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Journal of Convex Analysis 13 (2006), No. 3, 695--709
Copyright Heldermann Verlag 2006

ε-Fréchet Differentiability of Lipschitz Functions and Applications

Marián Fabian
Mathematical Institute, Academy of Sciences, Zitná 25, 115 67 Prague 1, Czech Republic

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

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

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.

Keywords: epsilon-Frechet differentiability, mean-value theorem, local epsilon-support, intermediate differentiability, Asplund generated space, boundedly Asplund set, separable reduction.

MSC: 46G05; 58C20, 49J50

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