On first and second-order perturbed differential inclusions governed by maximal monotone operators
DOI:
https://doi.org/10.12775/TMNA.2023.032Keywords
Absolutely continuous variation, differential inclusion, Lipschitz mapping, maximal monotone operator, pseudo-distance, second-orderAbstract
In this paper we establish, in a separable Hilbert space, a result asserting the existence of absolutely continuous solutions for a system made up of a first-order differential inclusion governed by time and state-dependent maximal monotone operators; and an ordinary differential equation. From this result, we derive existence of absolutely continuous solutions to a second-order differential inclusion governed by time and state-dependent maximal monotone operators.References
L. Adam and J. Outrata, On optimal control of a sweeping process coupled with an ordinary differential equation, Discrete Contin. Dyn. Syst. Ser. B 19 (2014), no. 9, 2709–2738.
S. Adly, A. Hantoute and B.K. Le, Nonsmooth Lur’e dynamical systems in Hilbert spaces, Set-Valued Var. Anal. 24 (2016), 13–35.
S. Adly and B.K. Le, Unbounded second-order state-dependent Moreau’s sweeping processes in Hilbert spaces, J. Optim. Theory Appl. 169 (2016), no. 2, 407–423.
S. Adly and B.K. Le, Second-order state-dependent sweeping process with unbounded and nonconvex constraints, Pure Appl. Funct. Anal. 3 (2018), 271–285.
S. Adly and F. Nacry, An existence result for discontinuous second-order nonconvex state-dependent sweeping processes, Appl. Math. Opt. 79 (2019), 515–546.
F. Aliouane and D. Azzam-Laouir, Second order sweeping process with a Lipschitz perturbation, J. Math. Anal. Appl. 452 (2017), 729–746.
F. Aliouane and D. Azzam-Laouir, A second order differential inclusion with proximal normal cone in Banach spaces, Topol. Methods in Nonlinear Analysis 44 (2014), no. 1, 143–160.
F. Aliouane, D. Azzam-Laouir, C. Castaing and M.D.P. Monteiro Marques, Second order time and state dependent sweeping process, J. Optim. Theory Appl. 182 (2019), 153–188.
D. Azzam-Laouir, W. Belhoula, C. Castaing and M.D.P. Monteiro Marques, Multivalued perturbation to evolution problems involving time dependent maximal monotone operators, Evol. Equ. Control Theory 9 (2020), no. 1, 219–254.
D. Azzam-Laouir, W. Belhoula, C. Castaing and M.D.P. Monteiro Marques, Perturbed evolution problems with absolutely continuous variation in time and applications, J. Fixed Point Theory Appl. 21 (2019), no. 40.
D. Azzam-Laouir and M. Benguessoum, Multi-valued perturbations to a couple of differential inclusions governed by maximal monotone operators, Filomat 35 (2021), no. 13, 4369–4380.
D. Azzam-Laouir and I. Boutana-Harid, Mixed semicontinuous perturbation to an evolution problem with time-dependent maximal monotone operator, J. Nonlinear Convex Anal. 20 (2018), no. 1, 39–52.
D. Azzam-Laouir, C. Castaing and M.D.P. Monteiro Marques, Perturbed evolution problems with continuous bounded variation in time and applications, Set-Valued Var Anal. 26 (2018), no. 3, 693–728.
D. Azzam-Laouir and S. Izza, Existence of solutions for second-order perturbed nonconvex sweeping process, Comput. Math. Appl. 62 (2001), 1736–1744.
V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff. Int. Publ. Leyden, 1976.
M. Benguessoum, D. Azzam-Laouir and C. Castaing, On a time and state dependent maximal monotone operator coupled with a sweeping process with perturbations, Set-Valued Var. Anal. 29 (2021), 191–219.
N. Bouhali, D. Azzam-Laouir and M.D.P. Monteiro Marques, Optimal control of an evolution problem involving time-dependent maximal monotone operators, J. Optim. Theory Appl. 194 (2022), 59–91.
M. Bounkhel, General existence results for second order non convex sweeping process with unbounded perturbations, Port. Math. 60 (2003), no. 3, 269–304.
M. Bounkhel and D. Azzam-Laouir, Existence results for second order nonconvex sweeping processes, Set-Valued Anal. 12 (2004), no. 3, 291–318.
H. Brézis, Opérateurs Maximaux Monotones et Semi-Groupes de Contraction dans un Espace de Hilbert, North Holland, 1979.
B. Brogliato and D. Goeleven, Existence, uniqueness of solutions and stability of nonsmooth multivalued Lur’e dynamical systems, J. Convex Anal. 20 (2013), no. 3, 881–900.
