In this paper, we study the topological structure of a solution set to the Cauchy problem for semilinear differential inclusions of fractional order $\alpha(1, 2)$ in Banach spaces. It is assumed that the linear part of the inclusions is a linear closed operator generating a strongly continuous and uniformly bounded family of cosine operator functions. The nonlinear part is represented by a upper semicontinuous multivalued operator of Caratheodory type. It is established that the set of solutions to the problem is an $R_\delta$-set.The paper has the following structure. After the introduction, we present the necessary preliminary information from the theories of multivalued mappings, measures of noncompactness, fractional analysis, and the family of cosine operator functions. The third section is devoted to auxiliary results. In the next section, a number of lemmas and the main result of the paper (Theorem 2) are proved. In the final section, as an example of applying the obtained results, a generalized periodic problem for semilinear differential inclusions of fractional order $\alpha\in (1,2)$ is considered.
Keywords: topological structure, $R_\delta$-set, differential inclusion, fractional derivative, family of cosine operator functions, multivalued map, condensing multioperator
Received August 12, 2025
Revised September 7, 2025
Accepted September 15, 2025
Funding Agency: This work was supported by Russian Science Foundation, project № 25-11-00056, https://rscf.ru/project/25-11-00056/.
Garik Gagikovich Petrosyan, Cand. Sci. (Phys.-Math.), Leading Researcher, Research Institute of Mathematics, Voronezh State University, Voronezh, 394018 Russia, e-mail: garikpetrosyan@yandex.ru
REFERENCES
1. 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.
2. Podlubny I. Fractional differential equations. San Diego, Acad. Press, 1999, 340 p. ISBN-10: 0125588402 .
3. Mainardi F., Rionero S., Ruggeri T. Probability distributions generated by fractional diffusion equations, 2007, 46 p. https://doi.org/10.48550/arXiv.0704.0320
4. Obukhovskii V.V., Petrosyan G.G., Soroka M.S. On an initial value problem for nonconvex-valued fractional differential inclusions in a Banach space. Math. Notes, 2024, vol. 115, no. 3–4, pp. 358–370. https://doi.org/10.1134/S0001434624030088
5. Petrosyan G.G. On a boundary value problem for a class of fractional Langevin type differential equations in a Banach space. Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2022, vol. 32, iss. 3, pp. 415–432 (in Russian). https://doi.org/10.35634/vm220305
6. Benedetti I., Obukhovskii V., Taddei V. On generalized boundary value problems for a class of fractional differential inclusions. Fract. Calc. Appl. Anal., 2017, vol. 20, pp. 1424–1446. https://doi.org/10.1515/fca-2017-0075
7. Gomoyunov M.I. To the theory of differential inclusions with Caputo fractional derivatives. Diff. Equat., 2020, vol. 56, no. 11, pp. 1387–1401. https://doi.org/10.1134/S00122661200110014
8. Gomoyunov M.I., Lukoyanov N.Yu. Optimal feedback in a linear–quadratic optimal control problem for a fractional-order system. Diff. Equat., 2023, vol. 59, no. 8, pp. 1117–1129. https://doi.org/10.1134/S0012266123080104
9. Gomoyunov M.I., Lukoyanov N.Yu. Construction of solutions to control problems for fractional-order linear systems based on approximation models. Proc. Steklov Inst. Math., 2021, vol. 313, no. 1, pp. S73–S82. https://doi.org/10.1007/s13235-018-0281-7
10. Ke T.D., Obukhovskii V., Wong N.C., Yao J.C. On a class of fractional order differential inclusions with infinite delays. Applicable Anal., 2013, vol. 92, no. 1, pp. 115–137. https://doi.org/10.1080/00036811.2011.601454
11. Kamenskii M.I., Obukhoskii V.V., Petrosyan G.G., Yao J.C. Boundary value problems for semilinear differential inclusions of fractional order in a Banach space. Applicable Anal., 2017, vol. 97, no. 4, pp. 571–591. https://doi.org/10.1080/00036811.2016.1277583
12. Aronszajn N. Le correspondant topologique de l’unicité dans la théorie des équations différentielles. Annals Math., 1942, vol. 43, no. 4, pp. 730–738. https://doi.org/10.2307/1968963
13. Górniewicz L. On the solution sets of differential inclusions. J. Math. Anal. Appl., 1986, vol. 113, no. 1, pp. 235–244. https://doi.org/10.1016/0022-247X(86)90347-1
