Many nonlinear analysis problems have their roots in geometry，astronomy，fluid and elastic mechanics，physics，chemistry，biology，control theory，image processing and economics. The theories and methods in nonlinear analysis stem from many areas of mathematics： Ordinary drfferential equations，partial differential equations，the calculus of variations，dynamical systems，differential geometry，Lie groups，algebraic topology，linear and nonlinear functional analysis，measure theory，harmonic analysis，convex analysis，game theory，optimization theory，etc. Amidst solving these problems，many branches are intertwined，thereby advancing each other.
The book is the result of many years of revision of the authors lecture notes. Some of the more involved sections were originally used in seminars as introductory parts of some new subjects. However，due to their importance，the materials have been reorganized and supplemented，so that they may be more valuable to the readers.

1 Linearization
1.1 Differential Calculus in Banach Spaces
1.1.1 Frechet Derivatives and Gateaux Derivatives
1.1.2 Nemytscki Operator
1.1.3 High-Order Derivatives
1.2 Implicit Function Theorem and Continuity Method
1.2.1 Inverse Function Theorem
1.2.2 Applications
1.2.3 Continuity Method
1.3 Lyapunov-Schmidt Reduction and Bifurcation
1.3.1 Bifurcation
1.3.2 Lyapunov-Schmidt Reduction
1.3.3 A Perturbation Problem
1.3.4 Gluing
1.3.5 Transversality 1  Linearization
  1.1  Differential Calculus in Banach Spaces
    1.1.1  Frechet Derivatives and Gateaux Derivatives
    1.1.2  Nemytscki Operator
    1.1.3  High-Order Derivatives
  1.2  Implicit Function Theorem and Continuity Method
    1.2.1  Inverse Function Theorem
    1.2.2  Applications
    1.2.3  Continuity Method
  1.3  Lyapunov-Schmidt Reduction and Bifurcation
    1.3.1  Bifurcation
    1.3.2  Lyapunov-Schmidt Reduction
    1.3.3  A Perturbation Problem
    1.3.4  Gluing
    1.3.5  Transversality
  1.4  Hard Implicit Function Theorem
    1.4.1  The Small Divisor Problem
    1.4.2 Nash-Moser Iteration
2  Fixed-Point Theorems
  2.1  Order Method
  2.2  Convex Function and Its Subdifferentials
    2.2.1  Convex Functions
    2.2.2  Subdifferentials
  2.3  Convexity and Compactness
  2.4  Nonexpansive Maps
  2.5  Monotone Mappings
  2.6  Maximal Monotone Mapping
3  Degree Theory and Applications
  3.1  The Notion of Topological Degree
  3.2  Fundamental Properties and Calculations of Brouwer Degrees
  3.3  Applications of Brouwer Degree
    3.3.1  Brouwer Fixed-Point Theorem
    3.3.2  The Borsuk-Ulam Theorem and Its Consequences
    3.3.3  Degrees for S1 Equivariant Mappings
    3.3.4  Intersection
  3.4  Leray-Schauder Degrees
  3.5  The Global Bifurcation
  3.6  Applications
    3.6.1  Degree Theory on Closed Convex Sets
    3.6.2  Positive Solutions and the Scaling Method
    3.6.3  Krein-Rutman Theory for Positive Linear Operators
    3.6.4  Multiple Solutions
    3.6.5  A Free Boundary Problem
    3.6.6  Bridging
  3.7  Extensions
    3.7.1  Set-Valued Mappings
    3.7.2  Strict Set Contraction Mappings and Condensing Mappings
    3.7.3  Fredholm Mappings
4  Minimization Methods
  4.1  Variational Principles
    4.1.1  Constraint Problems
    4.1.2  Euler-Lagrange Equation
    4.1.3  Dual Variational Principle
  4.2  Direct Method
    4.2.1  Fundamental Principle
    4.2.2  Examples
    4.2.3  The Prescribing Gaussian Curvature Problem and the Schwarz Symmetric Rearrangement
  4.3  Quasi-Convexity
    4.3.1  Weak Continuity and Quasi-Convexity
    4.3.2  Morrey Theorem
    4.3.3  Nonlinear Elasticity
  4.4  Relaxation and Young Measure
    4.4.1  Relaxations
    4.4.2  Young M

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