Links between directional derivatives through multidirectional mean value inequalities
The aim of the article is to extend certain inequalities between Rockafellar's directional derivatives and the mixed lower limit of the lower Dini's derivatives to the general case of lower semi-continuous (lsc) functions on Banach spaces and also the inclusions between the corresponding tangent cones to closed subsets of Banach spaces. To this end, the authors prove first the so called ``multidirectional mean value inequality'' \[ d^\uparrow f(\overline x;\overline v)\leq \lim \sup_{x\to_f\overline x}\inf_{v\to \overline v} d^-f(x;v) \] between the Rockafellar's directional derivative, \(d^\uparrow f(.;.)\), and the lower Dini's derivatives, \(d^-f(.;.)\), and a similar inequality for the weak lower Dini's derivatives, \(d_w^-f(.;.)\); next, as an easy corollary, one obtains in this context the known inclusion \[ \liminf_{x\to_f\overline x}T_B(S;x)\subset T_C(S;\overline x) \] between Bouligand (contingent) cones and Clarke's tangent cones to a closed subset of a Banach space. In the last part of the paper one studies several classes of functions for which the mean value equality holds.
scientific article; zbMATH DE number 5382668
scientific article