Approximate Rolle's theorems for the proximal subgradient and the generalized gradient.
The authors obtain an approximative Rolle's theorem for proximinal subgradients in real Hilbert spaces and for the generalized gradients \(\partial f\): Theorem. Let \(U\) be a bounded connected open subset of a real Banach space \(X\) and \(f:\overline U\to {\mathbb R}\) be a bounded locally Lipschitz function satisfying \(\sup f(u) > \sup f (\partial U)\). Then, for every \(\alpha>0\), there exist \(x\in U\) and \(\xi\in \partial f(x)\) such that \(\| \xi\| <\alpha\). It is also proved that an exact Rolle's theorem for the generalized gradients \(\partial f\) is false in every infinite-dimensional Banach spaces \(X\): Theorem. For every~\(X\) there exists a bounded Lipschitz function \(f\) defined on a bounded convex body~\(U\) such that \(f\) vanishes at~\(\partial U\) and yet \(y\notin \partial f(x)\). Here, given a locally Lipschitz \(f:X\to {\mathbb R}\), \(\partial f(x)=\{\xi \in X^*\mid f^0(x,v)\geq \langle \xi,v\rangle\) \(\forall v\in X\}\), \(f^0(x,v)=\limsup (f(y+tv)-f(y))/t\) as \((y,t)\to (x,0)\).
scientific article; zbMATH DE number 1970517