V.E. Fedorov, K.V. Boyko. Quasilinear equations with a sectorial set of operators at Gerasimov–Caputo derivatives ... P. 248-259

The issues of unique solvability of the Cauchy problem are studied for a quasilinear equation solved with respect to the highest fractional Gerasimov–Caputo derivative in a Banach space with closed operators from the class Aα,Gn in the linear part and with a nonlinear operator continuous in the graph norm. A theorem on the local existence and uniqueness of a solution to the Cauchy problem is proved in the case of a locally Lipschitz nonlinear operator. Under the nonlocal Lipschitz condition for the nonlinear operator, the existence of a unique solution on a predetermined interval is shown. Abstract results are illustrated by examples of initial–boundary value problems for partial differential equations with Gerasimov–Caputo time derivatives.

Keywords: Gerasimov–Caputo fractional derivative, Cauchy problem, sectorial set of operators, resolving family of operators, quasilinear equation, local solution, nonlocal solution, initial–boundary value problem

Received February 28, 2023

Revised March 15, 2023

Accepted March 20, 2023

Funding Agency: This work was supported by the RF President’s Grant for State Support of Leading Scientific Schools (project no. 2708.2022.1.1).

Vladimir Evgenyevich Fedorov, Dr. Phys.-Math. Sci., Prof., Chelyabinsk State University, Chelya-binsk, 454001 Russia, e-mail: kar@csu.ru

Kseniya Vladimirovna Boyko, doctoral student, Chelyabinsk State University, Chelyabinsk, 454001 Russia, e-mail: kvboyko@mail.ru

REFERENCES

1.   Uchaikin V.V. Metod drobnykh proizvodnykh [Method of fractional derivatives], Ulyanovsk, ArteShock Publ., 2008, 512 p. ISBN: 978-5-904198-01-5.

2.   Tarasov V.E. Fractional dynamics: Applications of fractional calculus to dynamics of particles, fields and media, NY, Springer, 2011, 505 p. ISBN 978-3-642-14003-7.

3.   Samko S.G., Kilbas A.A., Marichev O.I. Fractional integrals and derivatives. Theory and applications. Philadelphia, Gordon and Breach Science Publ., 1993, 976 p. ISBN: 9782881248641. Original Russian text was published in Samko S.G., Kilbas A.A., Marichev O.I. Integraly i proizvodnye drobnogo poryadka i nekotorye ikh prilozheniya, Minsk, Nauka i Tekhnika Publ., 1987, 638 p.

4.   Prüss J. Evolutionary integral equations and applications, Basel, Springer, 1993, 366 p. doi: 10.1007/978-3-0348-8570-6

5.   Podlubny I. Fractional differential equations, San Diego, Boston, Academic Press, 1999, 340 p. ISBN: 9780080531984.

6.   Pskhu A.V. Uravneniya v chastnykh proizvodnykh drobnogo poryadka [Equations in partial derivatives of fractional order], Moscow, Nauka Publ., 2005, 199 p. ISBN: 5-02-033721-8.

7.   Kilbas A.A., Srivastava H.M., Trujillo J.J. Theory and applications of fractional differential equations, Amsterdam, Elsevier Science Publ., 2006, 540 p. ISBN: 978-0-444-51832-3.

8.   Kostić M. Abstract Volterra integro-differential equations, Boca Raton, CRC Press, 2015, 484 p. doi: 10.1201/b18463

9.   Fedorov V.E., Boyko K.V., Fuong T.D. Initial problems for some classes of linear evolutionary equations with several fractional derivatives. Mathematical Notes of NEFU, 2021, vol. 28, no. 3, pp. 85–104 (in Russian). doi: 10.25587/SVFU.2021.75.46.006

10.   Boyko K.V., Fedorov V.E. The Cauchy problem for a class of multi-term equations with Gerasimov — Caputo derivatives. Lobachevskii J. Math., 2022, vol. 43, no. 6, pp. 1293–1302. doi: 10.1134/S1995080222090049

11.   Boyko K.V., Fedorov V.E. Inverse problem for a class of degenerate evolutionary equations with several Gerasimov–Caputo derivatives. Itogi Nauki i Tekhniki. Ser. Sovremennaya Matematika i ee Prilozheniya. Tematicheskie Obzory, 2022, vol. 213, pp. 38–66 (in Russian). doi: 10.36535/0233-6723-2022-213-38-46

12.   Fedorov V.E., Turov M.M. The defect of a Cauchy type problem for linear equations with several Riemann–Liouville derivatives. Sib. Math. J., 2021, vol. 62, no. 5, pp. 925–942. doi: 10.1134/S0037446621050141

13.   Turov M.M. Quasilinear equations with several Riemann–Liouville derivatives of arbitrary orders. Chelyabinskiy Fiz.-Mаt. Zhurn.., 2022, vol. 7, no. 4, pp. 434–446 (in Russian). doi: 10.47475/2500-0101-2022-17404

14.   Gerasimov A.N. Generalization of linear laws of deformation and its applications to problems of inner friction. Prikladnaya Matematika i Mekhanika, 1948, vol. 12, no. 3, pp. 251–260 (in Russian).

15.   Caputo M. Linear model of dissipation whose Q is almost frequency independent II. Geophysical J. International, 1967, vol. 13, no. 5, pp. 529–539. doi: 10.1111/j.1365-246X.1967.tb02303.x

16.   Novozhenova O.G. Life and science of Alexey Gerasimov, one of the pioneers of fractional calculus in Soviet Union. Frac. Calcul. Appl. Anal., 2017, vol. 20, no. 3, pp. 790–809. doi: 10.1515/fca-2017-0040

17.   Bajlekova E.G. Fractional Evolution Equations in Banach Spaces: PhD thesis. Eindhoven, Eindhoven University of Technology, 2001, 107 p.

18.   Kato T. Perturbation Theory for Linear Operators. Berlin, Heidelberg, Springer-Verlag, 1966. doi: 10.1007/978-3-642-66282-9 . Translated to Russian under the title Teoriya vozmushcheniya lineinykh operatorov, Moscow, Mir Publ., 1972, 740 p.

19.   Plekhanova M.V., Baybulatova G.D. Semilinear equations in Banach spaces with lower fractional derivatives. In: Springer Proc. Math. Stat., 2019, pp. 81–93. doi: 10.1007/978-3-030-26987-6_6

20.   Hassard B.D., Kazarinoff N.D., Wan Y.-H. Theory and applications of hopf bifurcation. Cambridge, Cambridge Univ. Press, 1981, 311 p. doi: 10.1002/zamm.19820621221 . Translated to Russian under the title Teoriya i prilozheniya bifurkatsii rozhdeniya tsikla, Moscow, Mir Publ., 1985, 280 p.

Cite this article as: V.E. Fedorov, K.V. Boyko. Quasilinear equations with a sectorial set of operators at Gerasimov–Caputo derivatives. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2023, vol. 29, no. 2, pp. 248–259; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2023, Vol. 321, Suppl. 1, pp. S78–S89.