Skip to main content
SpringerLink
Log in

BV Continuous Solutions of an Evolution Inclusion with Maximal Monotone Operator and Nonconvex-Valued Perturbation. Existence Theorem

  • Published:
Set-Valued and Variational Analysis Aims and scope Submit manuscript

Abstract

An evolution inclusion with the right-hand side containing a time-dependent maximal monotone operator and a multivalued mapping with closed nonconvex values is studied in a separable Hilbert space. The dependence of the maximal monotone operator on time is described with the help of the distance between maximal monotone operators in the sense of Vladimirov. This distance as a function of time has bounded variation with an upper bound given by a nonatomic positive Radon measure. By a solution of the inclusion one means a continuous function of bounded variation whose differential measure (Stieltjes measure) is absolutely continuous with respect to the positive Radon measure above, and the values of the density of this differential measure with respect to the Radon measure belong to the right-hand side of the inclusion almost everywhere. Under the traditional assumptions on the perturbation (measurability, Lipschitzianity in the phase variable in the Hausdorff metric, linear growth condition), the existence of solutions is proven and some properties of the solution set are established.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Dinculeanu, N.: Vector Measures. Veb Deutscher Verlag der Wissenschaften, Berlin (1966)

    MATH  Google Scholar 

  2. Moreau, J.J.: Sur les mesures differentielles des fonctions vectorielles a variation bornee. Sem. Anal. Convexe Montpellier, expose 17 (1975)

  3. Thibault, L.: Moreau sweeping process with bounded truncated retraction. J. Convex Anal. 23(4), 1051–1098 (2016)

    MathSciNet  MATH  Google Scholar 

  4. Vladimirov, A.A.: Nonstationary dissipative evolution equation in Hilbert space. Nonlinear Anal. 19, 499–518 (1991)

    Article  MathSciNet  Google Scholar 

  5. Kunze, M., Monteiro Marques, M.D.P.: BV solutions to evolution problems with time dependent domains. Set-valued Anal. 5, 57–72 (1994)

    Article  MathSciNet  Google Scholar 

  6. Azzam-Laouir, D., Castaing, C., Monteiro Marques, M.D.P.: Perturbed evolution problem with continuous bounded variation in time and applications. Set-valued Var. Anal. 26(3), 693–728 (2018)

    Article  MathSciNet  Google Scholar 

  7. Himmelberg, C.J.: Measurable relations. Fundamenta Math. 87, 53–72 (1975)

    Article  MathSciNet  Google Scholar 

  8. Mattila, P.: Geometry of sets and measures in Euclidean Spaces. Cambridge Univ. Press, London (1995)

    Book  Google Scholar 

  9. Hou, S.-H.: On property (Q) and other semicontinuity properties of multifunctions. Pacific J. Math. 103(1), 39–56 (1982)

    Article  MathSciNet  Google Scholar 

  10. Schwartz, L.: Analysis, v. 1, Mir. Moscow (1972)

  11. Diestel, J., Uhl, J.J. Jr.: Vector measures. Amer. Math. Soc. (1977)

  12. Monteiro Marques, M.D.P.: Differential Inclusions in Nonsmooths Mechanical Problems. Shocks and Dry Friction. Birkhäuser, Basel (1993)

    Book  Google Scholar 

  13. Moreau, J.J., Valadier, M.: A chain rule involving vector functions of bounded variation. J. Funct. Anal. 74, 333–345 (1987)

    Article  MathSciNet  Google Scholar 

  14. Azzam-Laouir, D., Belhoula, W., Castaing, C., Monteiro Marques, M.D.P.: Perturbed evolution problems with absolutely continuous variation in time and applications. J. Fixed Point. Theory Appl. 21 Art. 40, 32 (2019)

    MathSciNet  MATH  Google Scholar 

  15. Azzam-Laouir, D., Belhoula, W., Castaing, C., Monteiro Marques, M.D.P.: Multi-valued perturbation to evolution problems involving time dependent maximal monotone operators. Evol. Equa. Contr. Theory 9(1), 219–254 (2020)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The author thanks the referees for their valuable remarks which helped to improve the manuscript.

Author information

Authors and Affiliations

  1. Matrosov Institute for System Dynamics and Control Theory of Siberian Branch of Russian Academy of Sciences, Lermontov str., 134, Irkutsk, 664033, Russia

    A. A. Tolstonogov

Authors
  1. A. A. Tolstonogov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. A. Tolstonogov.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tolstonogov, A.A. BV Continuous Solutions of an Evolution Inclusion with Maximal Monotone Operator and Nonconvex-Valued Perturbation. Existence Theorem. Set-Valued Var. Anal 29, 29–60 (2021). https://doi.org/10.1007/s11228-020-00535-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11228-020-00535-3

Keywords

Mathematics Subject Classification (2010)

Search

Navigation