Saïdi, Soumia Some results associated to first-order set-valued evolution problems with subdifferentials. (English) Zbl 1517.47119 J. Nonlinear Var. Anal. 5, No. 2, 227-250 (2021). Summary: In Hilbert spaces, we establish some results related to a class of first-order differential inclusions governed by time-dependent subdifferential operators. Based on the existence result, we study a system governed by a couple of an evolution inclusion involving the time-dependent subdifferential operator and a differential equation with the Caputo fractional derivative. Cited in 3 Documents MSC: 47J35 Nonlinear evolution equations 47J22 Variational and other types of inclusions 47H05 Monotone operators and generalizations 34A60 Ordinary differential inclusions Keywords:differential inclusion; subdifferential; maximal monotone operator; Caputo fractional derivative; fixed point PDFBibTeX XMLCite \textit{S. Saïdi}, J. Nonlinear Var. Anal. 5, No. 2, 227--250 (2021; Zbl 1517.47119) Full Text: DOI References: [1] J.J. Moreau, Rafle par un convexe variable, I, Sém. Anal. Convexe, Montpellier, Vol. 1 (1971), Exposé No. 15. · Zbl 0343.49019 [2] M.D.P. Monteiro Marques, Differential Inclusions in Nonsmooth Mechanical Problems -Shocks and Dry Friction, Birkhauser Verlag, Basel, 1993. · Zbl 0802.73003 [3] B. Cornet, Existence of slow solutions for a class of differential inclusions, J. Math. Anal. Appl. 96 (1983), 130-147. · Zbl 0558.34011 [4] C. Henry, An existence theorem for a class of differential equations with multivalued right-hand side, J. Math. Anal. Appl. 41 (1973), 179-186. · Zbl 0262.49019 [5] K. Addy, S. Adly, B. Brogliato, D. Goeleven, A method using the approach of Moreau and Panagiotopoulos for the mathematical formulation of non-regular circuits in electronics, Nonlinear Anal. Hybrid Syst. 1 (2007), 30-43. · Zbl 1172.94650 [6] B. Maury, J. Venel, A mathematical framework for a crowd motion model, C. R. Math. Acad. Sci. Paris 346 (2008), 1245-1250. · Zbl 1168.34333 [7] H. Attouch, A. Damlamian, On multivalued evolution equations in Hilbert spaces, Israel J. Math., 12 (1972), 373-390. · Zbl 0243.35080 [8] H. Benabdellah, C. Castaing, A. Salvadori, Compactness and discretization methods for differential inclusions and evolution problems, Atti. Sem. Math. Fis. Univ. Modena, XLV (1997), 9-51. · Zbl 0876.34014 [9] H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, Lecture Notes in Math., North-Holland, 1973. · Zbl 0252.47055 [10] C. Castaing, A. Faik, A. Salvadori, Evolution equations governed by m-accretive and subdifferential operators with delay, Int. J. Appl. Math. 2 (2000), 1005-1026. · Zbl 1058.34511 [11] C. Castaing, M.D.P. Monteiro Marques, Evolution problems associated with nonconvex closed moving sets with bounded variation, Port. Math. 53 (1996), 73-78. · Zbl 0848.35052 [12] C. Castaing, M.D.P. Monteiro Marques, S. Saïdi, Evolution problems with time-dependent subdifferential operators, Adv. Math. Econ. 23 (2019), 1-39. · Zbl 1448.34125 [13] C. Castaing, A. Salvadori, L. Thibault, Functional evolution equations governed by nonconvex sweeping process, J. Nonlinear Convex Anal. 2 (2001), 217-241. · Zbl 0999.34062 [14] J.F. Edmond, L. Thibault, BV solution of nonconvex sweeping process differential inclusion with perturbation, J. Differential Equations 226 (2006), 135-159. · Zbl 1110.34038 [15] S. Hu, N.S. Papageorgiou, Handbook of multivalued analysis. Vol. II, volume 500 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 2000. · Zbl 0943.47037 [16] N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Educ. Chiba Univ. 30 (1981) 1-87. · Zbl 0662.35054 [17] M. Kubo, Characterisation of a class of evolution operators generated by time dependent subdifferential, Funkcial. Ekvac. 32 (1989), 301-321. · Zbl 0689.35048 [18] J.C. Peralba,Équations d’évolution dans un espace de Hilbert, associéesà des opérateurs sous-différentiels, Thèse de doctorat de spécialité, Montpellier, (1973). [19] S. Saïdi, L. Thibault, M. Yarou, Relaxation of optimal control problems involving time dependent subdifferen-tial operators, Numer. Funct. Anal. Optim. 