldstein, Polugruppy lineinykh operatorov i ikh prilozheniya, Vyscha shkola, Kiev, 1989 | MR | Zbl

[19] A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integraly i ryady. Elementarnye funktsii, Nauka, M., 1981 | MR | Zbl