On perturbations of abstract fractional differential equations by nonlinear operators
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We prove the unique solvability of a Cauchy-type problem for an abstract parabolic equation containing fractional derivatives and a nonlinear perturbation term. The result is applied to establish the solvability of the inverse coefficient problem for a fractional-order equation.
- W. Arendt, C. Batty, M. Hieber, and F. Neubrander, Laplace Transforms and Cauchy Problems, Birkhauser-Verlag, Basel–Boston–Berlin (2001).
- M. M. Dzhrbashyan, Integral Transformations and the Presentation of Functions in a Complex Domain [in Russian], Nauka, Moscow (1966).
- M. M. El-Borai, “Some probability densities and fundamental solutions of fractional evolution equations,” Chaos, Solitons and Fractals, 14, 433–440 (2002). CrossRef
- A. V. Glushak, “On the Cauchy-type problem for abstract fractional-order differential equations,” Vestnik Voronezh. Univ. Ser. Fiz. Mat., No. 2, 74–77 (2001).
- A. V. Glushak and H. K. Avad, “On perturbations of abstract fractional differential equations,” Dokl. Adyg (Cherkes) Int. Akad. Nauk, 10, No. 1, 25–31 (2008).
- A. V. Glushak and Yu.V. Povalyaeva, “On properties of solutions of Cauchy-type problems for abstract fractional differential equations,” Spectral and Evolution Problems, 14, 163–172 (2004).
- K. Iosida, Functional Analysis [Russian translation], Mir, Moscow (1967).
- T. Kato, Perturbation Theory for Linear Operators [Russian translation], Mir, Moscow (1972).
- A. A. Kilbas, H.M. Srivastava, and J. J. Trujillo, Theory and Application of Fractional Differential Equations, Elsevier Science B.V., Amsterdam (2006).
- V. A. Kostin, “The Cauchy problem for an abstract differential equation with fractional derivatives,” Russian Acad. Sci. Dokl. Math., 46, No. 2, 316–319 (1992).
- M. A. Krasnosel’skiĭ, P. P. Zabreĭko, E. I. Pustyl’nik, and P.E. Sobolevskĭ, Integral Operators in Spaces of Summable Functions [in Russian], Nauka, Moscow (1966).
- A. M. Nakhushev, Equations of Mathematical Biology [in Russian], Vysshaya Shkola, Moscow (1995).
- A. M. Nakhushev, Fractional Calculus and Applications [in Russian], Fizmatlit, Moscow (2003).
- A. I. Prilepko, D. G. Orlovsky, and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker, New York–Basel (2000).
- A.P. Prudnikov, Yu.A. Brychkov, and O. I. Marichev, Integrals and Series. Elementary Functions [in Russian], Nauka, Moscow (1981).
- A. V. Pskhu, Boundary-Value Problems for Fractional-Order and Continual-Order Partial Differential Equations [in Russian], Kabardino-Balkar Scientific Center, Nal’chik (2005).
- A. A. Samarskii and P.N. Vabishchevich, Numerical Methods for Solving Inverse Problems of Mathematical Physics, Walter de Gruyter, Berlin (2007).
- S. G. Samko, A. A. Kilbas, and O. I. Marichev, Integrals and Derivatives of Fractional Order and Some of Their Applications [in Russian], Nauka i Tekhnika, Minsk (1987).
- On perturbations of abstract fractional differential equations by nonlinear operators
Journal of Mathematical Sciences
Volume 170, Issue 3 , pp 306-323
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