Advanced Search
Article Contents
Article Contents

Set-valued problems under bounded variation assumptions involving the Hausdorff excess

This research has been supported by "Excellence in Advanced Research, Leadership in Innovation and Patenting for University and Regional Development" - EXCALIBUR, Grant Contract no. 18PFE / 10.16.2018 Institutional Development Project - Funding for Excellence in RDI, Program 1 - Development of the National R & D System, Subprogram 1.2 - Institutional Performance, National Plan for Research and Development and Innovation for the period 2015-2020 (PNCDI III)

Abstract Full Text(HTML) Related Papers Cited by
  • In the very general framework of a (possibly infinite dimensional) Banach space $ X $, we are concerned with the existence of bounded variation solutions for measure differential inclusions

    $ \begin{equation} \begin{split} &dx(t) \in G(t, x(t)) dg(t),\\ &x(0) = x_0, \end{split} \end{equation}\;\;\;\;\;\;(1) $

    where $ dg $ is the Stieltjes measure generated by a nondecreasing left-continuous function.

    This class of differential problems covers a wide variety of problems occuring when studying the behaviour of dynamical systems, such as: differential and difference inclusions, dynamic inclusions on time scales and impulsive differential problems. The connection between the solution set associated to a given measure $ dg $ and the solution sets associated to some sequence of measures $ dg_n $ strongly convergent to $ dg $ is also investigated.

    The multifunction $ G : [0,1] \times X \to \mathcal{P}(X) $ with compact values is assumed to satisfy excess bounded variation conditions, which are less restrictive comparing to bounded variation with respect to the Hausdorff-Pompeiu metric, thus the presented theory generalizes already known existence and continuous dependence results. The generalization is two-fold, since this is the first study in the setting of infinite dimensional spaces.

    Next, by using a set-valued selection principle under excess bounded variation hypotheses, we obtain solutions for a functional inclusion

    $ \begin{equation} \begin{split} &Y(t)\subset F(t,Y(t)),\\ &Y(0) = Y_0. \end{split} \end{equation}\;\;\;\;(2) $

    It is shown that a recent parametrized version of Banach's Contraction Theorem given by V.V. Chistyakov follows from our result.

