Abstract
We prove the existence of solutions of a differential variational inequality involving a prox-regular set in an infinite dimensional Hilbert space via a new existence result of a non-convex state-dependent sweeping process.
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Kunze, M., Monteiro Marques, M.D.P.: On parabolic quasi-variational inequalities and state-dependent sweeping processes. Topol. Methods Nonlinear Anal. 12, 179–191 (1998)
Chemetov, N., Monteiro Marques, M.D.P.: Non-convex quasi-variational differential inclusions. Set-Valued Anal. 15, 209–221 (2007)
Castaing, C., Ibrahim, A.G., Yarou, M.: Some contributions to nonconvex sweeping process. J. Nonlinear Convex Anal. 10, 1–20 (2009)
Bounkhel, M., Castaing, C.: State dependent sweeping process in p-uniformly smooth and q-uniformly convex Banach spaces. Set-Valued Anal. 3, 200–214 (2011)
Monteiro Marques, M.D.P.: Differential Inclusions in Nonsmooth Mechanical Problems: Shocks and Dry Friction. Progress in Nonlinear Differential Equations and Their Applications, vol. 9. Birkhäuser, Basel (1993)
Auricchio, F., Stefanelli, U.: Numerical analysis of a 3-D super-elastic constitutive model. Int. J. Numer. Methods Eng. 61, 142–155 (2004)
Auricchio, F., Stefanelli, U.: Well-posedness and approximation for a one-dimensional model for shape memory alloys. Math. Models Methods Appl. Sci. 15, 1301–1327 (2005)
Duvaut, D., Lions, J.L.: Inequalities in Mechanics and Physics. Springer, Berlin (1976)
Clarke, F.H., Stern, R.J., Wolenski, P.R.: Proximal smoothness and the lower-C2 property. J. Convex Anal. 2, 117–144 (1995)
Poliquin, R.A., Rockafellar, R.T., Thibault, L.: Local differentiability of distance functions. Trans. Am. Math. Soc. 352, 5231–5249 (2000)
Bounkhel, M., Thibault, L.: Nonconvex sweeping process and prox-regularity in Hilbert space. J. Nonlinear Convex Anal. 6, 359–374 (2005)
Haddad, T., Jourani, A., Thibault, L.: Reduction of sweeping process to unconstrained differential inclusion. Pac. J. Optim. 4, 493–512 (2008)
Thibault, L.: Proprietes des sous-differentiels de fonctions localement Lipschitziennes definies sur un espace de Banach separable. Applications. Thesis, Universite Montpellier II, France (1976)
Deimling, K.: Multivalued Differential Equations. Walter de Gruyter, Berlin (1992)
Castaing, C., Dúc Ha, T.X., Valadier, M.: Evolution equations governed by the sweeping process. Set-Valued Anal. 1, 109–139 (1993)
Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions. Springer, Berlin (1977)
Bounkhel, M.: Regularity Concepts in Nonsmooth Analysis: Theory and Applications. Springer, Berlin (2011)
Bounkhel, M., Jofre, A.: Subdifferential stability of the distance function and its applications to nonconvex economies and equilibrium. J. Nonlinear Convex Anal. 3, 331–347 (2004)
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The author would like to thank the editor and the anonymous referees for their valuable comments and suggestions.
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Communicated by Jean-Pierre Crouzeix.
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Haddad, T. Nonconvex Differential Variational Inequality and State-Dependent Sweeping Process. J Optim Theory Appl 159, 386–398 (2013). https://doi.org/10.1007/s10957-013-0353-1
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DOI: https://doi.org/10.1007/s10957-013-0353-1