Limit superior of subdifferentials of uniformly convergent functions
In this paper the author shows that the G-subdifferential of a lower semicontinuous function \(f : X \to R \cup \{+\infty\}\), where \(X\) is a Banach space, is contained in the superior limit of the G-subdifferentials of a family of lower semicontinuous functions uniformly convergent to \(f\). This result extends to the infinite dimensional case analogous ones established for \(C^2\)-functions by \textit{A. Ya. Kruger} and \textit{B. S. Mordukhovich} [Dokl. Akad. Nauk BSSR 24, 684-687 (1980; Zbl 0449.49015)] and \textit{A. D. Ioffe} [Trans. Am. Math. Soc. 281, 389-416 (1984; Zbl 0531.49014)]. Moreover, an equivalence between this result and the corresponding normal cones formulas for a family of sets which converges in the sense of the bounded Hausdorff distance is proved. Hence, this result also provides an extension of those by \textit{J. Benoist} [C. R. Acad. Sci., Paris, Sér. I 315, 941-944 (1992; Zbl 0772.49009), Set-Valued Anal. 2, No. 1-2, 95-115 (1994; Zbl 0803.49017)] to the infinite-dimensional case and G-normal cones. The approach is of analytic nature, based on the Ekeland variational principle. As a consequence, a characterization of the subdifferential of functions which are bounded from below by a negative quadratic form is derived in terms of their Moreau-Yosida proximal approximations.
scientific article; zbMATH DE number 1278716