General information

Course type AMUPIE
Module title Introduction to Nonlinear Analysis
Language English
Module lecturer prof. UAM dr hab. Daria Bugajewska
Lecturer's email dbw@amu.edu.pl
Lecturer position Prof. UAM
Faculty Faculty of Mathematics and Computer Science
Semester 2026/2027 (winter)
Duration 60
ECTS 6
USOS code 06-DWANLM0-E

Timetable

Module aim (aims)

The Introduction to Nonlinear Analysis course focuses primarily on the applications of mathematical analysis to selected problems in nonlinear analysis, including the Laplace transform, functions of bounded variation, periodic and almost periodic functions. In particular, the course provides a basis for studying more advanced topics in  nonlinear analysis as well as in partial differential equations, harmonic analysis and applied mathematics.

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

One has to have basic knowledge in mathematical analysis and ordinary differential equations.

Syllabus

  1. Laplace transform (definition, existence, continuity, examples).
  2. Basic properties of the Laplace transform.
  3. Further properties of the Laplace transform.
  4. Inverse of the Laplace transform (definition and properties).
  5. Applications of the Laplace transform to differential equations.
  6. Functions of bounded variation in the sense of Jordan (definition, basic properties).
  7. Jordan Decomposition Theorem, Helly's Selection Theorem, further properties.
  8. Space of functions of bounded variation in the sense of Jordan.
  9. Riemann-Stieltjes integral in context of bounded variation functions.
  10. Bounded variation in the sense of Wiener.
  11. Periodic functions and their properties.
  12. Approximation type theorems connected with periodic functions.
  13. Almost periodic functions in the sense of Bohr (definition and basic properties).
  14.  Bochner's Test of uniform almost periodicity.
  15. Mean value theorems for periodic and almost periodic functions.

Reading list

  1. 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.
  2. M. Borkowski, D. Bugajewska and P. Kasprzak, Selected topics in nonlinear analysis, Lecture Notes in Nonlinear Analysis, vol. 19, Nicolaus Copernicus University, Toruń, 2021.
  3. G.Doetsch, Introduction to the theory and application of the Laplace transformation, Springer-Verlag, 1970.
  4. J. Schiff, The Laplace transform, theory and applications, Springer-Verlag, New York, 1997.
  5. S. Stoiński, Almost periodic functions, Scientific Publisher UAM, Poznań, 2008.