Abstract
In a previous work the authors proved in a separable Banach space under the assumption of the global upper semicontinuity of the perturbation, the existence of Lipschitz solutions for second order non convex sweeping processes in a separable reflexive uniformly smooth Banach space. In the present paper we prove the same results, where the perturbation is assumed to be separately measurable and separately upper semicontinuous.
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References
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), 143–160.
F. Bernard and L. Thibault, Prox-regularity of functions and sets in Banach spaces. Set-valued Anal. 12 (2004), 25–47.
F. Bernard, L. Thibault and N. Zlateva, Characterizations of Prox-Regular sets in uniformaly convex Banach spaces. J. Convex Anal. 13 (2006), 525–560.
F. Bernard, L. Thibault and N. Zlateva, Prox-regular sets and epigraphs in uniformly convex Banach spaces: various regularities and other properties. Trans. Amer. Math. Soc. Volume 363, Number 4, (2010) 2211–2247.
F. Bernicot, J. Venel, Existence of sweeping process in Banach spaces under directional prox-regularity. J. Convex Anal. 17 (2010), 451–484.
H. Benabdellah and A. Faik, Perturbations convexes et nonconvexes des equations d’évolution, Portugal. Math, 53(2) (1996), 187–208.
A. Canino, On p-convex sets and geodesics. J. Diff. Euations 75 (1988), 118–157.
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.
C. Castaing and M. D. P. Monteiro Marques, A multivalued version of Scorza- Dragonis theorem with an application to normal integrands, Bull. Pol. Acad. Sci. Mathematics, 42 (1994) 133–140.
C. Castaing and M. Valadier, Convex analysis and measurable multifunctions. LNM 580, Springer Verlag, Berlin (1977).
F. H. Clarke, R. J. Stern and P. R. Wolenski, Proximal smoothness and the lower-C 2 property. J. Convex Anal. 2 (1995), 117–144.
F. H. Clarke, Y. S. Ledyaev, R. j. Stern and P. R. Wolenski, Nonsmooth Analysis and control theory. Springer-Verlag, (1998).
J. Diestel, Geometry of Banach spaces: selected topics, Springer-Verlag, New-York, (1975).
H. Federer, Curvature measures. Trans. Amer. Math. Soc. 93 (1959); 418–491.
R. A. Poliquin, R. T. Rockafellar and L. Thibault, Local differentiability of distance functions. Trans. Amer. Math. Soc. 352 (2000), 5231–5249.
A. S. Shapiro, Existence and differentiability of metric projections in Hilbert spaces. SIAM J. Optim. 4 (1994) 130–141.
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Aliouane, F., Azzam-Laouir, D. (2015). Existence of solutions of a class of second order sweeping process in Banach spaces. In: Jeribi, A., Hammami, M., Masmoudi, A. (eds) Applied Mathematics in Tunisia. Springer Proceedings in Mathematics & Statistics, vol 131. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-18041-0_20
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DOI: https://doi.org/10.1007/978-3-319-18041-0_20
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-18040-3
Online ISBN: 978-3-319-18041-0
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