A measure differential inclusion with time-dependent maximal monotone operators
KeywordsBounded variation, differential measure, Lipschitz mapping, maximal monotone operator, pseudo-distance, right continuous
AbstractIn this paper we establish the existence and uniqueness result of right continuous bounded variation solution for a perturbed differential inclusion governed by time-dependent maximal monotone operators.
S. Adly, T. Haddad and L. Thibault, Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities, Math. Program Ser. B 148 (2014), no. 1–2, 5–47.
S. Adly, F. Nacry and L. Thibault, Discontinuous sweeping process with prox-regular sets, ESAIM: Control, Optimizatios and Calculus of Variation 23 (2017), no. 4, 1293–1329.
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), 40.
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 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 proplems with continuous bounded variation in time and applications, Set-Valued Var. Anal. 26 (2018), no. 3, 693–728.
D. Azzam-Laouir, C. Castaing and M.D.P. Monteiro Marques, BV right continuous solutions of differential inclusions involving time dependent maximal monotone operators, ArXiv: 2103.01113v1 (2021), 1–43.
V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff Int. Publ. Leyden, 1976.
H. Benabdellah, C. Castaing and M.A. Gamal Ibrahim, BV Solutions of Multivalued Differential Equations on Closed Moving Sets in Banach Spaces, Banach Center Publications, vol. 32, Institute of Mathematics, Polish academy of Sciences, Warszawa, 1995.
H. Benabdellah, C. Castaing, A. Salvadori and A. Syam, Nonconvex sweeping process, J. Appl. Anal. 2 (1996), no. 2, 17–40.
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.
H. Brézis, Opérateurs Maximaux Monotones, North Holland Publ. Compagny, Amsterdam, London, 1973.
C. Castaing, and M.D.P. Monteiro Marques, Evolution problems associated with nonconvex closed moving sets with bounded variation, Portugal. Math. 53 (1996), 73–87.
J.F. Edmond and L. Thibault, BV solutions of non convex sweeping proccess differential inclusion with perturbation, J. Differential Equations 22 (2006), 135–179.
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 problems with time-dependent domains, Set-Valued Anal. 5 (1997), 57–72.
B.K. Le, Well-posedness and nonsmooth Lyapunov pairs for state-dependent maximal monotone differential inclusions, Optimization 69 (2020), no. 6, 1187–1217.
M.D.P. Monteiro Marques, Differential inclusions nonsmooth mechanical problems, shocks and dry friction, Progress in Nonlinear Differential Equations and Their Applications, vol. 9, Birkhäuser, 1993.
J.J. Moreau, Rafle par un convexe variable I, Sem. Anal. Convexe Montpellier (1971), Exposé 15.
J.J. Moreau, Sur les mesures différentielles des fonctions vectorielles à variation bornée, Sem. Anal. Convexe Montpellier (1975), Exposé 17.
J.J. Moreau, Evolution problem asssociated with a moving convex set in a Hilbert space, J. Differiential. Equations 26 (1977), 347–374.
J.J. Moreau, Bounded variation in time, Topics in Non-smooth Mechanics (J.J. Moreau, P.D. Panagiotopoulos and G. Strang, eds.), Birkhäuser Verlag, Basel, Boston, Berlin, 1988.
J.J. Moreau and M. Valadier, A chain rule involving vector functions of bounded variations, J. Funct. Anal. 74 (1987), no. 2, 333–345.
F. Nacry, J. Noel and L. Thibault, On first and second order statedependent prox-regular sweeping process (2020), preprint, fnacry.perso.math.cnrs.fr/Aricles/Nacry-Noel-Thibault forPAFA.pdf.
N.H. Pavel, Nonlinear evolution operators and semigroups, Lecture Notes in Mathematics, vol. 1260, Springer, New York, 1987.
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.
L. Thibault, Moreau sweeping process with bounded retraction, J. Convex Anal. 23 (2016), no. 4, 1051–1098.
A.A. Tolstonogov, BV continuous solutions of an evolution inclusion with maximal monotone operator and non-convex valued perturbation. Existence theorem, Set-Valued Var. Anal. 29 (2021), 29–60.
A.A. Vladimirov, Nonstationnary dissipative evolution equation in Hilbert space, Nonlinear Anal. 17 (1991), 499–518.
I.I. Vrabie, Compactness methods for nonlinear evolution equations, Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific and Technical, John Wiley and Sons, Inc. New York, vol. 32, 1987.
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