Nonlinear Functional Analysis

Gerald Teschl

Abstract
This manuscript provides a brief introduction to nonlinear functional analysis.

We start out with calculus in Banach spaces, review differentiation and integration, derive the implicit function theorem (using the uniform contraction principle) and apply the result to prove existence and uniqueness of solutions for ordinary differential equations in Banach spaces.

Next we introduce the mapping degree in both finite (Brouwer degree) and infinite dimensional (Leray-Schauder degree) Banach spaces. Several applications to game theory, integral equations, and ordinary differential equations are discussed.

As an application we consider partial differential equations and prove existence and uniqueness for solutions of the stationary Navier-Stokes equation.

Finally, we give a brief discussion of monotone operators.

MSC: 46-01, 47H10, 47H11, 58Fxx, 76D05
Keywords: Mapping degree, fixed point theorems, differential equations, Navier-Stokes equation.

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Table of contents
  1. Analysis in Banach spaces
    1. Differentiation and integration in Banach spaces
    2. Contraction principles
    3. Ordinary differential equations
  2. The Brouwer mapping degree
    1. Introduction
    2. Definition of the mapping degree and the determinant formula
    3. Extension of the determinant formula
    4. The Brouwer fixed point theorem
    5. Kakutani's fixed point theorem and applications to game theory
    6. Further properties and extensions
    7. The Jordan curve theorem
  3. The Leray-Schauder mapping degree
    1. The mapping degree on finite dimensional Banach spaces
    2. Compact operators
    3. The Leray-Schauder mapping degree
    4. The Leray-Schauder principle and the Schauder fixed point theorem
    5. Applications to integral and differential equations
  4. The stationary Navier-Stokes equation
    1. Introduction and motivation
    2. An insert on Sobolev spaces
    3. Existence and uniqueness of solutions
  5. Monotone operators
    1. Monotone operators
    2. The nonlinear Lax--Milgram theorem
    3. The main theorem of monotone operators

    Bibliography
    Glossary of notations
    Index