    Mathematics Subject Classification: 34A06, 34A12, 26A45, 54C65.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] J.-P. Aubin, Impulsive Differential Inclusions and Hybrid Systems: A Viability Approach, Lecture Notes, Univ. Paris, 2002.
    [2] J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990.
    [3] V. Barbu and T. Precupanu, Convexity and Optimization in Banach Spaces, 4$^{th}$ edition, Springer Monographs in Mathematics. Springer, Dordrecht, 2012. doi: 10.1007/978-94-007-2247-7.
    [4] S. A. Belov and V. V. Chistyakov, A selection principle for mappings of bounded variation, J. Math. Anal. Appl., 249 (2005), 351-366.  doi: 10.1006/jmaa.2000.6844.
    [5] P. Billingsley, Weak Convergence of Measures: Applications in Probability, CBMS-NSF Regional Conference Series in Applied Mathematics, 1971.
    [6] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math. 580, Springer-Verlag, Berlin-New York, 1977.
    [7] V. V. Chistyakov, Asymmetric variations of multifunctions with application to functional inclusions, J. Math. Anal. Appl., 478 (2019), 421-444.  doi: 10.1016/j.jmaa.2019.05.035.
    [8] V. V. Chistyakov and D. Repovš, Selections of bounded variation under the excess restrictions, J. Math. Anal. Appl., 331 (2007), 873-885.  doi: 10.1016/j.jmaa.2006.09.004.
    [9] M. Cichoń and B. Satco, Measure differential inclusions - between continuous and discrete, Adv. Diff. Equations, 2014 (2014), 1-18.  doi: 10.1186/1687-1847-2014-56.
    [10] M. CichońB. Satco and A. Sikorska-Nowak, Impulsive nonlocal differential equations through differential equations on time scales, Appl. Math. Comput., 218 (2011), 2449-2458.  doi: 10.1016/j.amc.2011.07.057.
    [11] L. Di PiazzaV. Marraffa and B. Satco, Closure properties for integral problems driven by regulated functions via convergence results, J. Math. Anal. Appl., 466 (2018), 690-710.  doi: 10.1016/j.jmaa.2018.06.012.
    [12] L. Di PiazzaV. Marraffa and B. Satco, Approximating the solutions of differential inclusions driven by measures, Ann. Mat. Pura Appl., 198 (2019), 2123-2140.  doi: 10.1007/s10231-019-00857-6.
    [13] L. Di PiazzaV. Marraffa and B. Satco, Measure differential inclusions: Existence results and minimum problems, Set-Valued Var. Anal., 29 (2021), 361-382.  doi: 10.1007/s11228-020-00559-9.
    [14] J. Diestel and J. J. Uhl Jr., Vector Measures, Math. Survey 15, Amer. Math. Soc., Providence, RI, 1977.
    [15] M. FedersonJ. G. Mesquita and A. Slavík, Measure functional differential equations and functional dynamic equations on time scales, J. Differential Equations, 252 (2012), 3816-3847.  doi: 10.1016/j.jde.2011.11.005.
    [16] D. Fraňková, Regulated functions, Math. Bohem., 116 (1991), 20-59.  doi: 10.21136/MB.1991.126195.
    [17] D. Fraňková, Regulated functions with values in Banach space, Math. Bohem., 144 (2019), 437-456.  doi: 10.21136/MB.2019.0124-19.
    [18] M. Frigon and R. López Pouso, Theory and applications of first-order systems of Stieltjes differential equations, Adv. Nonlinear Anal., 6 (2017), 13-36.  doi: 10.1515/anona-2015-0158.
    [19] J. Henrikson, Completeness and Total Boundedness of the Hausdorff Metric, http://citeseerx.ist.psu.edu.
    [20] J. Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter, Czechoslovak Math. J., 7 (1957), 418-449. 
    [21] B. Miller and E. Y. Rubinovitch, Impulsive Control in Continuous and Discrete-Continuous Systems, Kluwer Academic Publishers, Dordrecht, 2003. doi: 10.1007/978-1-4615-0095-7.
    [22] G. A. Monteiro and B. Satco, Distributional, differential and integral problems: Equivalence and existence results, Electron. J. Qual. Theory Differ. Equ., 2017 (2017), 1-26.  doi: 10.14232/ejqtde.2017.1.7.
    [23] G. A. Monteiro and B. Satco, Extremal solutions for measure differential inclusions via Stieltjes derivatives, Adv. Difference Equ., 2019 (2019), 1-18.  doi: 10.1186/s13662-019-2172-7.
    [24] G. A. Monteiro, A. Slavik and M. Tvrdy, Kurzweil-Stieltjes Integral and Its Applications, World Scientific, 2018.
    [25] R. López Pouso and A. Rodriguez, A new unification of continuous, discrete, and impulsive calculus through Stieltjes derivatives, Real Anal. Exch., 40 (2014/15), 319-353.  doi: 10.14321/realanalexch.40.2.0319.
    [26] B. Satco, Continuous dependence results for set-valued measure differential problems, Electron. J. Qual. Theory Differ. Equ., 2015 (2015), 1-15.  doi: 10.14232/ejqtde.2015.1.79.
    [27] B. Satco, Nonlinear Volterra integral equations in Henstock integrability setting, Electr. J. Diff. Equ., 39 (2008), 9pp.
    [28] Š. Schwabik, Generalized Ordinary Differential Equations, World Scientific, 1992. doi: 10.1142/1875.
    [29] Š. Schwabik and G. Ye, Topics in Banach Space Integration, World Scientific, 2005. doi: 10.1142/5905.
    [30] R. Serfozo, Convergence of Lebesgue integrals with varying measures, Sankhyā Ser., 44 (1982), 380-402. 
    [31] A. N. Sesekin and S. T. Zavalishchin, Dynamic Impulse Systems, Dordrecht, Kluwer Academic, 1997. doi: 10.1007/978-94-015-8893-5.
    [32] G. N. Silv and R. B. Vinter, Measure driven differential inclusions, J. Math. Anal. Appl., 202 (1996), 727-746.  doi: 10.1006/jmaa.1996.0344.
    [33] A. J. Ward, The Perron–Stieltjes integral, Math. Z., 41 (1936), 578-604.  doi: 10.1007/BF01180442.
  • 加载中

Article Metrics

HTML views(1070) PDF downloads(163) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint