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A note on an approximate mean value theorem for Fréchet subgradients. (English) Zbl 1238.49021

Summary: We derive a new characterization of Asplund spaces and give a clarification of the proof of the approximate mean value theorem in B. S. Mordukhovich [Variational analysis and generalized differentiation. I: Basic theory. II: Applications, Grundlehren der Mathematischen Wissenschaften 331/332. Berlin: Springer (ISBN 3-540-25437-4/hbk; 3-540-25438-2/hbk). xxii, 579 p./v.1, xxii, 610 p. (2006; Zbl 1100.49002]] and B.S. Mordukhovich and Y. Shao [”Extremal characterizations of Asplund spaces”, Proc. Am. Math. Soc. 124, No.1, 197-205 (1996; Zbl 0849.46010)].

MSC:

49J40 Variational inequalities
49J45 Methods involving semicontinuity and convergence; relaxation
49J52 Nonsmooth analysis
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[1] Mordukhovich, B. S.; Shao, Y., Nonsmooth sequential analysis in Asplund spaces, Trans. Amer. Math. Soc., 124, 1235-1280 (1996) · Zbl 0881.49009
[2] Loewen, P. D., A mean value theorem for Fréchet subgradients, Nonlinear Anal., 23, 1365-1381 (1994) · Zbl 0824.46047
[3] Thibault, L., A note on the Zagrodny mean value theorem, Optimization, 35, 127-130 (1995) · Zbl 0837.49012
[4] Zagrodny, D., Approximate mean value theorem for upper subderivatives, Nonlinear Anal., 12, 1413-1428 (1988) · Zbl 0689.49017
[5] Mordukhovich, B. S., Variational Analysis and Generalized Differentiation, Vol. I: Basic Theory (2006), Springer: Springer Berlin
[6] Mordukhovich, B. S.; Nam, N. M.; Yen, N. D., Fréchet subdifferential calculus and optimality conditions in mathematical programming, Optimization, 55, 685-708 (2006) · Zbl 1121.49017
[7] Fabian, M., Gâteaux Differentiability of Convex Functions and Topology. Weak Asplund Spaces (1997), Wiley: Wiley New York · Zbl 0883.46011
[8] Mordukhovich, B. S.; Shao, Y., Extremal characterizations of Asplund spaces, Proc. Amer. Math. Soc., 124, 197-205 (1996) · Zbl 0849.46010
[9] Phelps, R. R., Convex Functions, Monotone Operators and Differentiability (1993), Springer: Springer Berlin · Zbl 0921.46039
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