Nonlinear analysis
type: Item
instance of: scholarly article
zbMATH Open document ID: 1275.46001
publication date: 2012-07-18T00:00:00Z
Mathematics Subject Classification ID: 46-01
Mathematics Subject Classification ID: 47-01
Mathematics Subject Classification ID: 49-01
Mathematics Subject Classification ID: 49J40
Mathematics Subject Classification ID: 26B25
Mathematics Subject Classification ID: 90C33
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 41A50
zbMATH DE Number: 6058193
zbMATH Keywords: nonlinear analysis
zbMATH Keywords: functional analysis
zbMATH Keywords: convex analysis
zbMATH Keywords: operator theory
zbMATH Keywords: variational inequalities
zbMATH Keywords: fixed point theory
zbMATH Keywords: complementarity problem
author: Sudarsan Nanda
MaRDI profile type: Publication
title: Nonlinear analysis
review text: 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 Chapte…
reviewed by: Werner Kleinert
description: scientific article; zbMATH DE number 6058193