Smooth variational principles with constraints
The authors consider a Banach space \(X\), a finite-dimensional subspace \(Y \subseteq X\) and a lower semicontinuous function \(f:X \to {\mathbb R} \cup \{+\infty\}\) such that \(0 = f(0) = \min_{y \in Y} f(y)\). Under the assumption that \(X\) admits \(C^{1}\)-bump functions, they show that for every \(\varepsilon >0\) there exist \(x \in X\) and \(p \in D^{-}f(x)\) such that \(\| x\| < \varepsilon\), \(| f(x)| < \varepsilon\), \(\| p\| d(x,Y) < \varepsilon\) and \(\sup \{\langle p,y \rangle: y \in Y, \| y\| \leq 1\} < \varepsilon\), where \(D^{-}\) denotes the so-called viscosity subdifferential introduced in \textit{M. G. Crandall} and \textit{P. L. Lions} [Trans. Am. Math. Soc. 277, 1-42 (1983; Zbl 0599.35024)]. If \(X\) admits \(C^{2}\)-bump functions, informations on the second subdifferential are also given. An application to a fuzzy sum rule is provided. The techniques are strictly related to those of \textit{R. Deville, G. Godefroy} and \textit{V. Zizler} [``Smoothness and renorming in Banach spaces'' (1993; Zbl 0782.46019)].
scientific article; zbMATH DE number 1145772