B. Brogliato and A. Tanwani, Dynamical systems coupled with monotone set-valued operators Formalisms, applications, well-posedness, and stability, SIAM Rev. 62 (2020), 3–129.
M. Brokate and P. Krejčı́, Optimal control of ODE systems involving a rate independent variational inequality, Discrete Contn. Dyn. Syst. Ser. B 18 (2012), no. 2, 331–348.
M.K. Camlibel, L. Ianelli and A. Tanwani, Convergence of proximal solutions for evolution inclusions with time-dependent maximal monotone operators, Math Program Ser. A 194 (2022), 1017–1059.
M.K. Camlibel and J.M. Schumacher, Linear passive systems and maximal monotone mappings, Math. Program 157 (2016), no. 2, 397–420.
C. Castaing, Quelques problèmes d’évolution du second ordre, Sém. Anal. Convexe Montpellier (1988), Exposé 5.
C. Castaing, A.G. Ibrahim and M.F. Yarou, Existence problems in second order evolution inclusions: Discretization and variational approach, Taiwanese J. Math. 12 (2008), 1433–1475.
C. Castaing, C. Godet-Thobie and S. Saidi, On fractional evolution inclusion coupled with a time and state dependent maximal monotone operator, Set-Valued Var. Anal. 30 (2022), no. 2, 621–656.
C. Castaing, C. Godet-Thobie and L.X. Truong, Fractional order of evolution inclusion coupled with a time and state dependent maximal monotone operator, Mathematics 8 (2020), no. 9, 1–31.
C. Castaing, M.D.P. Monteiro Marques and P. Raynaud de Fitte, Second-order evolution problems with time-dependent maximal monotone operator and applications, Adv. Math. Econ. 22 (2018), 25–77.
C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes on Math., vol. 580, Springer–Verlag, Berlin, 1977.
K. Dib and D. Azzam-Laouir, Existence solutions for a couple of differential inclusions involving maximal monotone operators, Appl. Anal. 102 (2023), no. 9, 2628–2650.
J. Diestel and J.J. Uhl, Vector Measures, Math. Surveys Monogr., vol. 5, Amer. Math. Soc., 1977.
N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Educ. Chiba Univ. 30 (1981), 1–81.
M. Kunze and M.D.P. Monteiro Marques, BV solutions to evolution proplems with time-dependent domains, Set-Valued. Anal. 5 (1997), 57–72.
B.K. Le, Well-posedeness and nonsmooth Lyapunov pairs for state-dependent maximal monotone differential inclusions, Optimization 69 (2020), no. 6, 1187–1217.
B.K. Le, On a class of Lur’e dynamical systems with state-dependent set-valued feedback, Set-Valued Var. Anal. 28 (2020), 537–557.
M.D.P. Monteiro Marques, Differential Inclusions in Nonsmooth Mechanical Problems, Shocks and Dry Friction, vol. 9, Birkhäuser, 1993.
J.J. Moreau, Rafle par un convexe variable I, Sém. Anal. Convexe Montpellier (1971), Exposé 15.
J.J. Moreau, Evolution problem associated with a moving convex set in a Hilbert Space, J. Differential Equations 26 (1977), 347–374.
J. Noel, Second-order general perturbed sweeping process differential inclusion, J. Fixed Point Theory Appl. 20 (2018), no. 133, 1–21.
S. Saidi, A perturbed second-order problem with time and state-dependent maximal monotone operators, Discuss. Math. Differ. Incl. Control Opti. 41 (2021), no. 1, 61–86.
S. Saidi, Second-order evolution problems by time and state-dependent maximal monotone operators and set-valued perturbations, Int. J. Nonlinear Anal. Appl. 14 (2023), 699–715.
F. Selamnia, D. Azzam-Laouir and M.D.P. Monteiro Marques, Evolution problems involving state-dependent maximal monotone operators, Appl. Anal. 101 (2022), no. 1, 297–313.
A.A. Tolstonogov, BV Continuous solutions of an evolution inclusion with maximal monotone operator and nonconvex-valued perturbation. Existence theorem, Set-Valued Var. Anal. 29 (2021), 29–60.
E. Vilches and B.T. Nguyen, Evolution inclusions governed by time-dependent maximal monotone operators with a full domain, Set-Valued Var Anal. 20 (2020), 569–581.
A.A. Vladimirov, Nonstationary dissipative evolution equation in Hilbert space, Nonlinear Anal. 17 (1991), 499–518.
Published
How to Cite
Issue
Section
License
Copyright (c) 2024 Messaouda Benguessoum, Dalila Azzam-Laouir
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
Stats
Number of views and downloads: 0
Number of citations: 0
