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The contingent and the paratingent as generalized derivatives for vector- valued and set-valued mappings. (English) Zbl 0529.26010


MSC:

26E25 Set-valued functions
26B12 Calculus of vector functions
49J50 Fréchet and Gateaux differentiability in optimization
26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
Full Text: DOI

References:

[1] Artstein, Z., On the calculus of closed set-valued functions, Indiana Univ. math. J., 24, 433-441 (1974) · Zbl 0296.28015
[2] Aubin, J.-P., Gradients generalisés de Clarke, Ann. Soc. math. Québec, 2, 197-252 (1078) · Zbl 0411.49001
[3] Banks, H. T.; Jacobs, M. Q., A differential calculus for multifunctions, J. math. analysis Applic., 29, 246-272 (1970) · Zbl 0191.43302
[4] Berge, C., Espaces Topologiques. Fonctions Multivoques. (1959), Dunod: Dunod Paris · Zbl 0088.14703
[5] Blagodatskih, V. I., On differentiability of solutions with respect to initial conditions, Diff. Uravn., 9, 2136-2140 (1973) · Zbl 0274.34006
[6] Bouligand, G., Sur les surfaces dépourvues de points hyperlimites, Ann. Soc. pol. Math., 9, 32-41 (1930) · JFM 57.0097.01
[7] Bourbaki, N., Topologie Générale (1970), Hermann: Hermann Paris · Zbl 0085.37103
[8] Bourbaki, N., Variétés Différentielles et Analytiques (1967), Hermann: Hermann Paris · Zbl 0171.22004
[9] Bradley, M.; Datko, R., Some analytic and measure theoretic properties of set-valued mappings, SIAM J. Control Optim., 15, 625-635 (1977) · Zbl 0354.28001
[10] Bridgland, T. F., Trajectory integrals for set-valued functions, Pacific J. Math., 33, 43-68 (1970) · Zbl 0193.01201
[11] Cimpu, E., Approximations des solutions des équations différentielles an paratingent à retardement, dans les espaces de Banach, Revue Roum. Math. pures appl., 25, 509-518 (1980) · Zbl 0441.34057
[12] Clarke, F. H., Generalized gradients and applications, Trans. Am. Math. Soc., 205, 247-262 (1975) · Zbl 0307.26012
[13] Clarke, F. H., On the inverse function theorem, Pacific J. Math., 64, 97-102 (1976) · Zbl 0331.26013
[14] Clarke, F. H., Optimal control and the true Hamiltonian, SIAM Rev., 21, 157-166 (1979) · Zbl 0408.49025
[15] De Blasi, F. S., On the differentiability of multifunctions, Pacific J. Math., 66, 67-81 (1976) · Zbl 0348.58004
[16] Dieudonne, J., Foundations of Modern Analysis (1969), Academic Press: Academic Press New York · Zbl 0176.00502
[17] Halkin, H., Necessary conditions for optimal control problems with differentiable and nondifferentiable data, Lecture Notes in Mathematics, 680, 77-119 (1978) · Zbl 0388.49011
[18] Hiriart-Urruty, J.-B., New concepts in nondifferentiable programming, Bull. Soc. math. France, Mémoire, 60, 57-83 (1979) · Zbl 0469.90071
[20] Hiriart-Urruty, J.-B.; Thibault, L., Existence et caractérisation de différentielles généralisées d’applications localement Lipschitziennes d’un espace de Banach séparable dans un espace de Banach réflexif séparable, C. hebd. Séanc. Acad. Sci. Paris, 290, 1091-1094 (1980) · Zbl 0441.46035
[21] Hukuhara, M., Integration des applications mésurables dont la valeur est un compact convexe, Funkcialaj Ekvacioj, 10, 205-223 (1967) · Zbl 0161.24701
[22] Ioffe, A. D., Différentielles généralisées d’applications localement lipschitziennes d’un espace de Banach dans un autre, C.r. hebd. Séanc. Acad. Sci. Paris, 289, 637-640 (1979) · Zbl 0421.46039
