Locally Lipschitz set-valued maps on topological vector spaces and surjectivity theorems (Q1373074)

A subject of great importance in optimization theory is the study of set-valued maps and their properties. Being intermediate between continuity and differentiability, the Lipschitz properties take on a special significance. In Section 1, a definition of Lipschitz set-valued maps in topological vector spaces is given. In Section 2, a surjectivity theorem is proved for Lipschitz approachable set-valued maps defined on a locally convex space (i.e., for those maps which can be Lipschitz approximated by a continuous map having a positive constant of surjection). As consequences of the theorem, we obtain again the known results on surjectivity of \textit{M. Fabian} and \textit{D. Preiss} [Commentat. Math. Univ. Carol. 28, 311-324 (1987; Zbl 0625.46052)] and \textit{P. C. Duong} and \textit{H. Tuy} [Acta Math. Vietnam 3, 89-105 (1978), per. bibl.] in Banach spaces.