Differentiability and regularity of Lipschitzian mappings

Mordukhovich, Boris; Wang, Bingwu

doi:10.1090/S0002-9939-02-06836-3

Differentiability and regularity of Lipschitzian mappings
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by Boris S. Mordukhovich and Bingwu Wang PDF

Proc. Amer. Math. Soc. 131 (2003), 389-399 Request permission

Abstract:

We introduce new differentiability properties of functions between Banach spaces and establish their relationships with graphical regularity of Lipschitzian single-valued and set-valued mappings. The proofs are based on advanced tools of nonsmooth variational analysis including new results on coderivative scalarization and normal cone calculus.

References

Frank H. Clarke, Optimization and nonsmooth analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1983. A Wiley-Interscience Publication. MR 709590
Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
Marián Fabián, Subdifferentiability and trustworthiness in the light of a new variational principle of Borwein and Preiss, Acta Univ. Carolin. Math. Phys. 30 (1989), no. 2, 51–56. 17th Winter School on Abstract Analysis (Srní, 1989). MR 1046445
Aleksandr D. Ioffe, On subdifferentiability spaces, Fifth international conference on collective phenomena, Ann. New York Acad. Sci., vol. 410, New York Acad. Sci., New York, 1983, pp. 107–119. MR 775520, DOI 10.1111/j.1749-6632.1983.tb23308.x
A. D. Ioffe, Directional compactness, scalarization and nonsmooth semi-Fredholm mappings, Nonlinear Anal. 29 (1997), no. 2, 201–219. MR 1446225, DOI 10.1016/S0362-546X(96)00046-6
B. S. Mordukhovich, Coderivatives of set-valued mappings: calculus and applications, Nonlinear Anal. 30 (1997), 3059–3070.
Boris S. Mordukhovich and Yong Heng Shao, Nonsmooth sequential analysis in Asplund spaces, Trans. Amer. Math. Soc. 348 (1996), no. 4, 1235–1280. MR 1333396, DOI 10.1090/S0002-9947-96-01543-7
B. S. Mordukhovich and B. Wang, Restrictive metric regularity and generalized differential calculus in Banach spaces, preprint, Department of Mathematics, Wayne State University, 2002.
R. T. Rockafellar, Maximal monotone relations and the second derivatives of nonsmooth functions, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), no. 3, 167–184 (English, with French summary). MR 797269
R. Tyrrell Rockafellar and Roger J.-B. Wets, Variational analysis, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 317, Springer-Verlag, Berlin, 1998. MR 1491362, DOI 10.1007/978-3-642-02431-3
Lionel Thibault, Sous-différentiels de fonctions vectorielles compactement lipschitziennes, C. R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 21, A995–A998 (French, with English summary). MR 497892
Lionel Thibault, On compactly Lipschitzian mappings, Recent advances in optimization (Trier, 1996) Lecture Notes in Econom. and Math. Systems, vol. 452, Springer, Berlin, 1997, pp. 356–364. MR 1467041, DOI 10.1007/978-3-642-59073-3_{2}5