Regularity concepts in nonsmooth analysis. Theory and applications
The book is devoted to the study of the various regularity notions in nonsmooth analysis and encompasses the most recent collection of various concepts, results and applications in this field. The term nonsmooth analysis has been used in the 1970th by F. Clarke in the study of the differential properties of functions and sets, which are not differentiable in the usual sense. The work is divided into three accessible parts. In the first part is presented an interesting introduction to nonsmooth analysis theory. More precisely, in Chapter 1 the main concepts of the theory, as the subdifferential and the tangent and normal cones, are given. Examples and exercises to explain the main concepts and to describe the results are also considered. The second part concerns the most recent results of various regularity concepts of sets, functions, and set-valued mappings. More precisely, in Chapter 2 various concepts of regularity for sets and the relationships between them are studied. In particular, the crucial role of uniform prox-regularity is highlighted. Then, Chapter 3 is dedicated to the different concepts of regularity of functions and their connections. Chapter 4 is devoted to the main concepts of regularity for set-valued mappings, which are of great interest in variational analysis and optimization. In the four chapters of the final section, different applications of the nonsmooth analysis are presented, with the interaction to the existing literature. Chapter 5 is devoted to problems described by first order differential inclusions under many kinds of regularity of the right term and existence of solution theorems are provided. In Chapter 6 some special types of second order differential inclusion, called Second Order Sweeping Processes (SOSP) are studied, both in finite dimensions and in Hilbert spaces under regularity of the right term. The existence for SOSP with delay is also examined. The main purpose of Chapter 7 is to show how the previous regularity concepts can be used in the study of nonconvex variational and quasi-variational inequalities. Finally, in Chapter 8 some applications to economic problems and to the equilibrium theory are presented. The book is supplemented by a precise and almost exhaustive bibliography on the subject. The text is addressed to graduate students and to researchers interested in the applications of nonsmooth analysis.
scientific article; zbMATH DE number 5944083