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On the properties of a Cauchy-type problem for an abstract differential equation with fractional derivatives. (English. Russian original) Zbl 1160.34053
Math. Notes 82, No. 5, 596-607 (2007); translation from Mat. Zametki 82, No. 5, 665-677 (2007).
The author studies Cauchy-type problems for fractional differential equations of Riemann-Liouville type of the form $D^\alpha D^\beta u(t) = Au(t).$ His particular interest is in the stability of solutions under perturbations of $$A$$, the right hand side of the equation. He considers perturbations which take the form of a bounded operator $$P$$. The paper also considers the effect of an inhomogeneous forcing term $$f$$ on the right hand side and of a fractional power cosine generator for $$A$$. Proofs of well-posedness of the problem and expressions for the solution are given.

##### MSC:
 34G10 Linear differential equations in abstract spaces 34A05 Explicit solutions, first integrals of ordinary differential equations 26A33 Fractional derivatives and integrals 47D09 Operator sine and cosine functions and higher-order Cauchy problems 47N20 Applications of operator theory to differential and integral equations 35R20 Operator partial differential equations (= PDEs on finite-dimensional spaces for abstract space valued functions)
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##### References:
 [1] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Integrals and Derivatives of Fractional Order and Some of Their Applications (Nauka i Tekhnika, Minsk, 1987) [in Russian]. · Zbl 0617.26004 [2] A. V. Glushak, ”Cauchy-type problem for an abstract differential equation with fractional derivatives,” Mat. Zametki 77(1), 28–41 (2005) [Math. Notes 77 (1–2), 26–38 (2005)]. [J] 2001, No. 2, 74–77 (2001). ISSN 1609-0705 [3] A. V. Glushak, ”The problem of Cauchy-type for an abstract differential equation with fractional derivative,” Vestn. Voronezh. Gos. Univ., Ser. Fiz. Mat., No. 2, 74–77 (2001). · Zbl 1061.34041 [4] A. V. Glushak, ”On a problem of Cauchy-type for an inhomogeneous abstract differential equation with fractional derivative,” Vestn. Voronezh. Gos. Univ., Ser. Fiz. Mat., No. 1, 121–123 (2002). · Zbl 1088.34531 [5] A. V. Glushak, ”On the relationship between the solutions of abstract differential equations containing fractional derivatives” Vestn. Voronezh. Gos. Univ., Ser. Fiz. Mat., No. 2, 61–63 (2002). [6] A. V. Glushak, ”On periodic solutions of abstract differential equations with fractional derivatives,” Vestn. Voronezh. Gos. Univ., Ser. Fiz. Mat., No. 1, 96–98 (2003). · Zbl 1418.34121 [7] V. V. Vasil’ev, S. G. Krein, and S. I. Piskarev, ”Semigroups of operators, cosine operator functions, and linear differential equations,” in Itogi Nauki Tekh., Ser. Mat. Anal. (VINITI, Moscow, 1990), Vol. 28, pp. 87–202 [J. Sov. Math. 54 (4), 1042–1129 (1991)]. [8] A. N. Kochubei, ”A Cauchy problem for evolution equations of fractional order,” Differentsial’nye Uraveniya 25(8), 1359–1368 (1989) [Differential Equations 25 (8), 967–974 (1989)]. [9] V. A. Kostin, ”The Cauchy problem for an abstract differential equation with fractional derivatives,” Dokl. Ross. Akad. Nauk 326(4), 597–600 (1992) [Dokl. Math. 46 (2), 316–319 (1992)]. [10] Ph. Clement, G. Gripenberg, and S.-O. Loden, ”Regularity properties of solution of fractional evolution equation,” in Evolution Equations and Their Applications in Physical and Life Sciences, Bad Herrenalb, 1998, Lecture Notes in Pure and Appl. Math. (Dekker, New York, 2001), Vol. 215, pp. 235–246. [11] H. Bateman and A. Erdélyi, Higher Transcendental Functions, Vol. 3: Elliptic and Modular Functions, Lamé and Mathieu Functions (McGraw-Hill, New York-Toronto-London, 1955; Nauka, Moscow, 1967). [12] S. A. Tersenov, Introduction to the Theory of Equations Degenerate on the Boundary (Novosibirsk Gos. Univ., Novosibirsk, 1973) [in Russian]. · Zbl 0289.35044 [13] A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series. Special Functions (Nauka, Moscow, 1983) [in Russian]. · Zbl 0626.00033 [14] A. G. Zemanyan, Integral Transformations of Generalized Functions (Nauka, Moscow, 1974) [in Russian]. [15] H. Bateman and A. Erdélyi, Higher Transcendental Functions, Vol. 1: The Hypergeometric Function, Legendre Functions (McGraw-Hill, New York-Toronto-London, 1953; Nauka, Moscow, 1965). [16] A. G. Sveshnikov and A. N. Tihonov, Theory of Functions of a Complex Variable, in Course in Higher Mathematics and Mathematical Physics (Nauka, Moscow, 1967), Vol. 4 [in Russian]. [17] C. C. Travis and G. F. Webb, ”Perturbation of strongly continuous cosine family generators,” Colloq. Math. 45(2), 277–285 (1981). · Zbl 0496.47039 [18] J. Goldstein, Semigroups of Linear Operators and Applications (Oxford University Press, New York, 1985; Vyshcha Shkola, Kiev, 1989). · Zbl 0592.47034 [19] A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series. Elementary Functions (Nauka, Moscow, 1981) [in Russian]. · Zbl 0511.00044
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