# American Institute of Mathematical Sciences

doi: 10.3934/dcdss.2021154
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## Set-valued problems under bounded variation assumptions involving the Hausdorff excess

 Stefan cel Mare University of Suceava, 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), Universitatii 13 - Suceava, Romania

Received  October 2019 Revised  March 2021 Early access December 2021

Fund Project: 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)

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
 $$$\begin{split} &dx(t) \in G(t, x(t)) dg(t),\\ &x(0) = x_0, \end{split}$$\;\;\;\;\;\;(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
 $$$\begin{split} &Y(t)\subset F(t,Y(t)),\\ &Y(0) = Y_0. \end{split}$$\;\;\;\;(2)$
It is shown that a recent parametrized version of Banach's Contraction Theorem given by V.V. Chistyakov follows from our result.
Citation: Bianca Satco. Set-valued problems under bounded variation assumptions involving the Hausdorff excess. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021154
##### References:
 [1] J.-P. Aubin, Impulsive Differential Inclusions and Hybrid Systems: A Viability Approach, Lecture Notes, Univ. Paris, 2002. Google Scholar [2] J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] P. Billingsley, Weak Convergence of Measures: Applications in Probability, CBMS-NSF Regional Conference Series in Applied Mathematics, 1971.  Google Scholar [6] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math. 580, Springer-Verlag, Berlin-New York, 1977.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] M. Cichoń, B. 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.  Google Scholar [11] L. Di Piazza, V. 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.  Google Scholar [12] L. Di Piazza, V. 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.  Google Scholar [13] L. Di Piazza, V. 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.  Google Scholar [14] J. Diestel and J. J. Uhl Jr., Vector Measures, Math. Survey 15, Amer. Math. Soc., Providence, RI, 1977.  Google Scholar [15] M. Federson, J. 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.  Google Scholar [16] D. Fraňková, Regulated functions, Math. Bohem., 116 (1991), 20-59.  doi: 10.21136/MB.1991.126195.  Google Scholar [17] D. Fraňková, Regulated functions with values in Banach space, Math. Bohem., 144 (2019), 437-456.  doi: 10.21136/MB.2019.0124-19.  Google Scholar [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.  Google Scholar [19] J. Henrikson, Completeness and Total Boundedness of the Hausdorff Metric, http://citeseerx.ist.psu.edu. Google Scholar [20] J. Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter, Czechoslovak Math. J., 7 (1957), 418-449.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [24] G. A. Monteiro, A. Slavik and M. Tvrdy, Kurzweil-Stieltjes Integral and Its Applications, World Scientific, 2018. Google Scholar [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.  Google Scholar [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.  Google Scholar [27] B. Satco, Nonlinear Volterra integral equations in Henstock integrability setting, Electr. J. Diff. Equ., 39 (2008), 9pp.  Google Scholar [28] Š. Schwabik, Generalized Ordinary Differential Equations, World Scientific, 1992. doi: 10.1142/1875.  Google Scholar [29] Š. Schwabik and G. Ye, Topics in Banach Space Integration, World Scientific, 2005. doi: 10.1142/5905.  Google Scholar [30] R. Serfozo, Convergence of Lebesgue integrals with varying measures, Sankhyā Ser., 44 (1982), 380-402.   Google Scholar [31] A. N. Sesekin and S. T. Zavalishchin, Dynamic Impulse Systems, Dordrecht, Kluwer Academic, 1997. doi: 10.1007/978-94-015-8893-5.  Google Scholar [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.  Google Scholar [33] A. J. Ward, The Perron–Stieltjes integral, Math. Z., 41 (1936), 578-604.  doi: 10.1007/BF01180442.  Google Scholar

show all references

##### References:
 [1] J.-P. Aubin, Impulsive Differential Inclusions and Hybrid Systems: A Viability Approach, Lecture Notes, Univ. Paris, 2002. Google Scholar [2] J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] P. Billingsley, Weak Convergence of Measures: Applications in Probability, CBMS-NSF Regional Conference Series in Applied Mathematics, 1971.  Google Scholar [6] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math. 580, Springer-Verlag, Berlin-New York, 1977.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] M. Cichoń, B. 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.  Google Scholar [11] L. Di Piazza, V. 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.  Google Scholar [12] L. Di Piazza, V. 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.  Google Scholar [13] L. Di Piazza, V. 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.  Google Scholar [14] J. Diestel and J. J. Uhl Jr., Vector Measures, Math. Survey 15, Amer. Math. Soc., Providence, RI, 1977.  Google Scholar [15] M. Federson, J. 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.  Google Scholar [16] D. Fraňková, Regulated functions, Math. Bohem., 116 (1991), 20-59.  doi: 10.21136/MB.1991.126195.  Google Scholar [17] D. Fraňková, Regulated functions with values in Banach space, Math. Bohem., 144 (2019), 437-456.  doi: 10.21136/MB.2019.0124-19.  Google Scholar [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.  Google Scholar [19] J. Henrikson, Completeness and Total Boundedness of the Hausdorff Metric, http://citeseerx.ist.psu.edu. Google Scholar [20] J. Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter, Czechoslovak Math. J., 7 (1957), 418-449.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [24] G. A. Monteiro, A. Slavik and M. Tvrdy, Kurzweil-Stieltjes Integral and Its Applications, World Scientific, 2018. Google Scholar [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.  Google Scholar [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.  Google Scholar [27] B. Satco, Nonlinear Volterra integral equations in Henstock integrability setting, Electr. J. Diff. Equ., 39 (2008), 9pp.  Google Scholar [28] Š. Schwabik, Generalized Ordinary Differential Equations, World Scientific, 1992. doi: 10.1142/1875.  Google Scholar [29] Š. Schwabik and G. Ye, Topics in Banach Space Integration, World Scientific, 2005. doi: 10.1142/5905.  Google Scholar [30] R. Serfozo, Convergence of Lebesgue integrals with varying measures, Sankhyā Ser., 44 (1982), 380-402.   Google Scholar [31] A. N. Sesekin and S. T. Zavalishchin, Dynamic Impulse Systems, Dordrecht, Kluwer Academic, 1997. doi: 10.1007/978-94-015-8893-5.  Google Scholar [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.  Google Scholar [33] A. J. Ward, The Perron–Stieltjes integral, Math. Z., 41 (1936), 578-604.  doi: 10.1007/BF01180442.  Google Scholar
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