0000000001005907
AUTHOR
Taron Zakaryan
Contribution to variational analysis : stability of tangent and normal cones and convexity of Chebyshev sets
The aim of this thesis is to study the following three problems: 1) We are concerned with the behavior of normal cones and subdifferentials with respect to two types of convergence of sets and functions: Mosco and Attouch-Wets convergences. Our analysis is devoted to proximal, Fréchet, and Mordukhovich limiting normal cones and subdifferentials. The results obtained can be seen as extensions of Attouch theorem to the context of non-convex functions on locally uniformly convex Banach space. 2) For a given bornology β on a Banach space X we are interested in the validity of the following "lim inf" formula (…).Here Tβ(C; x) and Tc(C; x) denote the β-tangent cone and the Clarke tangent cone to …
The validity of the “liminf” formula and a characterization of Asplund spaces
Abstract We show that for a given bornology β on a Banach space X the following “ lim inf ” formula lim inf x ′ ⟶ C x T β ( C ; x ′ ) ⊂ T c ( C ; x ) holds true for every closed set C ⊂ X and any x ∈ C , provided that the space X × X is ∂ β -trusted. Here T β ( C ; x ) and T c ( C ; x ) denote the β-tangent cone and the Clarke tangent cone to C at x. The trustworthiness includes spaces with an equivalent β-differentiable norm or more generally with a Lipschitz β-differentiable bump function. As a consequence, we show that for the Frechet bornology, this “ lim inf ” formula characterizes in fact the Asplund property of X. We use our results to obtain new characterizations of T β -pseudoconve…