Math 640: Nonlinear Analysis

Catalog Information

Title

Nonlinear Analysis.

(Credit Hours:Lecture Hours:Lab Hours)

(3:3:0)

Offered

W

Recommended(?)

Math 540.

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):

Additional topics

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
     • Rabinowitz saddle point theorem
     • Rabinowitz linking 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.