Abstract

We study the ε-Fréchet differentiability of Lipschitz functions on Asplund generated Banach spaces. We prove a mean valued theorem and its equivalent, a formula for Clarke's subdifferential, in terms of this concept. We inspect proofs of several statements based on the deep Preiss's theorem on Fréchet differentiability of Lipschitz functions and we recognize that it is enough to use a simpler lemma on ε-Fréchet differentiability due to Fabian and Preiss. We do so for generic differentiability results of Giles and Sciffer, for the existence of nearest points of Borwein and Fitzpatrick, etc. We also show that the ε-Fréchet differentiability is separably reducible.

Author information

Contact details are reproduced from the original publication and may be historical.

Marián Fabian

Mathematical Institute, Academy of Sciences, Žitná 25, 115 67 Prague 1, Czech Republic

fabian@math.cas.cz

Philip D. Loewen

Dept. of Mathematics, University of British Columbia, Vancouver, B. C., Canada V6T 1Z2

loew@math.ubc.ca

Xianfu Wang

Dept. of Mathematics & Statistics, Univ. of B. C. at Okanagan, 3333 University Way, Kelowna, B. C., Canada V1V 1V7

Shawn.Wang@ubc.ca

Keywords

  • epsilon-Fréchet differentiability
  • mean-value theorem
  • local epsilon-support
  • intermediate differentiability
  • Asplund generated space
  • boundedly Asplund set
  • separable reduction

Mathematics Subject Classification

46G05; 58C20, 49J50

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Suggested citation

M. Fabian, P. D. Loewen, X. Wang. “ε-Fréchet Differentiability of Lipschitz Functions and Applications.” Journal of Convex Analysis 13 (2006), No. 3/4, 695–709. https://doi.org/10.68381/jca13052

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Published by Heldermann Verlag, 2006. Rights now held by Banach Press.