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Fuzzy calculus for coderivatives of multifunctions. (English) Zbl 0879.58006
From the introduction: “This paper is concerned with generalized differentiation of set-valued mappings (multifunctions) between Banach spaces.
The main goal of this paper is to obtain comprehensive calculus results for Fréchet $$\varepsilon$$-coderivatives including the basic case of $$\varepsilon =0$$. In contrast to the exact calculus for the limiting constructions, here we are able to derive a full fuzzy calculus for Fréchet coderivatives that involves quantitative estimates”.

##### MSC:
 58C06 Set-valued and function-space-valued mappings on manifolds 47H04 Set-valued operators
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##### References:
 [1] Aubin, J.-P., Contingent derivatives of setvalued maps and existence of solutions to nonlinear inclusions and differential inclusions, (), 159-229 [2] Aubin, J.-P., Lipschitz behavior of solutions to convex minimization problems, Math. oper. res., 9, 87-111, (1984) · Zbl 0539.90085 [3] Aubin, J.-P.; Frankowska, H., () [4] Borwein, J.M.; Ioffe, A.D., Proximal analysis in smooth spaces, Set-valued analysis, 4, 1-24, (1996) · Zbl 0858.49015 [5] Borwein J. M.& Zhu Q. J., Viscosity solutions and viscosity subderivatives in smooth Banach spaces with applications to metric regularity, SIAM J. Control Optim.{\bf34}, 1568-1591. · Zbl 0882.49020 [6] Borwein, J.M.; Zhuang, D.M., Verifiable necessary and sufficient conditions for regularity of set-valued and single-valued maps, J. math. anal. appl., 134, 441-459, (1988) · Zbl 0654.49004 [7] Clarke, F.H., () [8] Deville R. & El Haddad E. M., The subdifferential of the sum of two functions in Banach spaces I. First order case, Convex Analysis (to appear). · Zbl 0935.49011 [9] Dontchev, A.L.; Rockafellar, R.T., Characterizations of strong regularity for variational inequalities over polyhedral convex sets, SIAM J. optim., 6, 1087-1105, (1996) · Zbl 0899.49004 [10] Fabian, M., Subdifferentiability and trustworthiness in the light of a new variational principle of Borwein and preiss, Acta univ. carolinae, 30, 51-56, (1989) · Zbl 0714.49022 [11] Ioffe, A.D., Sous-differentielles approaches de fonctions numériques, C. R. acad. sc. Paris, 292, 675-678, (1981) · Zbl 0482.46029 [12] Ioffe, A.D., On subdifferentiability spaces, Ann. New York acad. sci., 410, 107-119, (1983) [13] Ioffe, A.D., Calculus of dini subdifferentials and contingent coderivatives of set-valued maps, Nonlinear analysis, 8, 517-539, (1984) · Zbl 0542.46023 [14] Ioffe, A.D., Proximal analysis and approximate subdifferentials, J. London math. soc., 41, 175-192, (1990) · Zbl 0725.46045 [15] Kruger, A.Y., Properties of generalized differentials, Sib. math. J., 26, 822-832, (1985) · Zbl 0596.46038 [16] Kruger, A.Y., A covering theorem for set-valued mappings, Optimization, 19, 763-780, (1988) · Zbl 0666.49003 [17] Kruger, A.Y.; Mordukhovich, B.S., Extremal points and the Euler equation in nonsmooth optimization, Dokl. akad. nauk BSSR, 24, 684-687, (1980) · Zbl 0449.49015 [18] Leach, E.B., A note on inverse function theorem, (), 694-697 · Zbl 0191.15003 [19] Loewen, P.D., A Mean value theorem for Fréchet subgradients, Nonlinear analysis, 23, 1365-1381, (1994) · Zbl 0824.46047 [20] Loewen P. D. & Rockafellar R. T., New necessary conditions for the generalized problem of Bolza, SIAM J. Control Optim.{\bf34}, 1496-1511. · Zbl 0871.49023 [21] Mordukhovich, B.S., Maximum principle in problems of time optimal control with nonsmooth constraints, J. appl. math. mech., 40, 960-969, (1976) · Zbl 0362.49017 [22] Mordukhovich, B.S., Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problems, Soviet math. dokl., 22, 526-530, (1980) · Zbl 0491.49011 [23] Mordukhovich, B.S., () [24] Mordukhovich, B.S., Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions, Trans. amer. math. soc., 340, 1-35, (1993) · Zbl 0791.49018 [25] Mordukhovich, B.S., Generalized differential calculus for nonsmooth and set-valued mappings, J. math. anal. appl., 183, 250-288, (1994) · Zbl 0807.49016 [26] Mordukhovich, B.S., Lipschitzian stability of constraint systems and generalized equations, Nonlinear analysis, 22, 173-206, (1994) · Zbl 0805.93044 [27] Mordukhovich, B.S., Stability theory for parametric generalized equations and variational inequalities via nonsmooth analysis, Trans. am. math. soc., 343, 609-658, (1994) · Zbl 0826.49008 [28] Mordukhovich, B.S., Discrete approximations and refined Euler-Lagrange conditions for nonconvex differential inclusions, SIAM J. control optim., 33, 882-915, (1995) · Zbl 0844.49017 [29] Mordukhovich, B.S.; Shao, Y., Differential characterizations of covering, metric regularity, and Lipschitzian properties of multifunctions between Banach spaces, Nonlinear analysis, 24, 1401-1424, (1995) · Zbl 0863.47030 [30] Mordukhovich, B.S.; Shao, Y., Extremal characterizations of asplund spaces, (), 197-205 · Zbl 0849.46010 [31] Mordukhovich, B.S.; Shao, Y., Nonsmooth sequential analysis in asplund spaces, Trans. am. math. soc., 348, 1235-1280, (1996) · Zbl 0881.49009 [32] Mordukhovich, B.S.; Shao, Y., Stability of set-valued mappings in infinite dimensions: point criteria, applications, preprint 1994, SIAM J. control optim., 35, 285-314, (1997) · Zbl 0895.49011 [33] Mordukhovich, B.S.; Shao, Y., Nonconvex differential calculus for infinite dimensional multifunctions, Set-valued analysis, 4, 205-236, (1996) · Zbl 0866.49024 [34] Penot, J.-P.; Metric, regularity, Openness and Lipschitzian behavior of multifunctions, Nonlinear analysis, 13, 629-643, (1989) · Zbl 0687.54015 [35] Phelps, R.R., Convex functions, monotone operators and differentiability, () · Zbl 0932.47040 [36] Rockafellar, R.T., Proto-differentiability of setvalued mappings and its applications in optimization, (), 449-482 · Zbl 0674.90082 [37] Zhu, Q.J., Calculus rules for subderivatives in smooth Banach spaces, (1995), preprint [38] Jourani, A.; Thibault, L., Coderivatives of multivalued mappings, locally compact cones and metric regularity, (1995), preprint · Zbl 0919.46031 [39] Penot, J.-P., Compactness properties, openness criteria and coderivatives, (1995), preprint [40] Ioffe, A.D., Coderivative compactness, metric regularity and subdifferential calculus, (1996), preprint
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