# Math 640: Nonlinear Analysis

## Contents

## Catalog Information

### Title

Nonlinear Analysis.

### (Credit Hours:Lecture Hours:Lab Hours)

(3:3:0)

### Prerequisite

### Description

Differential calculus in normed spaces, fixed point theory, and abstract critical point theory.

## Desired Learning Outcomes

This course is intended as a natural nonlinear sequel to Math 540. Like its prequel, the focus would be on operators on abstract Banach spaces.

### Prerequisites

Students need to have a good understanding of basic linear analysis, whether this comes from taking the Math 540 or some other way.

### Minimal learning outcomes

Students should obtain a thorough understanding of the topics listed below. In particular they should be able to define and use relevant terminology, compare and contrast closely-related concepts, and state (and, where feasible, prove) major theorems.

- Differential calculus on normed spaces
- Fréchet derivatives
- Gâteaux derivatives
- Inverse Function theorem
- Implicit Function theorem
- Lyapunov-Schmidt reduction

- Fixed point theory
- Metric spaces
- Banach’s contraction mapping principle
- Parametrized contraction mapping principle

- Finite-dimensional spaces
- Brouwer fixed point theorem

- Normed spaces
- Schauder fixed point theorem
- Leray-Schauder principle
- Leray-Schauder alternative

- Ordered Banach spaces
- Monotone iterative method

- Monotone operators
- Browder-Minty theorem

- Metric spaces
- Abstract critical point theory
- Functional properties
- Convexity
- Coercivity
- Lower semi-continuity

- Existence of global minimizers
- Existence of constrained minimizers
- Minimax results
- Ambrosetti-Rabinowitz mountain pass theorem

- Functional properties

### Additional topics

In addition to the minimal learning outcomes above, instructors should give serious consideration to covering the following specific topics:

- Differential calculus on normed spaces
- Nash-Moser theorem

- Fixed point theory
- Metric spaces
- Caristi fixed point theorem

- Hilbert spaces
- Browder-Göhde-Kirk theorem

- Ordered Banach spaces
- Krasnoselski’s fixed point theorem
- Krein-Rutman theorem

- Monotone operators
- Hartman-Stampacchia theorem

- Metric spaces
- Abstract critical point theory
- Minimax results
- Ky Fan’s minimax inequality
- Ekeland’s variational principle
- Schechter’s bounded mountain pass theorem
- Rabinowitz saddle point theorem
- Rabinowitz linking theorem

- Minimax results

Furthermore, it is anticipated that instructors will want to motivate the abstract theory by considering appropriate concrete examples.

### Courses for which this course is prerequisite

It is proposed that this course be a prerequisite for Math 647.