Geodesic
Quasilinear Equations with a Sectorial Set of Operators at Gerasimov–Caputo Derivatives
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 29 (2023) no. 2, pp. 248-259 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $A_{\alpha,G}^{n}$ 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, initial–boundary value problem.
Mots-clés : nonlocal solution 
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V. E. Fedorov; K. V. Boyko. Quasilinear Equations with a Sectorial Set of Operators at Gerasimov–Caputo Derivatives. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 29 (2023) no. 2, pp. 248-259. http://geodesic.mathdoc.fr/item/TIMM_2023_29_2_a17/

[1] Uchaikin V.V., Metod drobnykh proizvodnykh, Artishok, Ulyanovsk, 2008, 512 pp.

[2] Tarasov V.E., Fractional dynamics: Applications of fractional calculus to dynamics of particles, fields and media, Springer, NY, 2011, 505 pp. | MR

[3] Samko S.G., Kilbas A.A., Marichev O.I., Fractional integrals and derivatives. Theory and applications, Gordon and Breach Science Publ., Philadelphia, 1993, 976 pp. | MR | Zbl

[4] Prüss J., Evolutionary integral equations and applications, Springer, Basel, 1993, 366 pp. | MR

[5] Podlubny I., Fractional differential equations, Acad. Press, San Diego; Boston, 1999, 340 pp. | MR | Zbl

[6] Pskhu A.V., Uravneniya v chastnykh proizvodnykh drobnogo poryadka, Nauka, M., 2005, 199 pp.

[7] Kilbas A.A., Srivastava H.M., Trujillo J.J., Theory and applications of fractional differential equations, Elsevier Science Publ., Amsterdam, 2006, 540 pp. | MR | Zbl

[8] Kostić M., Abstract Volterra integro-differential equations, CRC Press, Boca Raton, 2015, 484 pp. | MR | Zbl

[9] Fedorov V.E., Boiko K.V., Fuong T.D., “Nachalnye zadachi dlya nekotorykh klassov lineinykh evolyutsionnykh uravnenii s neskolkimi drobnymi proizvodnymi”, Mat. zametki SVFU, 28:3 (2021), 85–104 | DOI

[10] Boyko K.V., Fedorov V.E., “The Cauchy problem for a class of multi-term equations with Gerasimov — Caputo derivatives”, Lobachevskii J. Math., 43:6 (2022), 1293–1302 | DOI | MR

[11] Boiko K.V., Fedorov V.E., “Obratnaya zadacha dlya odnogo klassa vyrozhdennykh evolyutsionnykh uravnenii s neskolkimi proizvodnymi Gerasimova — Kaputo”, Itogi nauki i tekhniki. Ser. Sovrem. matematika i ee prilozheniya. Temat. obzory, 213 (2022), 38–46 | DOI

[12] Fedorov V.E., Turov M.M., “Defekt zadachi tipa Koshi dlya lineinykh uravnenii s neskolkimi proizvodnymi Rimana — Liuvillya ”, Sib. mat. zhurn., 62:5 (2021), 1143–1162 | DOI

[13] Turov M.M., “Kvazilineinye uravneniya s neskolkimi proizvodnymi Rimana — Liuvillya proizvolnykh poryadkov”, Chelyab. fiz.-mat. zhurn., 7:4 (2022), 434–446 | DOI | MR

[14] Gerasimov A.N., “Obobschenie lineinykh zakonov deformatsii i ikh prilozhenie k zadacham vnutrennego treniya”, Priklad. matematika i mekhanika, 12:3 (1948), 251–260 | Zbl

[15] Caputo M., “Linear model of dissipation whose $Q$ is almost frequancy independent. II”, Geophysical Journal of the Royal Astronomical Society, 13:5 (1967), 529–539 | DOI

[16] Novozhenova O.G., “Life and science of Alexey Gerasimov, one of the pioneers of fractional calculus in Soviet Union”, Frac. Calcul. Appl. Anal., 20:3 (2017), 790–809 | DOI | MR | Zbl

[17] Bajlekova E.G., Fractional evolution equations in banach spaces, PhD thesis, Eindhoven University of Technology, Eindhoven, 2001, 107 pp. | MR | Zbl

[18] Kato T., Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972, 740 pp. | MR

[19] Plekhanova M.V., Baybulatova G.D., “Semilinear equations in Banach spaces with lower fractional derivatives”, Springer Proc. Math. Stat., 292 (2019), 81–93 | DOI | MR | Zbl

[20] Khessard B., Kazarinov N., Ven I., Teoriya i prilozheniya bifurkatsii rozhdeniya tsikla, Mir, M., 1985, 280 pp.