Generic differentiability of locally Lipschitz functions on product spaces

Although it is known that locally Lipschitz functions are densely differentiable on certain classes of Banach spaces, it is a minimality condition on the subdifferential mapping of the function which enables us to guarantee that the set of points of differentiability is a residual set. We characterise such minimality by a quasi continuity property of the Dini derivatives of the function and derive sufficiency conditions for the generic differentiability of locally Lipschitz functions on a product space.

Minimal Convex Uscos and Monotone Operators on Small Sets

We generalize the generic single-valuedness and continuity of monotone operators defined on open subsets of Banach spaces of class (S) and Asplund spaces to monotone operators defined on convex subsets of such spaces which may even fail to have non-support points. This yields differentiability theorems for convex Lipschitzian functions on such sets. From a result about minimal convex uscos which are densely single-valued we obtain generic differentiability results for certain Lipschitzian realvalued functions.

The smooth variational principle and generic differentiability

A modified version of the smooth variational principle of Borwein and Preiss is proved. By its help it is shown that in a Banach space with uniformly Gâteaux differentiable norm every continuous function, which is directionally differentiable on a dense Gδ subset of the space, is Gâteaux differentiable on a dense Gδ subset of the space.

Generic differentiability of order-bounded convex oparators

We give sufficient conditions for order-bounded convex operators to be generically differentiable (Gâteaux or Fréchet). When the range space is a countably order-complete Banach lattice, these conditions are also necessary. In particular, every order-bounded convex operator from an Asplund space into such a lattice is generically Fréchet differentiable, if and only if the lattice has weakly-compact order intervals, if and only if the lattice has strongly-exposed order intervals. Applications are given which indicate how such results relate to optimization theory.