Porosity and differentiability in smooth Banach spaces

Author:
Pando Gr. Georgiev

Journal:
Proc. Amer. Math. Soc. **133** (2005), 1621-1628

MSC (2000):
Primary 49J53; Secondary 49J50

DOI:
https://doi.org/10.1090/S0002-9939-05-07736-1

Published electronically:
January 14, 2005

MathSciNet review:
2120263

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Abstract | References | Similar Articles | Additional Information

Abstract: We improve a result of Preiss, Phelps and Namioka, showing that every submonotone mapping in a Gateaux smooth Banach space is single-valued on the complement of a -cone porous subset. If a Banach space has a uniformly -differentiable Lipschitz bump function (with respect to some bornology ), then we show with a much simpler argument (localization of -minimum of a perturbed function) that every continuous convex function on is -differentiable on the complement of a -uniformly porous set.

**1.**J.M. Borwein and Q. Zhu,*Viscosity solutions and viscosity subderivatives in smooth Banach spaces with applications to metric regularity*, SIAM J. Control Optim.**34**(1996), no. 5, 1568-1591. MR**1404847 (97g:49037)****2.**J.M. Borwein, J.S. Treiman and Q.J. Zhu,*Partially smooth variational principles and applications*, Nonlinear Anal.**35**(1999), no. 8, Ser. B: Real World Applications, 1031-1059. MR**1707806 (2000j:49028)****3.**R. Deville, G. Godefroy and V. Zizler,*A smooth variational principle with applications to Hamilton-Jacobi equations in infinite dimensions*, J. Funct. Anal.**111**(1993), 197-212. MR**1200641 (94b:49010)****4.**I. Ekeland and G. Lebourg,*Generic Fréchet differentiabili ty and perturbed optimization problems in Banach spaces*, Trans. Amer. Math. Soc.**224**(1979), 193-216. MR**0431253 (55:4254)****5.**P.G. Georgiev,*Submonotone mappings in Banach spaces and applications*, Set-Valued Anal.**5**(1997), no. 1, 1-35. MR**1451845 (98d:49021)****6.**P.G. Georgiev and N.P. Zlateva,*Generic Gateaux differentiability via smooth perturbations*, Bull. Austral. Math. Soc.**56**(1997), 421-428. MR**1490659 (98i:46039)****7.**P.S. Kenderov, W.B. Moors and S. Sciffer,*A weak Asplund space whose dual is not weak* fragmentable*, Proc. Amer. Math. Soc.**129**(2001), 3741-3747. MR**1860511 (2002h:54014)****8.**R.R. Phelps,*Convex Functions, Monotone Operators and Differentiability*, Lecture Notes in Mathematics, No.**1364**. MR**0984602 (90g:46063)****9.**D. Preiss, R.R. Phelps and I.Namioka,*Smooth Banach spaces, weak Asplund spaces and monotone or usco mappings*, Israel J. Math.**72**(1990), 257-279. MR**1120220 (92h:46021)****10.**N.K. Ribarska,*Internal characterization of fragmentable spaces*, Mathematika**34**(1987), 243-257. MR**0933503 (89e:54063)****11.**N.K. Ribarska,*The dual of a Gateaux smooth Banach space is weak star fragmentable*, Proc. Amer. Math. Soc.**114**(1992), 1003-1008. MR**1101992 (92g:46020)****12.**W.K. Tang,*Uniformly Differentiable Bump Functions*, Arch. Math.**68**(1997), 55-59. MR**1421846 (97j:46010)****13.**L. Zajicek,*Smallnesss of sets of nondifferentiability of convex functions in non-separable Banach spaces*, Czech. Math. J.**41**(116) (1991), 288-296. MR**1105445 (92d:46112)****14.**L. Zajicek,*On differentiability properties of Lipschitz functions on a Banach space with uniformly Gateaux differentiable bump function*, Comment. Math. Univ. Carolinae**38**(1997), 329-336. MR**1455499 (99b:46066)**

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Additional Information

**Pando Gr. Georgiev**

Affiliation:
Department of Mathematics and Informatics, Sofia University “St. Kl. Ohridski", 5 James Bourchier Boulevard, 1126 Sofia, Bulgaria

Address at time of publication:
Electrical & Computer Engineering and Computer Science Department, University of Cincinnati, ML 0030, Cincinnati, Ohio 45220

Email:
pgeorgie@ececs.uc.edu

DOI:
https://doi.org/10.1090/S0002-9939-05-07736-1

Keywords:
Porous set,
submonotone mappings,
differentiability

Received by editor(s):
July 31, 2002

Published electronically:
January 14, 2005

Dedicated:
Dedicated to Professor Petar Kenderov on the occasion of his 60th anniversary

Communicated by:
Jonathan M. Borwein

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.