Math 640: Nonlinear Analysis - Revision history

Ls5: /* Catalog Information */

2019-11-14T17:35:47Z

‎Catalog Information

Ls5: moved Math 640 to Math 640: Nonlinear Analysis

2013-04-03T22:44:24Z

moved Math 640 to Math 640: Nonlinear Analysis

Cpg at 21:48, 3 August 2011

2011-08-03T21:48:35Z

Cpg: /* Minimal learning outcomes */

2010-07-28T16:37:43Z

‎Minimal learning outcomes

Cpg at 18:57, 4 June 2009

2009-06-04T18:57:10Z

Cpg at 00:16, 16 August 2008

2008-08-16T00:16:37Z

Cpg: New page: == Catalog Information == === Title === Nonlinear Analysis. === (Credit Hours:Lecture Hours:Lab Hours) === (3:3:0) === Prerequisite === Math 540. === Description === Differential ca...

2008-08-16T00:15:51Z

New page: == Catalog Information == === Title === Nonlinear Analysis. === (Credit Hours:Lecture Hours:Lab Hours) === (3:3:0) === Prerequisite === Math 540. === Description === Differential ca...

New page

== Catalog Information ==

=== Title ===
Nonlinear Analysis.
=== (Credit Hours:Lecture Hours:Lab Hours) ===
(3:3:0)

=== Prerequisite ===
[[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.

<div style="-moz-column-count:2; column-count:2;">

# 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
# 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

</div>

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

<div style="-moz-column-count:2; column-count:2;">

# 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
# 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

</div>

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]].

[[Category:Courses]]

← Older revision		Revision as of 00:16, 16 August 2008
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	It is proposed that this course be a prerequisite for [[Math 647]].		It is proposed that this course be a prerequisite for [[Math 647]].

−	[[Category:Courses]]	+	[[Category:Courses\|640]]

@@ Line 3: / Line 3: @@
 === Title ===
 Nonlinear Analysis.
 === (Credit Hours:Lecture Hours:Lab Hours) ===
 (3:3:0)
 === Recommended(?) ===
@@ Line 11: / Line 15: @@
 === 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.

← Older revision	Revision as of 22:44, 3 April 2013
(No difference)

@@ Line 37: / Line 37: @@
 #* Normed spaces
 #** Schauder fixed point theorem
 #** Leray-Schauder alternative
 #* Ordered Banach spaces

@@ Line 52: / Line 52: @@
 #* Minimax results
 #** Ambrosetti-Rabinowitz mountain pass theorem
 </div>
 === Additional topics ===