Selected Topics Of Nonlinear Analysis

Timetable

Module aim (aims)

Nonlinear functional analysis is a broad part of modern mathematics which is still being
 rapidly developed. That development is closely connected with its numerous applications
 in various branches of science. The main goal of this course is to present five
selected topics of nonlinear functional analysis which are currently investigated by
many mathematicians. These topics contain fixed points theorems, measures of
noncompactness, hyperconvex metric spaces, functions of bounded variation and almost
periodic functions.

Some classical results as well as some new achievements connected with these topics will
 be presented.

Pre-requisites in terms of knowledge, skills and social competences (where relevant)

One has to have basic knowledge in mathematical analysis, theory of differential
equations as well as topology, especially metric topology. Moreover, some knowledge of
basic definitions and facts of functional analysis would be also useful.

Syllabus

Week 1:            Fixed point theorems: Schauder’s theorem.       

Week 2:            Fixed point theorems: Krasnoselski’s theorem, nonlinear alternative,
Leray- Schauder’s       alternative.   

Week 3:            Fixed point theorems: nonlinear alternative, Leray- Schauder’s
alternative.           

Week 4:            Measures of noncompactness: Kuratowski’s measures of noncompactness
and its properties.

Week 5:            Applications of measures of noncompactness in the theory of ordinary
differential equations.       

Week 6:            Applications of measures of noncompactness in the fixed point theory.

Week 7:            Hyperconvex metric spaces: basic properties.   

Week 8:            Hyperconvex metric spaces: examples.

Week 9:            Fixed point theorems in hyperconvex metric spaces.      

Week 10: Functions of bounded variation: ?-variation and its properties.  

Week 11: Functions of bounded variation: convolution operators and superposition
operators.      

Week 12: Functions of bounded variation:  ?BV-solutions to differential and integral
equations.

Week 13: Almost periodic functions: properties.

Week 14: Almost periodic functions: superposition operators and convolution operators.   

Week 15: Mean value of almost periodic functions           and applications of such
functions.

 

Reading list

- J. Appell, J. Banaś, and N. Merentes, Bounded Variation and Around, De Gruyter Studies
 in Nonlinear Analysis and Applications, no. 17, De Gruyter, Berlin, 2014.
- M. Borkowski, Theory of Hyperconvex Metric Spaces. A Beginner's Guide, Lecture Notes
in Nonlinear Analysis, vol. 14, 2015.
-M.Borkowski, D.Bugajewska and P.Kasprzak, Selected Topics in Nonlinear Analysis, Lecture Notes in Nonlinear Analysis, vol.19, Toruń,  2021.
- J. Dugundji and A. Granas, Fixed point theory, Springer, 2003.
- S. Stoiński, Almost periodic functions, Scientific Publisher UAM, Poznań, 2008.