-
Previous Article
Necessary optimality conditions of a reaction-diffusion SIR model with ABC fractional derivatives
- DCDS-S Home
- This Issue
-
Next Article
An optimal control problem applied to a wastewater treatment plant
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 |
$ X $ |
$ \begin{equation} \begin{split} &dx(t) \in G(t, x(t)) dg(t),\\ &x(0) = x_0, \end{split} \end{equation}\;\;\;\;\;\;(1) $ |
$ dg $ |
$ dg $ |
$ dg_n $ |
$ dg $ |
$ G : [0,1] \times X \to \mathcal{P}(X) $ |
$ \begin{equation} \begin{split} &Y(t)\subset F(t,Y(t)),\\ &Y(0) = Y_0. \end{split} \end{equation}\;\;\;\;(2) $ |
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. |
show all references
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. |
[1] |
Francesca Faraci, Antonio Iannizzotto. Three nonzero periodic solutions for a differential inclusion. Discrete and Continuous Dynamical Systems - S, 2012, 5 (4) : 779-788. doi: 10.3934/dcdss.2012.5.779 |
[2] |
Yongjian Liu, Zhenhai Liu, Dumitru Motreanu. Differential inclusion problems with convolution and discontinuous nonlinearities. Evolution Equations and Control Theory, 2020, 9 (4) : 1057-1071. doi: 10.3934/eect.2020056 |
[3] |
Piotr Kowalski. The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2569-2580. doi: 10.3934/dcdsb.2014.19.2569 |
[4] |
T. Caraballo, J. A. Langa, J. Valero. Structure of the pullback attractor for a non-autonomous scalar differential inclusion. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 979-994. doi: 10.3934/dcdss.2016037 |
[5] |
Ziqing Yuana, Jianshe Yu. Existence and multiplicity of nontrivial solutions of biharmonic equations via differential inclusion. Communications on Pure and Applied Analysis, 2020, 19 (1) : 391-405. doi: 10.3934/cpaa.2020020 |
[6] |
Zengyun Wang, Jinde Cao, Zuowei Cai, Lihong Huang. Finite-time stability of impulsive differential inclusion: Applications to discontinuous impulsive neural networks. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2677-2692. doi: 10.3934/dcdsb.2020200 |
[7] |
Clara Carlota, António Ornelas. The DuBois-Reymond differential inclusion for autonomous optimal control problems with pointwise-constrained derivatives. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 467-484. doi: 10.3934/dcds.2011.29.467 |
[8] |
Antonia Chinnì, Roberto Livrea. Multiple solutions for a Neumann-type differential inclusion problem involving the $p(\cdot)$-Laplacian. Discrete and Continuous Dynamical Systems - S, 2012, 5 (4) : 753-764. doi: 10.3934/dcdss.2012.5.753 |
[9] |
Mingqi Xiang, Binlin Zhang, Die Hu. Kirchhoff-type differential inclusion problems involving the fractional Laplacian and strong damping. Electronic Research Archive, 2020, 28 (2) : 651-669. doi: 10.3934/era.2020034 |
[10] |
Dina Kalinichenko, Volker Reitmann, Sergey Skopinov. Asymptotic behavior of solutions to a coupled system of Maxwell's equations and a controlled differential inclusion. Conference Publications, 2013, 2013 (special) : 407-414. doi: 10.3934/proc.2013.2013.407 |
[11] |
Yang Xu, Zheng-Hai Huang. Pareto eigenvalue inclusion intervals for tensors. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022035 |
[12] |
Avadhesh Kumar, Ankit Kumar, Ramesh Kumar Vats, Parveen Kumar. Approximate controllability of neutral delay integro-differential inclusion of order $ \alpha\in (1, 2) $ with non-instantaneous impulses. Evolution Equations and Control Theory, 2021 doi: 10.3934/eect.2021058 |
[13] |
Yaotang Li, Suhua Li. Exclusion sets in the Δ-type eigenvalue inclusion set for tensors. Journal of Industrial and Management Optimization, 2019, 15 (2) : 507-516. doi: 10.3934/jimo.2018054 |
[14] |
Gang Wang, Guanglu Zhou, Louis Caccetta. Z-Eigenvalue Inclusion Theorems for Tensors. Discrete and Continuous Dynamical Systems - B, 2017, 22 (1) : 187-198. doi: 10.3934/dcdsb.2017009 |
[15] |
Ram U. Verma. On the generalized proximal point algorithm with applications to inclusion problems. Journal of Industrial and Management Optimization, 2009, 5 (2) : 381-390. doi: 10.3934/jimo.2009.5.381 |
[16] |
Tomasz Szarek, Mariusz Urbański, Anna Zdunik. Continuity of Hausdorff measure for conformal dynamical systems. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4647-4692. doi: 10.3934/dcds.2013.33.4647 |
[17] |
Luis Barreira and Jorg Schmeling. Invariant sets with zero measure and full Hausdorff dimension. Electronic Research Announcements, 1997, 3: 114-118. |
[18] |
Mansour Shrahili, Ravi Shanker Dubey, Ahmed Shafay. Inclusion of fading memory to Banister model of changes in physical condition. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 881-888. doi: 10.3934/dcdss.2020051 |
[19] |
Chaoqian Li, Yajun Liu, Yaotang Li. Note on $ Z $-eigenvalue inclusion theorems for tensors. Journal of Industrial and Management Optimization, 2021, 17 (2) : 687-693. doi: 10.3934/jimo.2019129 |
[20] |
Xue-Ping Luo, Yi-Bin Xiao, Wei Li. Strict feasibility of variational inclusion problems in reflexive Banach spaces. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2495-2502. doi: 10.3934/jimo.2019065 |
2021 Impact Factor: 1.865
Tools
Metrics
Other articles
by authors
[Back to Top]