The issues of existence and uniqueness of a solution to the Cauchy problem are studied for a linear equation in a Banach space with a closed operator at the unknown function that is resolved with respect to a first-order integro-differential operator of the Gerasimov type. The properties of resolving families of operators of the homogeneous equations are investigated. It is shown that sectoriality, i.e., belonging to the class of operators $\mathcal A_K$ introduced here, is a necessary and sufficient condition for the existence of an analytical resolving family of operators in a sector. A theorem on the perturbation of operators of the class $\mathcal A_K$ is obtained, and two versions of the theorem on the existence and uniqueness of a solution to a linear inhomogeneous equation are proved. Abstract results are used to study initial—boundary value problems for an equation with the Prabhakar time derivative and for a system of partial differential equations with Gerasimov—Caputo time derivatives of different orders.
Keywords: integro-differential equation, Gerasimov–Caputo derivative, Cauchy problem, sectorial operator, resolving family of operators, initial–boundary value problem
Received March 11, 2024
Revised March 14, 2024
Accepted March 18, 2024
Funding Agency: This work was supported by the Russian Science Foundation (project no. 24-21-20015, https://rscf.ru/project/24-21-20015/) and by the Government of the Chelyabinsk region.
Vladimir Evgenyevich Fedorov, Dr. Phys.-Math. Sci., Prof., Chelyabinsk State University, Chelyabinsk, 454001 Russia, e-mail: kar@csu.ru
Aleksandra Danilovna Godova, doctoral student, Chelyabinsk State University, Chelyabinsk, 454001 Russia, e-mail: sasha.godova97@mail.ru
REFERENCES
1. Samko S.G., Kilbas A.A., Marichev O.I. Fractional integrals and derivatives. Theory and applications, Yverdon, Gordon and Breach Publ., 1993, 976 p. ISBN-13: 9782881248641. Original Russian text 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, 688 p.
2. Nakhushev A.M. Uravneniya matematicheskoy biologii [Equations of mathematical biology]. Moscow, Vys. Shk. Publ., 1995, 301 p. ISBN: 5-06-002670-1 .
3. Nakhushev A.M. Drobnoye ischisleniye i yego primeneniye [Fractional calculus and its applications]. Moscow, Fizmatlit Publ., 2003, 272 p. ISBN: 5-9221-0440-3 .
4. Pskhu A.V. Uravneniya v chastnykh proizvodnykh drobnogo poryadka [Partial differential equations of fractional order]. Moscow, Nauka Publ., 2005, 199 p. ISBN: 5-02-033721-8 .
5. Kilbas A.A., Srivastava H.M., Trujillo J.J. Theory and applications of fractional differential equations. Amsterdam, Elsevier Science Publ., 2006, 540 p. ISBN-13: 978-0444518323 .
6. Uchaykin V.V. Metod drobnykh proizvodnykh [Method of fractional derivatives]. Ul’yanovsk, Artichoke Publ., 2008, 512 p. ISBN: 978-5-904198-01-5 .
7. Tarasov V.E. Fractional dynamics: applications of fractional calculus to dynamics of particles, fields and media. NY, Springer Publ., 2011, 505 p. ISBN: 978-3-642-14003-7 .
8. Dа Prato G., Iannelli M. Linear integro-differential equations in Banach spaces. Rendiconti del seminario matematico della università di Padova, 1980, vol. 62, pp. 207–219.
9. Prüss J. Evolutionary integral equations and applications. Basel, Springer, 1993, 366 p. doi: 10.1007/978-3-0348-0499-8
10. Kostić M. Abstract Volterra integro-differential equations. Boca Raton, CRC Press., 2015, 484 p. doi: 10.1201/b18463
11. Caputo M., Fabrizio M. A new definition of fractional derivative without singular kernel. Progress in fractional differentiation and applications, 2015, vol. 1, no. 2, pp. 73–85. doi: 10.12785/pfda/010201
12. Atangana A., Baleanu D. New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model. Thermal Science, 2016, vol. 20, pp. 763–769. doi: 10.2298/TSCI160111018A
13. Fedorov V.E., Godova A.D., Kien B.T. Integro-differential equations with bounded operators in Banach spaces. Bull. Karaganda Univ. Math. Ser., 2022, no. 2 (106), pp. 93–107. doi: 10.31489/2022M2/93-107
14. Fedorov V.E., Godova A.D. Integro-differential equations in Banach spaces and analytic resolving families of operators. Sovrem. Matematika. Fundament. Napravleniya, 2023, vol. 69, no. 1, pp. 166–184 (in Russian). doi: 10.22363/2413-3639-2023-69-1-166-184
15. Arendt W., Batty C.J.K., Hieber M., Neubrander F. Vector-valued Laplace transforms and Cauchy problems. Basel, Springer Publ., 2011, 539 p. doi: 10.1007/978-3-0348-0087-7
16. Pazy A. Semigroups and linear operators and applications to partial differential equations. NY, Springer Publ., 1983, 279 p. doi: 10.1007/978-1-4612-5561-1
17. Bajlekova E.G. Fractional evolution equations in Banach spaces. PhD thesis, Eindhoven, Eindhoven University of Technology, Netherlands, 2001, 107 p. doi: 10.6100/IR549476
18. Fedorov V.E., Filin N.V. On strongly continuous resolving families of operators for fractional distributed order equations. Fractal and Fractional, 2021, vol. 5, no. 1, art. no. 20, 14 p. doi: 10.3390/fractalfract5010020
19. Sitnik S.M., Fedorov V.E., Filin N.V., Polunin V.A. On the solvability of equations with a distributed fractional derivative given by the Stieltjes integral. Mathematics, 2022, vol. 10, no. 16, art. no. 2979, 20 p. doi: 10.3390/math10162979
20. Fedorov V.E., Plekhanova M.V., Izhberdeeva E.M. Analytic resolving families for equations with the Dzhrbashyan–Nersesyan fractional derivative. Fractal and Fractional, 2022, vol. 6, no. 10, art. no. 541, 16 p. doi: 10.3390/fractalfract6100541
21. Boyko K.V. Linear and quasilinear equations with several derivatives Gerasimov–Caputo. Chelyab. Fiz.-Mat. Zhurn., 2024, vol. 9, iss. 1, pp. 5–22 (in Russian). doi: 10.47475/2500-0101-2024-9-1-5-22
22. Kato T. Perturbation theory for linear operators. Berlin, Heidelberg, Springer-Verlag, 1966, 623 p. doi: 10.1007/978-3-642-66282-9. Translated to Russian under the title Teoriya vozmushcheniy lineynykh operatorov, Moscow, Mir Publ., 1972, 740 p.
23. Prabhakar T.R. A singular integral equation with a generalized Mittag–Leffler function in the kernel. Yokohama Math. J., 1971, vol. 19, pp. 7–15.
24. Triebel H. Interpolation theory. Function spaces. Differential operators. Amsterdam: North-Holland Publ. co., 1978, 528 p. Translated to Russian under the title Teoriya interpolyatsii. Funktsional’nyye prostranstva. Differentsial’nyye operatory, Moscow, Mir Publ., 1980, 664 p.
doi: 10.1002/zamm.19790591227
Cite this article as: V.E. Fedorov, A.D. Godova. Integro-differential equations of Gerasimov type with sectorial operators. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2024, vol. 30, no. 2, pp. 243–258. Proceedings of the Steklov Institute of Mathematics, 2024, Vol. 325, Suppl. 1, pp. S99-S113.
