Variational pairs and applications to stability in nonsmooth analysis

Implicit multifunction theorems in complete metric spaces

“…Theorem 12 yields the following corollary whose first part has been established by Azé, Corvellec and Lucchetti in [5] (see also [6], Corollary 5.7).…”

“…This concept goes back to the surjectivity of a linear continuous mapping in the Banach Open Mapping Theorem and to its extension to nonlinear operators known as the Lyusternik & Graves Theorem ( [40], [27], see also [15]) and [21]). For a detailed account the reader is referred to the books or works of many researchers, [3], [5], [9], [10], [11], [12], [13], [17], [18], [20], [22], [29], [30], [32], [34], [36], [37], [40], [42], [43], [41], [44], [45], [46], [50], [51], [52], [57] and the references given therein for many theoretical results on the metric regularity as well as its various applications. Metric regularity or its equivalent notions (covering at a linear rate) [38] or Aubin property of the inverse [1] is now considered as a central concept in modern variational analysis (see the survey paper by Ioffe [34]).…”

“…The metric regularity of implicit multifunctions has been extensively studied by several authors (see, for example, [2], [5], [8], [25], [24], [39], [46], [56] and the references given therein). Almost all implicit multifunction results in the literature are based on conditions on some tangent cones or on some normal cones to the graphs of the multifunctions involved.…”

“…The approach based on error bounds to investigate the metric regularity has been recently used in [6] and in [46] to study implicit multifunctions in smooth spaces. Especially in the survey paper [5], it was shown that this approach is powerful to provide a unified theory of the metric regularity. In this work, we use a different way for deriving implicit multifunction results.…”

Infinitesimal calculus in metric spaces

“…Such a notion has been used with much success for dealing with existence of curves satisfying decrease properties, error bounds estimates and metric regularity (see [8]- [10], [22], [23] and their references). The following comparison is easy.…”