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

Abstract

The paper concerns a new method to obtain a direct proof of the openness at linear rate/metric regularity of composite set-valued maps on metric spaces by the unification and refinement of several methods developed somehow separately in several works of the authors. In fact, this work is a synthesis and a precise specialization to a general situation of some techniques explored in the last years in the literature. In turn, these techniques are based on several important concepts (like error bounds, lower semicontinuous envelope of a set-valued map, local composition stability of multifunctions) and allow us to obtain two new proofs of a recent result having deep roots in the topic of regularity of mappings. Moreover, we make clear the idea that it is possible to use (co)derivative conditions as tools of proof for openness results in very general situations.

Metric regularity, fixed points and some associated problems of variational analysis

“…The following corollary which is an essence of the main result of [9] is immediate from Proposition 7.8. Corollary 7.9.…”

“…By 6 It is claimed in [9] that the main result of the paper allows to get fixed point/coincidence theorems similar to those in [1,5,6,12]. This does not seem to be true because the main theorem in [9] is a purely local result, while fixed point theorems in the quoted papers need regularity on a fixed set (with an exception of one theorem in [5]).…”

“…The reason is that a composition of two regular set-valued mappings need not be regular. This brings the question of regularity of a composition into the agenda of variational analysis (see [9,12]). 6 We prove some simple general sufficient conditions for regularity of a composition.…”

Convex Analysis of Minimal Time and Signed Minimal Time Functions

“…This extension of the classical distance function opens up the possibility of dealing with many classes of generalized distances. Various properties of the minimal time functions have been topics of extensive research over the years including very recent developments; see, e.g., [6,7,8,10,11,12,13,20,21,23,24,26,25]. Applications of the minimal time functions to variational analysis, set optimization, facility location problems, and optimal control have been largely addressed in the literature.…”

Stability of Implicit Multifunctions via Point-Based Criteria and Applications

“…Stability analysis in implicit multifunctions and generalized equations has been investigated intensively by many researchers. One of the main problems here is to find sufficient conditions in terms of original data for an implicit multifunction to have a certain stability property such as the lower (upper) semicontinuity/Lipschitzian property [1], the continuity [2,3], the calmness [4], the metric regularity [5][6][7][8][9], the metric subregularity [10,11], the Aubin (known also as Lipschitz-like or pseudo-Lipschitz) property [12][13][14][15], and the Hölder-like property [10,16].…”