Generalized subdifferentials: a Baire categorical approach (Q2723455)

The authors provide a general method for the construction of pathological cases regarding the differentiability of Lipschitz functions.NEWLINENEWLINENEWLINEAlready in 1931, Banach and Mazurkiewicz pointed out that almost every continuous real-valued function is nowhere differentiable. These results are extended in the paper showing that almost all Lipschitz functions have ``large'' generalized derivatives.NEWLINENEWLINENEWLINETo use the Baire category theorem the authors introduce the set \({\mathcal X}_T\) of all \(T\)-Lipschitz functions on a Banach space \(X\), i.e., the set NEWLINE\[NEWLINE{\mathcal X}_T= \{f: X\to \mathbb{R}\mid f\text{ locally Lipschitz and }\partial_0f(x)\subset T(x) \forall x\in X\},NEWLINE\]NEWLINE where \(T: X\Rightarrow X^*\) is a weak\(^*\) cusco (i.e., a convex-weak\(^*\) compact-valued multifunction) and \(\partial_0f\) denotes the Clarke subdifferential mapping of \(f\). Using the distance \(\rho\) given by \(\rho(f, g)= \min\{1,\sup\{|f(x)- g(x)|\mid x\in X\}\) it is proved that \(({\mathcal X}_T,\rho)\) is a complete metric space.NEWLINENEWLINENEWLINEFor the first result of the paper the Banach space \(X\) is assumed to be separable. It is shown that for each \(f\in{\mathcal X}_T\) the set \(\{g\in{\mathcal X}_T\mid \partial_0f(x)\subset \partial_0 g(x) \forall x\in X\}\) is residual in \(({\mathcal X}_T,\rho)\). The same result is proved for the approximate subdifferential mapping \(\partial_a f\).NEWLINENEWLINENEWLINEIn the second part, these results are extended to non-separable Banach spaces. The following corollaries can be derived as consequences: Almost every 1-Lipschitz function on \(X\) has a Clarke subdifferential mapping that is identically equal to the unit ball in \(X^*\); if \(\{T_1,\dots, T_n\}\) is a family of maximal cyclically monotone operators on \(X\) then there exists a Lipschitz function \(g\) such that \(\partial_0g(x)= \text{co}\{T_1(x),\dots, T_n(x)\}\) for all \(x\in X\).NEWLINENEWLINENEWLINEIn the last part of the paper equivalent conditions are presented for which \({\mathcal X}_T\neq\emptyset\). In contrast to the former remarks, this last section also examines the question of the existence of Lipschitz functions with ``minimal'' subdifferential mappings.