NONLINEAR FUNCTIONAL ANALYSIS

This book mainly provides the basic theories and methods of nonlinear functional analysis. The content includes Fréchet differentiation, the implicit function theorem, bifurcation problems, Brouwer degree, Leray–Schauder degree, fixed point theorems, topological degree of cone mappings, the global bifurcation theorem, extremum principles, Ekeland's variational principle, the minimax principle, index, and category.

It consists of three chapters: Nonlinear Operators on Banach Spaces, Topological Degree Theory, and Variational Methods. Some examples are given to explain the applications of these theories, and each chapter contains exercises.

This is a textbook for graduate students and senior undergraduate students.

Contents:

• Preface
• About the Author
• Nonlinear Operators on Banach Spaces:
  • Banach Spaces and Linear Operators
  • The Calculus of Abstract Functions
  • Fréchet Differentiation
  • Gâteaux Differentiation
  • Examples
  • Higher-Order Derivatives and Taylor Formulas
  • Implicit Function Theorem
  • Global Implicit Function Theorem
  • Bifurcation Problem
  • Ordered Banach Spaces
  • Super- and Sub-Solutions Method
  • Mixed Monotone Operators
• Topological Degree Theory:
  • Brouwer Degree
  • Properties of Brouwer Degree
  • Brouwer Fixed Point Theorem and Borsuk Theorem
  • Leray–Schauder Degree
  • Fixed Point Theorem
  • Topological Degree of Cone Mappings
  • The Coincidence Degree
  • Topological Degree of Condensing Fields
  • The Global Bifurcation Theorem
• Variational Methods:
  • Extremum Principle
  • Minimax Principle
  • ℤ₂ Index and Category
• Bibliography
• Index

Readership: Senior undergraduate students and graduate students studying nonlinear problems.