# American Institute of Mathematical Sciences

March  2022, 15(3): 603-620. doi: 10.3934/dcdss.2021154

## 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 Published  March 2022 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 and Continuous Dynamical Systems - S, 2022, 15 (3) : 603-620. doi: 10.3934/dcdss.2021154
##### References:
 [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. 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. [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. [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. [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. [14] J. Diestel and J. J. Uhl Jr., Vector Measures, Math. Survey 15, Amer. Math. Soc., Providence, RI, 1977. [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. [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.

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##### References:
 [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. 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. [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. [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. [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. [14] J. Diestel and J. J. Uhl Jr., Vector Measures, Math. Survey 15, Amer. Math. Soc., Providence, RI, 1977. [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. [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.
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