Advertisement

Approximating the solutions of differential inclusions driven by measures

  • L. Di Piazza
  • V. MarraffaEmail author
  • B. Satco
Article
  • 9 Downloads

Abstract

The matter of approximating the solutions of a differential problem driven by a rough measure by solutions of similar problems driven by “smoother” measures is considered under very general assumptions on the multifunction on the right-hand side. The key tool in our investigation is the notion of uniformly bounded \(\varepsilon \)-variations, which mixes the supremum norm with the uniformly bounded variation condition. Several examples to motivate the generality of our outcomes are included.

Keywords

Differential inclusions BV functions \(\varepsilon \)-Variations Regulated functions 

Mathematics Subject Classification

Primary 26A45 Secondary 34A60 28B20 34A12 26A42 

Notes

References

  1. 1.
    Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990)zbMATHGoogle Scholar
  2. 2.
    Aye, K.K., Lee, P.Y.: The dual of the space of functions of bounded variations. Math. Bohem. 131, 1–9 (2006)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Belov, S.A., Chistyakov, V.V.: A selection principle for mappings of bounded variation. J. Math. Anal. Appl. 249, 351–366 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Billingsley, P.: Weak conference of measures: application in probability. In: Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No 5, Society for Industrial and Applied Mathematics. Pa, Philadelphia (1971)Google Scholar
  5. 5.
    Brokate, M., Krejci, P.: Duality in the space of regulated functions and the play operator. Math. Z. 245, 667–688 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cao, Y., Sun, J.: On existence of nonlinear measure driven equations involving non-absolutely convergent integrals. Nonlinear Anal. Hybrid Syst. 20, 72–81 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions. Lecture Notes in Math. 580. Springer, Berlin (1977)Google Scholar
  8. 8.
    Cichoń, M., Satco, B.: Measure differential inclusions–between continuous and discrete. Adv. Differ. Equ. 56, 18 (2014)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Cichoń, M., Satco, B.: On the properties of solutions set for measure driven differential inclusions. In: Discrete and Continuous Dunamical Systems. Special Issue: SI, pp. 287–296 (2015)Google Scholar
  10. 10.
    Cichoń, M., Satco, B., Sikorska-Nowak, A.: Impulsive nonlocal differential equations through differential equations on time scales. Appl. Math. Comput. 218, 2449–2458 (2011)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Cichoń, M., Cichoń, K., Satco, B.: Measure differential inclusions through selection principles in the space of regulated functions. Mediterr. J. Math. 15(4), 148 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Di Piazza, L., Marraffa, V., Satco, B.: Closure properties for integral problems driven by regulated functions via convergence results. J. Math. Anal. Appl. 466, 690–710 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Federson, M., Mesquita, J.G., Slavík, A.: Measure functional differential equations and functional dynamic equations on time scales. J. Differ. Equ. 252, 3816–3847 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Fraňková, D.: Regulated functions. Math. Bohem. 116, 20–59 (1991)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Golubov, B.I.: On functions of bounded \(p\)-variation. Math. USSR Izv. 2, 799–819 (1968)CrossRefzbMATHGoogle Scholar
  16. 16.
    Halas, Z., Tvrdý, M.: Continuous dependence of solutions of generalized linear differential equations on a parameter. Funct. Differ. Equ. 16, 299–313 (2009)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Hönig, C.S.: Volterra Stieltjes—Integral Equations. North-Holland, Amsterdam (1975)zbMATHGoogle Scholar
  18. 18.
    Hu, S., Papageorgiou, N.S.: Handbook of Multivalued Analysis. Kluwer Academic Publisher, Dordrecht (1997)CrossRefzbMATHGoogle Scholar
  19. 19.
    Krejci, P.: Hysteresis in singularly perturbed problems. In: Mortell, M., O’Malley, R., Pokrovskii, A., Sobolev, V. (eds.) Singular Perturbations and Hysteresis, pp. 73–100. SIAM, Philadelphia (2005)CrossRefGoogle Scholar
  20. 20.
    Krejci, P., Laurencot, P.: Generalized variational inequalities. J. Convex Anal. 9, 159–183 (2002)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Kurzweil, J.: Generalized ordinary differential equations and continuous dependence on a parameter. Czechoslov. Math. J. 7, 418–449 (1957)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Miller, B., Rubinovitch, E.Y.: Impulsive Control in Continuous and Discrete-Continuous Systems. Kluwer Academic Publishers, Dordrecht (2003)CrossRefGoogle Scholar
  23. 23.
    Monteiro, G.A., Tvrdý, M.: Generalized linear differential equations in a Banach space: continuous dependence on a paramete. Discrete Contin. Dyn. Syst. 33, 283–303 (2013)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Monteiro, G.A., Slavik, A., Tvrdý, M.: Kurzweil–Stieltjes Integral: Theory and Applications, Series in Real Analysis, vol. 15. World-Scientific, Singapore (2018)zbMATHGoogle Scholar
  25. 25.
    Satco, B.: Continuous dependence results for set-valued measure differential problems. Electron. J. Qual. Theory Differ. Equ. 79, 1–15 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Schwabik, Š.: Generalized Ordinary Differential Equations. World Scientific, Singapore (1992)CrossRefzbMATHGoogle Scholar
  27. 27.
    Schwabik, Š., Tvrdý, M., Vejvoda, O.: Differential and Integral Equations. Boundary Problems and Adjoints. Praha, Dordrecht (1979)zbMATHGoogle Scholar
  28. 28.
    Serfozo, R.: Convergence of Lebesgue integrals with varying measures. Sankhya Ser. A 44(3), 380–402 (1982)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Sesekin, A.N., Zavalishchin, S.T.: Dynamic Impulse Systems. Kluwer Academic, Dordrecht (1997)zbMATHGoogle Scholar
  30. 30.
    Silva, G.N., Vinter, R.B.: Measure driven differential inclusions. J. Math. Anal. Appl. 202, 727–746 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Tvrdý, M.: Differential and integral equations in the space of regulated functions. Membr. Differ. Equ. Math. Phys. 25, 1–104 (2002)MathSciNetzbMATHGoogle Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of PalermoPalermoItaly
  2. 2.Faculty of Electrical Engineering and Computer Science, Integrated Center for Research, Development and Innovation in Advanced Materials, Nanotechnologies, and Distributed Systems for Fabrication and Control (MANSiD)Stefan cel Mare University of SuceavaSuceavaRomania

Personalised recommendations