Topics in nonlinear analysis and applications. (English) Zbl 0878.47040

Hyers, Donald H.; Isac, George; Rassias, Themistocles M.

Singapore: World Scientific. xiii, 699 p. (1997).

This book is a treatise devoted to five different but relating to each other branches of modern Nonlinear Analysis: Cones and Complementary Problems, Metrics on Convex Cones, Zero-Epi Mappings, Variational Principles, and Maximal Elements Principles. In each chapter, the authors describe numerous results related to the name of the chapter, the basic ideas and methods which lie in the base of these results, and some new mathematical models and new applied problems which are close to them. Each chapter begins with an introduction, and ends with a list of references; symbol and subject indices are common to all chapters.
Chapter 1 deals with the theory of cones in locally convex spaces and the so-called complementary problems, which is a cross-point of several branches in nonlinear analysis now. In the first part the concepts of normal, regular and completely regular, well-based (plasterable) etc. cones are presented, in the second one the big list of different mathematical and applied (engineering and economical) problems leading to complementary problems, different existence results for complementary problems, and so on. Chapter 2 is a nice account of different results concerning Hilbert’s and Tompson’s metrics for rays and elements of cones and some their modifications and generalizations including the analysis of operators which are contractions and general contractions with respect to these metrics and also the existence results (of fixed points and eigenvalues) for these operators without the compactness property (to big regret, the authors seem not to have acquaintance with some nice and important results of M. A. Krasnosel’skiĭ and his coauthors about focusing and acute operators). Chapter 3 presents different existence results in terms of the so-called zero-epi mappings; in particular, the existence theorem of Nash equilibrium for a couple of mappings and some existence results for nonlinear complementary problems are given. One can remark that the notion of zero-epi mapping can be reduced to the usual notion of a vector field with nonzero rotation (index or degree) on the boundary of a suitable domain (M. A. Krasnosel’skiĭ) and coincides with the notion of an essential mapping (A. Granas). Chapter 4 deals with the so-called variational principles and their modifications and generalizations. The authors explain the equivalence to each other of the Ekeland’s variational principle, the drop theorem, the Caristi-Kirk fixed point theorem, the flower petal theorem, and so on. Among the generalizations the reader of this chapter can find the Borwein \(\varepsilon\)-principle, the Borwein-Preiss principle, the Deville-Godefroy-Zizler principle, the variational principle proposed A. I. Ioffe, etc.
The second part of the chapter is devoted to applications to fixed point theory, some density results, theory of Fredholm solvability, geometry of Banach spaces, etc. The final chapter is a small but deep survey of different variants of maximal elements principles, containing a wide spectrum of results beginning from variations of Zorn’s lemma, and ending the existence results for Pareto extremal elements. In general, in this book, the authors have gathered numerous and interesting results in nonlinear analysis and presented them from the united point of view. The book will unboubtedly be useful to all specialists and beginners in nonlinear analysis as a rich reference book and well as a source of new problems and ideas.

Reviewer: P.Zabreiko (Minsk)

MSC:

47Hxx	Nonlinear operators and their properties
47-02	Research exposition (monographs, survey articles) pertaining to operator theory
49-02	Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
49J40	Variational inequalities

Keywords:

cones; complementary problems; metrics on convex cones; zero-epi mappings; variational principles; maximal elements principles; plasterable; index; completely regular; well-based; fixed points; eigenvalues; Nash equilibrium; vector field with nonzero rotation; degree; essential mapping; drop theorem; Caristi-Kirk fixed point theorem; flower petal theorem; Borwein-Preiss principle; Deville-Godefroy-Zizler principle; Zorn’s lemma