## General information

Course type | AMUPIE |

Module title | Selected Topics Of Nonlinear Analysis |

Language | English |

Module lecturer | prof. dr hab. Dariusz Bugajewski |

Lecturer's email | ddbb@amu.edu.pl |

Lecturer position | Professor |

Faculty | Faculty of Mathematics and Computer Science |

Semester | 2022/2023 (summer) |

Duration | 60 |

ECTS | 6 |

USOS code | 06-DINAUM0-E |

## 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.