# Math 640: Nonlinear Analysis

### Title

Nonlinear Analysis.

(3:3:0)

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### 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.

1. Differential calculus on normed spaces
• Fréchet derivatives
• Gâteaux derivatives
• Inverse Function theorem
• Implicit Function theorem
• Lyapunov-Schmidt reduction
2. 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 alternative
• Ordered Banach spaces
• Monotone iterative method
• Monotone operators
• Browder-Minty theorem
3. 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

### Textbooks

Possible textbooks for this course include (but are not limited to):

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

1. Differential calculus on normed spaces
• Nash-Moser theorem
2. 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
3. Abstract critical point theory
• Minimax results
• Ky Fan’s minimax inequality
• Ekeland’s variational principle
• Schechter’s bounded mountain pass theorem

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.