Skip to main content
SpringerLink
Log in

Viable solutions of lower semicontinuous quantum stochastic differential inclusions

  • Published:
Analysis and Mathematical Physics Aims and scope Submit manuscript

Abstract

We establish the existence and some properties of viable solutions of lower semicontinuous quantum stochastic differential inclusions within the framework of the Hudson–Parthasarathy formulations of quantum stochastic calculus. The main results here are accomplished by establishing a major auxiliary selection result. The results here extend the classical Nagumo viability theorems,valid on finite dimensional Euclidean spaces, to the present infinite dimensional locally convex space of noncommutative stochastic processes.

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. Aubin, J.: Viability Theory. Birkhauser, Basel (1991)

    MATH  Google Scholar 

  2. Aubin, J., Bayen, A., Pierre, P.: Viability Theory New Directions. Springer, Berlin (2011)

    Book  Google Scholar 

  3. Aubin, J.P., Cellina, A.: Differential Inclusions; Set-Valued Maps and Viability Theory, pp. 139–171. Springer, Berlin (1984)

  4. Ayoola, E.O.: Lipschitzian quantum stochastic differential equations and the associated Kurzweil equations. Stoch. Anal. Appl. 19(4), 581–603 (2001)

    Article  MathSciNet  Google Scholar 

  5. Ayoola, E.O.: On convergence of one-step schemes for weak solutions of quantum stochastic differential equations. Acta Appl. Math. 67(1), 19–58 (2001)

    Article  MathSciNet  Google Scholar 

  6. Ayoola, E.O.: Continuous selections of solution sets of Lipschitzian quantum stochastic differential inclusions. Int. J. Theor. Phys. 43(10), 2041–2059 (2004)

    Article  MathSciNet  Google Scholar 

  7. Ayoola, E.O.: Topological properties of solution sets of Lipschitzian quantum stochastic differential inclusions. Acta Appl. Math. 100(1), 15–37 (2008)

    Article  MathSciNet  Google Scholar 

  8. Ayoola, E.O.: Error estimates for discretized quantum stochastic differential inclusions. Stoch. Anal. Appl. 21(6), 1215–1230 (2003)

    Article  MathSciNet  Google Scholar 

  9. Ayoola, E.O.: Exponential formula for the reachable sets of quantum stochastic differential inclusions. Stoch. Anal. Appl. 21(3), 515–543 (2003)

    Article  MathSciNet  Google Scholar 

  10. Carja, O., Necula, M., Vrabie, I.I.: Viability, Invariance and Applications, vol. 207. Elsevier, Amsterdam (2007)

    Book  Google Scholar 

  11. Cârja, O., Vrabie, I.I.: Differential equations on closed sets. In: Canada, A., Drabek, P., Fonda, A. (eds.) Handbook of Differential Equations, vol. 2, pp. 147–238. Elsevier, North Holland (2005)

    MATH  Google Scholar 

  12. Ekhaguere, G.: The functional Ito formula in quantum stochastic calculus. J. Math. Phys. 31(12), 2921–2929 (1990)

    Article  MathSciNet  Google Scholar 

  13. Ekhaguere, G.: Lipschitzian quantum stochastic differential inclusions. Int. J. Theor. Phys. 31(11), 2003–2027 (1992)

    Article  MathSciNet  Google Scholar 

  14. Ekhaguere, G.: Quantum stochastic differential inclusions of hypermaximal monotone type. Int. J. Theor. Phys. 34(3), 323–353 (1995)

    Article  MathSciNet  Google Scholar 

  15. Ekhaguere, G.: Quantum stochastic evolutions. Int. J. Theor. Phys. 35(9), 1909–1946 (1996)

    Article  MathSciNet  Google Scholar 

  16. Ekhaguere, G.: Topological solutions of noncommutative stochastic differential equations. Stoch. Anal. Appl. 25(5), 961–993 (2007)

    Article  MathSciNet  Google Scholar 

  17. Hudson, R.L., Parthasarathy, K.R.: Quantum Ito’s formula and stochastic evolutions. Commun. Math. Phys. 93(3), 301–323 (1984)

    Article  MathSciNet  Google Scholar 

  18. Marques, M.D.M., et al.: Viability results for nonautonomous differential inclusions. J. Convex Anal. 7(2), 437–443 (2000)

    MathSciNet  MATH  Google Scholar 

  19. Ogundiran, M.O., Ayoola, E.O.: Upper semicontinuous quantum stochastic differential inclusions via Kakutani-fan fixed point theorem. Dyn. Syst. Appl. 21, 121–132 (2012)

    MathSciNet  MATH  Google Scholar 

  20. Ogundiran, M.O., Ayoola, E.O.: Lower semicontinuous quantum stochastic differential inclusions. Eur. J. Math. Sci. 2(1), 1–16 (2013)

    Google Scholar 

  21. Smirnov, G.V.: Introduction to the Theory of Differential Inclusions, vol. 41. American Mathematical Society, Providence (2002)

    MATH  Google Scholar 

  22. Vrabie, I.I.: A nagumo type viability theorem. An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (NS) 51, 293–308 (2005)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Elizade University, Ilara-Mokin, Ondo-State, Nigeria

    Titilayo O. Akinwumi

  2. Department of Mathematics, University of Ibadan, Ibadan, Oyo-State, Nigeria

    Ezekiel O. Ayoola

Authors
  1. Titilayo O. Akinwumi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ezekiel O. Ayoola
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Titilayo O. Akinwumi.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interest.

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

Akinwumi, T.O., Ayoola, E.O. Viable solutions of lower semicontinuous quantum stochastic differential inclusions. Anal.Math.Phys. 11, 7 (2021). https://doi.org/10.1007/s13324-020-00446-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13324-020-00446-4

Keywords

Mathematics Subject Classification

Search

Navigation