Local Lipschitz-constant functions and maximal subdifferentials
Let \(X\) be a Banach space. For the study of generalized subdifferentials, the authors introduce the notion of topologically robust upper semicontinuity for real-valued functions. So the function \(k(x)\) is called to be topologically robust usc if it is upper semicontinuous and quasi lower semicontinuous. After the discussion of basic properties of such functions it is shown that the set-valued mapping \(k(x) B_{X^*}\) (where \(B_{X^*}\) is the dual unit ball) is a Clarke subdifferential map and (if \(X\) is separable) also an approximate subdifferential map. This result is extended showing that even the mapping \[ T(x):= CSC(\Omega+ kB_{X^*})(x) \] (where \(k(x)\) is an lsc function, \(\Omega(x)\) is the union of Clarke subdifferentials of a countable family of equi-locally Lipschitz functions, and \(CSC\) means the cusco hull, i.e., the convex upper semi-continuous hull of a map) is a Clarke subdifferential map. In the second part the authors regard the associated local Lipschitz-constant function \(\text{Lip}_f\) to a given locally Lipschitz function \(f\) according to \[ \text{Lip}_f(x):= \limsup_{\delta\downarrow 0} \Biggl\{{|f(y)- f(z)|\over\|y- z\|}:\|y\|,\|z\|\leq \delta, y\neq z\Biggr\}. \] Topological properties but also special chain rules for this notion are presented. Moreover, it is shown that every nonnegative topologically robust usc function is a local Lipschitz-constant function. Additionally assumptions are presented which ensure that conversely every local Lipschitz-constant function is also topologically robust usc and hence that the mappings \(\text{Lip}_f(x) B_{X^*}\) are Clarke subdifferential maps. In the last part of the paper it is shown that one can always find a maximal \(\beta\)-subderivative whenever the one-sided Dini directional derivatives equal the local Lipschitz-constant.
topologically robust upper semicontinuity
scientific article; zbMATH DE number 1901530