Nonlinear functional analysis.

*(English)*Zbl 0559.47040
Berlin etc.: Springer-Verlag. XIV, 450 p. DM 98.00 (1985).

This interesting book is devoted to the main ideas, concepts and methods of nonlinear functional analysis. The book is written on an elementary level easily and clearly. It is comprehensible for beginners that wish to study nonlinear analysis and its applications either to differential and integral equations or in nonclassical fields (biology, chemistry, economics and so on). At the same time the book will be useful for specialists in the field undoubtedly.

The book contains 10 chapters. Every chapter begins with an introduction in which the author describes its contents and finishes with concluding remarks in which more special results, generalizations and sometimes applications are discussed. A number of exercises in every chapter will help different readers to study the field more completely. Here are the main problems about which the author tells in his book: the theory of topological degree in finite dimensional spaces based on smooth approximations, in particular, the Brower’s fixed point, hedgehog, Borsuk, Jordan’s theorems (chapter 1), the theory of topological degree in infinite dimensional spaces for compact and condensing maps (chapter 2), monotone and accretive operators in Hilbert and Banach spaces (chapter 3), the classical implicit, inverse and open mapping theorems and their recent generalizations for scales of Banach spaces, the Lyapunov-Schmidt methods in the resonance problems (chapter 4), the fixed point theorems for metric spaces and for normed linear spaces using convexity, in particular weakly inward maps (chapter 5); equations with increasing maps and fixed points in cones for positive and nonpositive operators (chapter 6), the theory of approximate solutions to the equations with A-proper maps (chapter 7), the theory of compact, monotone and accretive multivalued maps (chapter 8), necessary and sufficient conditions for a minimum of differentiable functionals under constraints of different types and the theory of critical points for smooth functionals (chapter 9), theorems on local and global bifurcations (chapter 10).

The book contains 10 chapters. Every chapter begins with an introduction in which the author describes its contents and finishes with concluding remarks in which more special results, generalizations and sometimes applications are discussed. A number of exercises in every chapter will help different readers to study the field more completely. Here are the main problems about which the author tells in his book: the theory of topological degree in finite dimensional spaces based on smooth approximations, in particular, the Brower’s fixed point, hedgehog, Borsuk, Jordan’s theorems (chapter 1), the theory of topological degree in infinite dimensional spaces for compact and condensing maps (chapter 2), monotone and accretive operators in Hilbert and Banach spaces (chapter 3), the classical implicit, inverse and open mapping theorems and their recent generalizations for scales of Banach spaces, the Lyapunov-Schmidt methods in the resonance problems (chapter 4), the fixed point theorems for metric spaces and for normed linear spaces using convexity, in particular weakly inward maps (chapter 5); equations with increasing maps and fixed points in cones for positive and nonpositive operators (chapter 6), the theory of approximate solutions to the equations with A-proper maps (chapter 7), the theory of compact, monotone and accretive multivalued maps (chapter 8), necessary and sufficient conditions for a minimum of differentiable functionals under constraints of different types and the theory of critical points for smooth functionals (chapter 9), theorems on local and global bifurcations (chapter 10).

Reviewer: P.Zabreiko

##### MSC:

47Hxx | Nonlinear operators and their properties |

47-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operator theory |

65J15 | Numerical solutions to equations with nonlinear operators (do not use 65Hxx) |