14. Filippov V.V. On a theorem of Aronszajn. Differ. Equ., 1997, vol. 33, no. 1, pp. 75–79 (in Russian).
15. Górniewicz L. Topological fixed point theory of multivalued mappings, 2nd edt. Ser. Topological Fixed Point Theory and Its Applications. Dordrecht, Springer, 2006, 538 p. https://doi.org/10.1007/1-4020-4666-9
16. Djebali S., Górniewicz L., Ouahab A. Solution sets for differential equations and inclusions. De Gruyter Ser. in Nonlinear Analysis and Applications, 18. Berlin, Walter de Gruyter, 2012, 453 p. https://doi.org/10.1515/9783110293562
17. Kamenskii M., Obukhovskii V., Zecca P. Condensing multivalued maps and semilinear differential inclusions in Banach spaces. Berlin, New York, Walter de Gruyter, 2001, 231 p. https://doi.org/10.1515/9783110870893
18. Petrosyan G.G. On a topological structure of a solution set to a Cauchy problem for fractional differential inclusions with a upper semicontinuous right-hand side. Dokl. RAN. Matem., Inform., Prots. Upr., 2025, vol. 522, pp. 33–39 (in Russian).
19. Kamenskii M.I., Obukhoskii V.V., Petrosyan G.G. A continuous dependence of a solution set for fractional differential inclusions of an order $q \in(1,2)$ on parameters and initial data. Lobachevskii J. Math., 2023, vol. 44, no. 8, pp. 3331–3342. https://doi.org/10.1134/S1995080223080243
20. Sova M. Cosine operator functions. Ser. Rozprawy Mat., vol. 49. Warszawa, Państwowe Wydawnictwo Naukowe, 1966, 47 p.
21. Borisovich Yu.G., Gel’man B.D., Myshkis A.D., Obukhovskii V.V. Vvedenie v teoriyu mnogoznachnykh otobrazhenii i differentsial’nykh vklyuchenii [Introduction to the theory of multivalued maps and differential inclusions], Moscow, Librokom, 2011, 224 p. ISBN: 978-5-397-01526-4 .
22. Hyman D.H. On decreasing sequences of compact absolute retracts. Fund. Math., 1969, vol. 64, no. 1, pp. 91–97.
23. Zhou Y., He J.W. New results on controllability of fractional evolution systems with order $\alpha \in (1,2)$. Evol. Eq. Control Theory, 2021, vol. 10, no. 3, pp. 491–509. https://doi.org/10.3934/eect.2020077
24. Petrosyan G.G. On feedback control systems governed by fractional differential inclusions. Diff. Equat., 2024, vol. 60, no. 11, pp. 1572–1591. https://doi.org/10.1134/S0012266124110077
25. He J.W., Liang Y., Ahmad B., Zhou Y. Nonlocal fractional evolution inclusions of order $\alpha \in (1,2)$. Mathematics, 2019, vol. 7, no. 2, pp. 1–17. https://doi.org/10.3390/math7020209
26. Gorenflo R., Kilbas A.A., Mainardi F., Rogosin S.V. Mittag—Leffler functions, related topics and applications. Berlin, Heidelberg, Springer-Verlag, 2014, 443 p. ISBN: 9783662439296 .
Cite this article as: G.G. Petrosyan. On the $R_{\delta}$-structure of the solution set of the Cauchy problem for semilinear differential inclusions of fractional order $\alpha\in (1,2)$ in Banach spaces. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2025, vol. 31, no. 4, pp. 260–280.
