Fundamentals of nonlinear analysis -- 2012

The course introduces abstract mathematical concepts and techniques that are crucial to the fundamental understanding of the behaviour of deterministic dynamical systems in metric spaces in general, finite dimensional Euclidean space and infinite dimensional Banach spaces in particular. This includes systems of nonlinear ordinary differential equations (ODEs), some types of partial differential equations (PDEs) and Volterra-type integral equations, from which we draw various examples, linked to applications. The course intends to provide a thorough background in abstract analysis for students in mathematics interested in continuing with a more detailed study in analysis of continuous-time dynamical systems defined by nonlinear evolutionary equations.

Some of the topics that will be addressed are:

  1. Concepts, tools and techniques for analysis in metric spaces (Lipschitz mappings, Tietze's Extention Theorem, measures of non-compactness, Hausdorff distance, measures on metric spaces, e.g. Hausdorff measures and Hausdorff dimension);
  2. Nonlinear operators (compact maps, inner- and outer superposition operators);
  3. Various fixed point theorems, e.g. Schauder's and Darbo's Fixed Point Theorem;
  4. Dynamical systems in metric spaces (semigroups of transformations, limit sets, attractors, map iterations, existence of solutions to particular type of equations).

Required background knowledge:
Knowledge of the basics of Banach space theory is beneficial (e.g. Linear Analysis or another introductory course in Functional Analysis). Most concepts will be developed self-contained in the course however.

Literature:
The course will be based on parts of several books. The reading material will be a combination of handouts and downloadable course notes. Participants need not purchase a book for the course.

Examination:
Exercises and oral exam.


Course material

An updated syllabus will become available in parts during the course. The final version of the 2009 lecture series is available as pdf here (May 28, 2009).

Assignments

Three sets of exercises will become available during the course. More information on the due-dates will be given later.

Overview discussion topics oral exam

These are listed in the file found here (Dutch).


This page was last updated: 31 May 2012