A new approach to local analysis of nonsmooth mappings from one Banach space into another is suggested. The approach is essentially based on the use of set-valued mappings of a special kind, called fans, for local approximation. Convex sets of linear operators provide an example of fans. Generally, fans can be considered a natural set-valued extension of linear operators. The first part of the paper presents a study of fans; the second is devoted to calculus and includes extensions of the main theorems of classical calculus.

Metric regularity theory lies at the very heart of variational analysis, a relatively new discipline whose appearance was, to a large extent, determined by the needs of modern optimization theory in which such phenomena as nondifferentiability and set-valued mappings naturally appear. The roots of the theory go back to such fundamental results of the classical analysis as the implicit function theorem, Sard theorem and some others. This paper offers a survey of the state of the art of some principal parts of the theory along with a variety of its applications in analysis and optimization.

This paper is devoted to noninfinitesimal methods in nonlocal regularity theory for set-valued mappings between metric spaces and concentrates on studying two main interconnected topics: noninfinitesimal regularity criteria and fixed-points of set-valued mappings. A number of new results are proved, in particular those which cover and extend to a general metric setting some theorems viewed specifically as Banach-space results. In addition, a special technical interest in studying these two topics together is determined by the fact that each of them exploits a certain sequential iteration scheme (connected with Ekeland's principle in the first and Newton-type iterations in the second) and the extent to which each of the schemes can be effectively applied to the study of the other topic is at least unclear.1. Introduction. This paper gives a systematic account of the phenomenological regularity theory in metric space. The word "phenomenological" is used to emphasize that we shall study regularity phenomena (linear openness, metric regularity, pseudo-Lipschitz property) as such without invoking any infinitesimal mechanisms to define or characterize the properties. The infinitesimal approach is the basis of the classical regularity/transversality theory of smooth single-valued mappings. First studies of the regularity of set-valued or nonsmooth mappings (e.g., [18,32]) were also built along these lines. Subsequent developments of subdifferential calculus placed the infinitesimal regularity theory in Banach spaces on a firm analytical ground; see, e.g., [4,14,22,28,29,34] which together contain much of the available information about the current state of the theory and its numerous applications in optimization and variational analysis.The study of regularity phenomenology as such was motivated by the understanding of the metric nature of the regularity phenomena. The seminal paper by Dmitruk, Milyutin, and Osmolovskii [10] was likely the first publication that stressed the point with full clarity. It dealt exclusively with nonlocal versions of the linear openness (covering), but it is clear that the authors were aware of its equivalence to (or at least very close connection with) what would later in [7] be called "metric regularity," stated in terms of distance estimates like those that, in the context of Banach spaces, had already appeared in [18,26,32]. A related equivalence result (which actually implies even more refined information about the equivalence of linear openness and metric regularity as was later understood), still for Banach spaces, was stated without proof [19, Proposition 11.12]. (For a proof see [21].) Shortly afterwards Aubin introduced the "pseudo-Lipschitz" property (also in Banach spaces) in [3], and a few years later the equivalence of the three regularity properties in a purely metric setting was proved

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Abstract. We introduce and study a new class of subdifferentials associated with arbitrary functions. Among the questions considered are: connection with other derivative-like objects (e.g. derivatives, convex subdifferentials, generalized gradients of Clarke and derívate containers of Warga), calculus of approximate subdifferentials and applications to analysis of set-valued maps and to optimization.It turns out that approximate subdifferentials are minimal (as sets) among other conceivable subdifferentials satisfying some natural requirements. This shows that certain results involving approximate subdifferentials are the best possible and, at the same time, marks certain limitations of nonsmooth analysis. Another important property of approximate subdifferentials is that, being essentially nonconvex, they admit a rich calculus that covers the calculus of convex subdifferentials and leads to more precise and sometimes new results for generalized gradients of Clarke.