[23] Lasota, A.; Strauss, A., Asymptotic behaviour for differential equations which cannot be locally linearized, J. diff. Eqns., 10, 152-171 (1971) · Zbl 0235.34076
[24] Lebourg, G., Generic differentiability of Lipschitzian functions, Cah. math. Décision, 7704 (1977)
[25] Marchaud, A., Sur les champs de demi-cônes et les équations différentielles du premier ordre, Bull. Soc. math. France, 62, 1-38 (1934) · JFM 60.0373.09
[26] Martelli, M.; Vignoli, A., On differentiability of multi-valued maps, Boll. Un. mat. ital., 10, 701-712 (1974) · Zbl 0311.46029
[27] Mirica, St., A note on the generalized differentiability of mappings, Nonlinear Analysis, 4, 567-575 (1980) · Zbl 0437.26004
[28] Nurminskii, E. A., On differentiability of multifunctions, Kybernetika, Kiev, 6, 46-48 (1978), (in Russian).
[29] Penot, J.-P., Calcul sous-différentiel et optimization, J. funct. Analysis, 27, 248-276 (1978) · Zbl 0404.90078
[30] Penot, J.-P., Utilisation des sous-differentielles généralisées en optimisation, Premier Colloque AFCET-SMF T.3, 69-85 (1978) · Zbl 0483.49023
[31] Pourciau, B. H., Analysis and optimization of Lipschitz continuous mappings, J. optim. Theory Appl., 22, 311-351 (1977) · Zbl 0336.26008
[32] Psenicinyi, B. N., Convex multifunctions and their conjugates, Kybernetika, Kiev, 3, 94-102 (1972), (in Russian). · Zbl 0261.90050
[33] Radstrom, H., An embedding theorem for spaces of convex sets, Proc. Am. math. Soc., 3, 165-169 (1952) · Zbl 0046.33304
[34] Rockafellar, R. T., Convex Analysis (1970), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0229.90020
[35] Rockafellar, R. T., Directionally Lipschitzian functions and subdifferential calculus, Proc. London math. Soc., 39, 331-355 (1979) · Zbl 0413.49015
[36] Rockafellar, R. T., Generalized directional derivatives and subgradients of non-convex functions, Can. J. Math., 32, 257-280 (1980) · Zbl 0447.49009
[37] Salinetti, G.; Wets, R. J.-B., On the convergence of sequences of convex sets in finite dimensions, SIAM Rev., 21, 18-34 (1979) · Zbl 0421.52003
[38] Severi, F., Su alcune questioni di topologia infinitesimale, Annls Soc. pol. Math., 9, 97-108 (1930) · JFM 57.0754.01
[39] Sobieszek, W.; Kowalski, P., On the different definitions of the lower semicontinuity, upper semicontinuity, upper semicompacity, closity and continuity of the point-to-set mappings, Demonst. math., 11, 1053-1063 (1978) · Zbl 0408.54001
[40] Stefani, G.; Zecca, P., Properties of convex sets with applications to differential theory of multivalued functions, Nonlinear Analysis, 2, 583-595 (1978) · Zbl 0384.49027
[41] Sweetser, T. H., A minimal set-valued strong derivative for vector-valued Lipschitz functions, J. optim. Theory Appl., 23, 549-562 (1977) · Zbl 0345.26005
[43] Thibault, L., On generalized differentials and subdifferentials of Lipschitz vector-valued functions (1980), Université de Pau, preprint
[44] Ursescu, C., Sur une généralisation de la notion de différentiabilité, Accad. Naz. Lincei, 8, 199-204 (1973) · Zbl 0281.46038
[45] Waewski, T., Sur une condition équivalente á l’équation au contingent, Bull. Acad. pol. Soc. Math., 9, 865-867 (1961) · Zbl 0101.06001
[46] Zaremba, S. C., Sur une extension de la notion d’équation différentielle, C.r. hebd. Séanc. Acad. Sci. Paris, 199, 545-548 (1934) · Zbl 0009.39701
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