34 (2013), 1156-1186. · Zbl 1288.34057 [20] S. Saïdi, M.F. Yarou, Set-valued perturbation for time dependent subdifferential operator, Topol. Methods Nonlinear Anal. 46 (2015), 447-470. · Zbl 1365.34107 [21] A.A. Tolstonogov, Properties of attainable sets of evolution inclusions and control systems of subdifferential type, Sib. Math. J. 45 (2004), 763-784. · Zbl 1049.34077 [22] Y. Yamada, On evolution equations generated by subdifferential operators, J. Math. Sci. Univ. Tokyo 23 (1976), 491-515. · Zbl 0343.34053 [23] D. Azzam-Laouir, W. Belhoula, C. Castaing, M.D.P. Monteiro Marques, Multi-valued perturbation to evolution problems involving time dependent maximal monotone operators, Evol. Equ. Control Theory 1 (2020), 219-254. · Zbl 1469.34084 [24] M. Kunze and M.D.P. Monteiro Marques, BV solutions to evolution problems with time-dependent domains, Set-Valued Anal. 5 (1997), 57-72. · Zbl 0880.34017 [25] E. Vilches, B.T. Nguyen, Evolution inclusions governed by time-dependent maximal monotone operators with a full domain, Set-Valued Var. Anal. 28 (2020), 569-581. · Zbl 1511.34072 [26] A.A. Vladimirov, Nonstationary dissipative evolution equations in Hilbert space, Nonlinear Anal. 17 (1991), 499-518. · Zbl 0756.34064 [27] C. Castaing, T. X. Dùc Ha and M. Valadier, Evolution equations governed by the sweeping process, Set-Valued Anal. 1 (1993), 109-139. · Zbl 0813.34018 [28] J.J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Differential Equations, 26 (1977), 347-374. · Zbl 0356.34067 [29] M. Valadier, Quelques résultats de base concernant le processus de la rafle, Sém. Anal. Convexe, Montpellier, 3 (1988), 1-30. · Zbl 0672.49013 [30] D. Azzam-Laouir, A. Makhlouf, L. Thibault, On perturbed sweeping process, Appl. Anal. 95 (2016), 303-322. · Zbl 1344.34072 [31] F. Nacry, Perturbed BV sweeping process involving prox-regular sets, J. Nonlinear Convex Anal. 18 (2017), 1619-1651. · Zbl 1470.34057 [32] F. Nacry, Truncated nonconvex state-dependent sweeping process: implicit and semi-implicit adapted Moreau’s catching-up algorithms, J. Fixed Point Theory Appl. 20 (2018). · Zbl 1446.34031 [33] C. Castaing, M. Valadier, Convex analysis and Measurable Multifunctions, Lecture Notes in Math., 580, Springer-Verlag Berlin Heidelberg, 1977. · Zbl 0346.46038 [34] J.J. Moreau, Rétraction d’une multiapplication, Sém. Anal. convexe, Montpellier (1972), Exposé 13. · Zbl 0353.49033 [35] A.A. Tolstonogov, Sweeping process with unbounded nonconvex perturbation, Nonlinear Anal. 108 (2014), 291-301. · Zbl 1307.34098 [36] E. Klein, A. C. Thompson, Theory of Correspondences, Wiley, New York, 1984. · Zbl 0556.28012 [37] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Math. Stud. 204, North Holland, 2006. · Zbl 1092.45003 [38] I. Podlubny, Fractional Differential Equation, Academic Press, San Diego (1999). · Zbl 0924.34008 [39] M. Bergounioux and L. Bourdin, Pontryagin maximum principle for general Caputo fractional optimal control problems with Bolza cost and terminal constraints, ESAIM Control Optim. Calc. Var. 26 (2020). · Zbl 1447.49035 [40] C. Castaing, C. Godet-Thobie, F.Z. Mostefai, On a fractional differential inclusion with boundary conditions and application to subdifferential operators, J. Nonlinear Convex Anal. 18 (2017), 1717-1752. · Zbl 1470.28012 [41] A.A. Kilbas, B Bonilla, J.J. Trukhillo, Existence and uniqueness theorems for nonlinear fractional differential equations, Demonstr. Math. XXXIII (2000), 583-602. · Zbl 0964.34004 [42] A. Idzik, Almost fixed points theorems, Proc. Amer. Math. Soc. 104 (1988), 779-784. · Zbl 0691.47046 [43] S. Park, Fixed points of approximable or Kakutani maps, J. Nonlinear Convex Anal. 7 (2006), 1-17. · Zbl 1102.47044 [44] C. Castaing, P. Raynaud de Fitte and M. Valadier, Young measures on topological spaces with applications in control theory and probability theory, Kluwer Academic Publishers, Dordrecht, 2004. · Zbl 1067.28001 [45] C. Castaing, C. Godet-Thobie, P.D. Phung, L.X. Truong, On fractional differential inclusions with nonlocal boundary conditions, Fract. Calc. Appl. Anal. 22 (2019), 444-478. · Zbl 1428.34012 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.