Nonlinear analysis (Q2901228)

The book under review contains the material of a second course in functional analysis, with particular emphasis on various nonlinear aspects. As such, this textbook offers a number of topics for a one-semester course in advanced analysis for graduate students and interested research scholars, assuming a profound knowledge of a first basic course in functional analysis as a general prerequisite. As for the precise contents, the text comprises ten chapters and two appendices, where the material is organized as follows. Chapter 1 introduces the basics of calculus in real Banach spaces, including Gâteaux and Fréchet derivatives of general operators between Banach spaces, the corresponding mean value theorems, the relationship between these two types of derivatives, and the concept of integration of a function on the interval \([0,1]\) with values in a Banach space. Chapter 2 briefly discusses convex sets in real vector spaces, the Minkowski functional, Mazur's geometric form of the Hahn-Banach theorem, supporting hyperplanes for convex subsets, topological properties of convex sets, and convex cones. Chapter 3 is devoted to convex functions in real analysis and their basic topological properties. Along the way, the author discusses convex functionals on convex sets in a normed real vector space and their conjugate functionals, subgradients and subdifferentials, the duality mapping, and the concepts of quasi-convex and pseudo-convex functions as important generalized convex functions. Chapter 4 is more specific and deals with another generalization of convex functions, the so-called invex functions and their related functions. Further topics touched upon in this chapter concern the more recent notions of logarithmic and harmonic convex functions as well as the author's concept of \(g\)-convex functions [J. Indian Math. Soc., New Ser. 72, No. 1--4, 67--74 (2005; Zbl 1134.49011)].NEWLINENEWLINE The theory of best approximation in normed vector spaces is explained in Chapter 5, especially from the viewpoint of strict convexity, whereas Chapter 6 presents some of the classical fixed point theorems and their applications, namely, the fixed point theorems of Banach, Brouwer and Schauder, respectively. At the end of Chapter 6, some of the various generalizations of these classical results are collected in tabular form. Chapter 7 reviews the theory of linear operators in functional analysis, with special regard to bounded linear operators on Hilbert spaces, their spectral theory, and compact linear operators.NEWLINENEWLINE Chapter 8 briefly describes some basic properties of nonlinear operators, emphasizing monotone operators for later use in the study of variational inequalities and the complementarity problem in mathematical optimization. Chapter 9 turns to some basic properties of variational inequalities in the context of certain monotone operators from a reflexive real Banach space \(X\) to its dual space \(X^*\), taking restrictions to closed convex subsets \(K\subset X\) into account. Finally, Chapter 10 gives the mathematical formulation of a so-called complementarity problem with respect to a closed convex cone \(K\) in a reflexive real Banach space \(X\) and its polar cone \(K^*\) in the dual space \(X^*\). Building upon the theory developed in the preceding Chapter 9, solutions to this particular complementarity problem are derived. These results were obtained by the author himself several decades ago, but many of them have not appeared in any textbook until now. An outlook to the finite-dimensional case, which has important applications in optimization, game theory and operations research, concludes the main text of the present book.NEWLINENEWLINE Appendix I provides some additional material on semi-inner-product spaces and variational inequalities on those special complex Banach spaces. This concept was introduced by \textit{G. Lumer} [Trans. Am. Math. Soc. 100, 29--43 (1961; Zbl 0102.32701)] and intensively studied by several authors in the sequel. Among the main results discussed here is a generalized Riesz-Fischer theorem for continuous semi-inner-product spaces.NEWLINENEWLINE Appendix II is meant to present some basic facts on topological vector spaces, in general, and to discuss the fundamental Hahn-Banach theorem in its various forms and generalizations in a thorough manner.NEWLINENEWLINE The entire text is enriched by numerous motivating and instructive examples, but unfortunately no accompanying exercises have been given to help the reader to test her/his understanding of the abstract matter. On the other hand, the presentation is very concise, utmost clear and streamlined, complete and rigorous, essentially self-contained and exceptionally diversified.NEWLINENEWLINE Altogether, the book under review offers a highly interesting, very special and certainly useful second course in (nonlinear) functional analysis. In this regard, the present text should be seen as a valuable complement to the existing literature in the field.