On approach to the study of the periodic problem for random differential equations

S. V. Kornev ; P. S. Korneva ; N. E. Iakusheva

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 5 (2023), pp. 82-88.

Résumé

Geometric and topological methods of analysis applied to problems of nonlinear oscillations of dynamical systems go back to the names of A. Poincare, L. Brauer, P.S. Alexandrov, G. Hopf, J. Leray, J. Schauder. Later, these methods were developed and demonstrated their effectiveness in the works of many mathematicians. Note, in particular, an extremely fruitful direction associated with the notion of a guiding function, whose base was laid by M.A. Krasnosel’skii and A.I. Perov. In this paper, to study the periodic problem of random differential equations, we use a modification of the classical notion of a guiding function - a random nonsmooth multivalent guiding function. A significant advantage over the classical approach is the ability to "localize" $ $ the verification of the main condition of "directionality" $ $ on a domain that depends on the guiding function itself and on the domain not of the whole space, but of its subspace of lower dimension. In classical works on the method of guiding functions, as a rule, it is assumed that these functions are smooth over the whole phase space. This condition may seem restrictive, for example, in situations where the guiding potentials are different in different domains of the space. To remove this restriction, the paper considers nonsmooth direction potentials and their generalized gradients.

Détail

Keywords: random nonsmooth multivalent guiding function, random differential equation, random periodic solution.

Bibliographie

[1] Krasnoselskii M.A., Perov A.I., “Ob odnom printsipe suschestvovaniya ogranichennykh, periodicheskikh i pochti-periodicheskikh reshenii u sistem obyknovennykh differentsialnykh uravnenii”, Dokl. AN SSSR, 123:2 (1958), 235–238 | Zbl

[2] Krasnoselskii M.A., Operator sdviga po traektoriyam differentsialnykh uravnenii, Nauka, M., 1966 | MR

[3] Zvyagin V.G., Kornev S.V., “Method of guiding functions for existence problems for periodic solutions of differential equations”, J. Math. Sci., 233:4 (2018), 578–601 | DOI | MR | Zbl

[4] Zvyagin V.G., Kornev S.V., Metod napravlyayuschikh funktsii i ego modifikatsii, URSS, M., 2018

[5] Mawhin J., “Periodic solutions of nonlinear functional differential equations”, J. Diff. Equat., 10 (1971), 240–261 | DOI | MR | Zbl

[6] Mawhin J., Ward James R. Jr., “Guiding-like functions for periodic or bounded solutions of ordinary differential equations”, Discrete and continuous dynamical systems, 8:1 (2002), 39–54 | DOI | MR | Zbl

[7] Mawhin J., Thompson H.B., “Periodic or bounded solutions of Caratheodory systems of ordinary differential equations”, J. Dynam. Diff. Equat., 15:2–3 (2003), 327–334 | DOI | MR | Zbl

[8] Fonda A., “Guiding functions and periodic solutions to functional differential equations”, Proc. Amer. Math. Soc., 99:1 (1987), 79–85 | DOI | MR | Zbl

[9] Rachinskii D.I., “Multivalent guiding functions in forced oscillation problems”, Nonlinear Anal., 26:3 (1996), 631–639 | DOI | MR | Zbl

[10] Borisovich Yu.G., Gelman B.D., Myshkis A.D., Obukhovskii V.V., Vvedenie v teoriyu mnogoznachnykh otobrazhenii i differentsialnykh vklyuchenii, Librokom, M., 2011 | MR

[11] Gorniewicz L., Topological Fixed Point Theory of Multivalued Mappings, Kluwer Academic Publishers, Dordrecht-Boston-London, 1999 | MR | Zbl

[12] Kornev S.V., “O metode mnogolistnykh napravlyayuschikh funktsii v zadache o periodicheskikh resheniyakh differentsialnykh vklyuchenii”, Avtomat. i telemekhan., 2003, no. 3, 72–83 | Zbl

[13] Kornev S.V., Obukhovskii V.V., “O negladkikh mnogolistnykh napravlyayuschikh funktsiyakh”, Differents. uravneniya, 39:11 (2003), 1497–1502 | MR | Zbl

[14] Kornev S.V., Obukhovskii V.V., “Negladkie napravlyayuschie potentsialy v zadachakh o vynuzhdennykh kolebaniyakh”, Avtomat. i telemekhan., 2007, no. 1, 3–10 | Zbl

[15] Kornev S.V., “Mnogolistnye napravlyayuschie funktsii v zadache o suschestvovanii periodicheskikh reshenii differentsialnykh vklyuchenii s nevypukloi pravoi chastyu”, Izv. vuzov. Matem., 2016, no. 11, 14–26 | Zbl

[16] Andres J., Górniewicz L., “Random topological degree and random differential inclusions”, Topolog. Methods Nonlinear Anal., 40:2 (2012), 337–358 | MR | Zbl

[17] Kornev S., Obukhovskii V., Zecca P., “On multivalent guiding functions method in the periodic problem for random differential equations”, J. Dynam. Diff. Equat., 31:2 (2019), 1017–1028 | DOI | MR | Zbl

[18] Hu S., Papageorgiou N., Handbook of multivalued analysis: Theory, Kluwer, Dordrecht-Boston-London, 1997 | MR | Zbl

[19] Klark F., Optimizatsiya i negladkii analiz, Nauka, M., 1988 | MR

[20] Obukhovskii V.V., Kornev S.V., Getmanova E.N., “Ob operatore sdviga po traektoriyam reshenii sluchainykh differentsialnykh vklyuchenii”, Vestn. Voronezhsk. gos. un-ta. Ser. Fiz. Matem., 2021, no. 4, 81–89 | DOI