On the existence of a solution for a periodic boundary value problem for semilinear fractional-order differential inclusions in Banach spaces

Cover Page

Cite item

Full Text

Abstract

In this paper, we study a periodic boundary value problem for a class of semilinear differential inclusions of fractional order in a Banach space for which the multivalued nonlinearity satisfies the regularity condition expressed in terms of measures of noncompactness. To prove the existence of solutions to the problem, we first construct the corresponding Green function. Then we introduce into consideration a multivalued resolving operator in the space of continuous functions and reduce the posed problem to the existence of fixed points of the resolving multioperator. To prove the existence of a fixed point, a generalized theorem of B.N. Sadovskii type for a condensing multivalued map is used.

About the authors

Mikhail I. Kamenskii

Voronezh State University

Email: mikhailkamenski@mail.ru
Doctor of Physics and Mathematics, Head of the Functional Analysis and Operator Equations Department 1 Universitetskaya Sq., Voronezh 394018, Russian Federation

Valeri V. Obukhovskii

Voronezh State Pedagogical University

Email: valerio-ob2000@mail.ru
Doctor of Physics and Mathematics, Head of the Higher Mathematics Department 86 Lenin Str., Voronezh 394043, Russian Federation

Garik G. Petrosyan

Voronezh State Pedagogical University

Email: garikpetrosyan@yandex.ru
Candidate of Physics and Mathematics, Docent of the Higher Mathematics Department 86 Lenin Str., Voronezh 394043, Russian Federation

References

  1. S.G. Samco, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publ., Amsterdam, 1993.
  2. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., North-Holland Mathematics Studies, Amsterdam, 2006.
  3. F. Mainardi, S. Rionero, T. Ruggeri, “On the initial value problem for the fractional diffusionwave equation”, Waves and Stability in Continuous Media, 1994, 246-251.
  4. M. Afanasova, Y. Ch. Liou, V. Obukhoskii, G. Petrosyan, “On controllability for a system governed by a fractional-order semilinear functional differential inclusion in a Banach space”, Journal of Nonlinear and Convex Analysis, 20:9 (2019), 1919-1935.
  5. J. Appell, B. Lopez, K. Sadarangani, “Existence and uniqueness of solutions for a nonlinear fractional initial value problem involving Caputo derivatives”, J. Nonlinear Var. Anal., 2018, №2, 25-33.
  6. T.D. Ke, N.V. Loi, V. Obukhovskii, “Decay solutions for a class of fractional differential variational inequalities”, Fract. Calc. Appl. Anal., 2015, №18, 531-553.
  7. М.С. Афанасова, Г.Г. Петросян, “О краевой задаче для функционально-дифференциального включения дробного порядка с обобщенным начальным условием в банаховом пространстве”, Известия вузов. Математика, 2019, №9, 3-15.
  8. I. Benedetti, V. Obukhovskii, V. Taddei, “On generalized boundary value problems for a class of fractional differential inclusions”, Fract. Calc. Appl. Anal., 2017, №20, 1424-1446.
  9. M. Kamenskii, V. Obukhovskii, G. Petrosyan, J. C. Yao, “Boundary value problems for semilinear differential inclusions of fractional order in a Banach space”, Applicable Analysis, 97:4 (2018), 571-591.
  10. M. Kamenskii, V. Obukhovskii, G. Petrosyan, J. C. Yao, “On a Periodic Boundary Value Problem for a Fractional-Order Semilinear Functional Differential Inclusions in a Banach Space”, Mathematics, 7:12, Special Issue “Fixed Point, Optimization, and Applications” (2019), 5-19.
  11. Г.Г. Петросян, “Об антипериодической краевой задаче для полулинейного дифференциального включения дробного порядка с отклоняющимся аргументом в банаховом пространстве”, Уфимский математический журнал, 12:3 (2020), 71-82.
  12. R. Agarwal, B. Ahmad, “Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions”, Comput. Math. Appl., 2011, №62, 1200-1214.
  13. M. Kamenskii, V. Obukhovskii, G. Petrosyan, J. C. Yao, “Existence and Approximation of Solutions to Nonlocal Boundary Value Problems for Fractional Differential Inclusions”, Fixed Point Theory and Applications, 2019, №2, 1-21.
  14. M. Kamenskii, V. Obukhovskii, G. Petrosyan, J.C. Yao, “On approximate solutions for a class of semilinear fractional-order differential equations in Banach spaces”, Fixed Point Theory and Applications, 28:4 (2017), 1-28.
  15. M. Belmekki, J.J. Nieto, R. Rodriguez-Lopez, “Existence of periodic solution for a nonlinear fractional differential equation”, Boundary Value Problems, 2009 (2009), 1-18, Article ID 324561.
  16. M. Belmekki, J.J. Nieto, R. Rodriguez-Lopez, “Existence of solution to a periodic boundary value problem for a nonlinear impulsive fractional differential equation”, Electronic Journal of Qualitative Theory of Differential Equations, 16 (2014), 1-27.
  17. R. Gorenflo, A.A. Kilbas, F. Mainardi, S.V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer-Verlag, Berlin; Heidelberg, 2014.
  18. V.M. Bogdan, Generalized Vectorial Lebesgue and Bochner Integration Theory, 2010, arXiv:1006.3881v1.
  19. Г.М. Фихтенгольц, Курс дифференциального и интегрального исчисления. Т. 1, Физматлит, М., 2006.
  20. M.I. Kamenskii, V.V. Obukhovskii, P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, Walter de Gruyter, Berlin; New-York, 2001.
  21. V.V. Obukhovskii, B. Gelman, Multivalued Maps and Differential Inclusions. Elements of Theory and Applications, World Scientific, Singapore, 2020.
  22. J. Diestel, W. M. Ruess, W. Schachermayer, “On weak compactness in ”, Proc. Amer. Math. Soc., 1993, №118, 447-453.

Supplementary files

Supplementary Files
Action
1. JATS XML
Download


Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

Согласие на обработку персональных данных

 

Используя сайт https://journals.rcsi.science, я (далее – «Пользователь» или «Субъект персональных данных») даю согласие на обработку персональных данных на этом сайте (текст Согласия) и на обработку персональных данных с помощью сервиса «Яндекс.Метрика» (текст Согласия).