Difference between revisions of "Math 640: Nonlinear Analysis"
m (moved Math 640 to Math 640: Nonlinear Analysis) 

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Revision as of 15:44, 3 April 2013
Contents
Catalog Information
Title
Nonlinear Analysis.
(Credit Hours:Lecture Hours:Lab Hours)
(3:3:0)
Recommended(?)
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 closelyrelated concepts, and state (and, where feasible, prove) major theorems.
 Differential calculus on normed spaces
 Fréchet derivatives
 Gâteaux derivatives
 Inverse Function theorem
 Implicit Function theorem
 LyapunovSchmidt reduction
 Fixed point theory
 Metric spaces
 Banach’s contraction mapping principle
 Parametrized contraction mapping principle
 Finitedimensional spaces
 Brouwer fixed point theorem
 Normed spaces
 Schauder fixed point theorem
 LeraySchauder alternative
 Ordered Banach spaces
 Monotone iterative method
 Monotone operators
 BrowderMinty theorem
 Metric spaces
 Abstract critical point theory
 Functional properties
 Convexity
 Coercivity
 Lower semicontinuity
 Existence of global minimizers
 Existence of constrained minimizers
 Minimax results
 AmbrosettiRabinowitz mountain pass theorem
 Functional properties
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:
 Differential calculus on normed spaces
 NashMoser theorem
 Fixed point theory
 Metric spaces
 Caristi fixed point theorem
 Hilbert spaces
 BrowderGöhdeKirk theorem
 Ordered Banach spaces
 Krasnoselski’s fixed point theorem
 KreinRutman theorem
 Monotone operators
 HartmanStampacchia 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.