Partial subdifferentials, derivates and Rademacher’s theorem. (English) Zbl 0924.49013
In this paper the authors are mainly interested in the set on which one has coincidence of the Dini subdifferential of a function defined on a product of two separable Hilbert spaces and the product of the partial Dini subdifferentials. So, it is shown that one has coincidence on a staunch set when one of the spaces is finite dimensional and the function is locally Lipschitz. One obtains Rademacher’s theorem as a corollary. One obtains also interesting results concerning Gâteaux and Fréchet differentiability of locally Lipschitz functions on separable Hilbert spaces.
Reviewer: C.Zălinescu (Iaşi)
MSC:
| 49J52 | Nonsmooth analysis |
| 49J50 | Fréchet and Gateaux differentiability in optimization |
| 46G05 | Derivatives of functions in infinite-dimensional spaces |
| 58C20 | Differentiation theory (Gateaux, Fréchet, etc.) on manifolds |
Keywords:
directional derivates; Gâteaux and Fréchet derivatives; locally Lipschitz functions; nonsmooth analysis; partial subdifferentials; staunch setsReferences:
| [1] | N. Aronszajn, Differentiability of Lipschitzian mappings between Banach spaces, Studia Math. 57 (1976), no. 2, 147 – 190. · Zbl 0342.46034 |
| [2] | Bessis D.N., Analyse contingentielle et sousdifférentielle, PhD thesis, Université de Montréal, 1997. |
| [3] | J. M. Borwein and H. M. Strójwas, Proximal analysis and boundaries of closed sets in Banach space. II. Applications, Canad. J. Math. 39 (1987), no. 2, 428 – 472. · Zbl 0621.46019 · doi:10.4153/CJM-1987-019-4 |
| [4] | Andrew Bruckner, Differentiation of real functions, 2nd ed., CRM Monograph Series, vol. 5, American Mathematical Society, Providence, RI, 1994. · Zbl 0796.26004 |
| [5] | Gustave Choquet, Outils topologiques et métriques de l’analyse mathématique, Cours rédigé par Claude Mayer, Centre de Documentation Universitaire, Paris, 1969 (French). · Zbl 0191.34501 |
| [6] | Frank H. Clarke, Generalized gradients and applications, Trans. Amer. Math. Soc. 205 (1975), 247 – 262. · Zbl 0307.26012 |
| [7] | F. H. Clarke, Optimization and nonsmooth analysis, 2nd ed., Classics in Applied Mathematics, vol. 5, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990. · Zbl 0696.49002 |
| [8] | Clarke F.H., Ledyaev Yu.S., Stern R.J., Wolenski P.R., Nonsmooth analysis and optimal control, to be published in Graduate Texts in Mathematics, Springer-Verlag. CMP 98:06 · Zbl 1047.49500 |
| [9] | F. H. Clarke, Yu. S. Ledyaev, and P. R. Wolenski, Proximal analysis and minimization principles, J. Math. Anal. Appl. 196 (1995), no. 2, 722 – 735. · Zbl 0865.49015 · doi:10.1006/jmaa.1995.1436 |
| [10] | E. P. Dolženko, Boundary properties of arbitrary functions, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 3 – 14 (Russian). |
| [11] | M. Fabián and D. Preiss, On intermediate differentiability of Lipschitz functions on certain Banach spaces, Proc. Amer. Math. Soc. 113 (1991), no. 3, 733 – 740. · Zbl 0743.46040 |
| [12] | Halmos P.R., Measure Theory, D.Van Nostrand Company, Inc., New York, 1986. |
| [13] | J. R. Giles and Scott Sciffer, Locally Lipschitz functions are generically pseudo-regular on separable Banach spaces, Bull. Austral. Math. Soc. 47 (1993), no. 2, 205 – 212. · Zbl 0789.58011 · doi:10.1017/S0004972700012430 |
| [14] | K. Kuratowski, Topology. Vol. I, New edition, revised and augmented. Translated from the French by J. Jaworowski, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe, Warsaw, 1966. · Zbl 0158.40901 |
| [15] | Jaroslav Lukeš, Jan Malý, and Luděk Zajíček, Fine topology methods in real analysis and potential theory, Lecture Notes in Mathematics, vol. 1189, Springer-Verlag, Berlin, 1986. · Zbl 0607.31001 |
| [16] | F. Mignot, Contrôle dans les inéquations variationelles elliptiques, J. Functional Analysis 22 (1976), no. 2, 130 – 185 (French). · Zbl 0364.49003 |
| [17] | de Mises M.R., La base géométrique du théorème de M. Mandelbrojt sur les points singuliers d’une fonction analytique, C.R. Acad. Sci. Paris Sér. I Math. 205 (1937), 1353-1355. · Zbl 0017.42601 |
| [18] | Aleš Nekvinda and Luděk Zajíček, A simple proof of the Rademacher theorem, Časopis Pěst. Mat. 113 (1988), no. 4, 337 – 341 (English, with Russian and Czech summaries). · Zbl 0669.26005 |
| [19] | Robert R. Phelps, Convex functions, monotone operators and differentiability, 2nd ed., Lecture Notes in Mathematics, vol. 1364, Springer-Verlag, Berlin, 1993. · Zbl 0921.46039 |
| [20] | R. R. Phelps, Gaussian null sets and differentiability of Lipschitz map on Banach spaces, Pacific J. Math. 77 (1978), no. 2, 523 – 531. · Zbl 0396.46041 |
| [21] | D. Preiss, Differentiability of Lipschitz functions on Banach spaces, J. Funct. Anal. 91 (1990), no. 2, 312 – 345. · Zbl 0711.46036 · doi:10.1016/0022-1236(90)90147-D |
| [22] | David Preiss, Gâteaux differentiable Lipschitz functions need not be Fréchet differentiable on a residual subset, Proceedings of the 10th Winter School on Abstract Analysis (Srní, 1982), 1982, pp. 217 – 222. · Zbl 0518.46032 |
| [23] | Rademacher H., Über partielle und totale Differenzierbarkeit I., Math. Ann. 89 (1919), 340-359. · JFM 47.0243.01 |
| [24] | Jean Saint-Pierre, Sur le théorème de Rademacher, Travaux Sém. Anal. Convexe 12 (1982), no. 1, exp. no. 2, 10 (French). |
| [25] | Stanisław Saks, Theory of the integral, Second revised edition. English translation by L. C. Young. With two additional notes by Stefan Banach, Dover Publications, Inc., New York, 1964. · Zbl 1196.28001 |
| [26] | Shu Zhong Shi, Choquet theorem and nonsmooth analysis, J. Math. Pures Appl. (9) 67 (1988), no. 4, 411 – 432. · Zbl 0679.49015 |
| [27] | Shu Zhong Shi, Différentiabilité d’une fonction localement lipschitzienne dans un espace de Banach séparable, J. Systems Sci. Math. Sci. 3 (1983), no. 2, 112 – 119 (French, with Chinese summary). |
| [28] | Luděk Zajíček, Differentiability of the distance function and points of multivaluedness of the metric projection in Banach space, Czechoslovak Math. J. 33(108) (1983), no. 2, 292 – 308. |
| [29] | Luděk Zajíček, Sets of \?-porosity and sets of \?-porosity (\?), Časopis Pěst. Mat. 101 (1976), no. 4, 350 – 359 (English, with Loose Russian summary). · Zbl 0341.30026 |
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