Chain Rules for Linear Openness in General Banach Spaces

Metric regularity of composition set-valued mappings: Metric setting and coderivative conditions

“…The paper contains a main result (Theorem 3.6) which is prepared by several propositions having their own interest. In that main result one obtains, under some already standard (hence expected) assumptions (see [23], [24]), the openness of an auxilliary multifunction associated to a composition set-valued map, and on this basis, a result of openness around the reference point for the considered composition. We want to emphasize here two main points both of them revealing the novelty and the relevance of our work.…”

“…In the last years, the study of openness at linear rate (or equivalently metric regularity) of multifunctions obtained as operations with set-valued maps has received a new impetus coming from at least three connected issues: the link between Lyusternik-Graves type theorems and fixed point assertions ( [1], [14]), the growing interest to generalized forms of compositions ( [31], [23]) and the new developments of metric regularity results obtained under assumptions based on generalized differentiation calculus and especially on coderivative conditions ( [52], [20]).…”

“…Secondly, the proof is obtained using Ekeland Variational Principle (EVP, for short), a fact that answers a question A. Ioffe raised in a discussion with the fourth author of this paper: how to get direct proofs for openness results (and, also, for coincidence/fixed points results), on complete metric spaces, using EVP and not arguing by contradiction. In our knowlelge (see, for instance, [63], [14], [23]), in many cases, the proofs relying on EVP are made on normed vector spaces, and reasoning by contradiction. The supplemental structure of the space (i.e., its linear structure, but also the norm), which seems at first glance a little surprising, it is used essentially in the construction of the contradiction.…”

An inverse map result and some applications to sensitivity of generalized equations

“…We refer the reader to the monographs [4,18,19], and to some very recent papers on the subject (see [20,6,7] and the references therein).…”

“…Let us mention, for instance, the study of covering mappings in metric spaces carried out by Arutyunov [5], and the study of openness stability of set-valued maps and the metric regularity behavior of the implicit set-valued map related to a generalized variational system by Durea and Strugariu [6]. Anyway, even if in [6] the viewpoint has been broadened to the case of the sum of set-valued maps, the analysis of the regularity is essentially of local type and it relies on the notion of sum-stable maps (see also [7,8] for very recent results in this setting).…”