Nonlinear Functional Analysis (NMNV402) (Summer Semester 2019/2020)
This course focuses on the basic analysis of nonlinear operators, and abstract numerical methods for solving them. Topics covered will include:
- Basics from functional analysis,
- Abstract existence theorem,
- Fixed point theorems,
- Monotone/Pseudomonotone operators — theory, existence, and abstract numerical methods,
- Potential operators — theory, and abstract numerical methods.
The final exam will consist of an oral examination on the topics covered.
- Lectures:
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Monday 14:00 – 15:30, K433KNM Sokolovská 83 Karlín
Recordings and whiteboard notes for online lectures available here.
- Practicals:
- Thursday 12:20 – 13:50, K5 Sokolovská 83 Karlín
Cancelled Practicals due to Covid-19
- 19.03.2020: Notes & Exercises; Exercise Solutions
- 26.03.2020: Notes & Exercises; Exercise Solutions
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09.04.2020: Notes & Exercises; Exercise Solutions
The sheet previously had a few errors, which have now been corrected:
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The inequality (3) in Theorem 2.17 (Stampacchia) should be:
a(u,v - u) = F(v - u)
Note, this should also be checked in your lectures notes, as it is possible the handwritten notes has a similar mistake
In Exercise 2 (end of page 2) the second statement of three statements to show are equivalent is incorrect. The statements should be:
- α(Au, v - u ≥ α(f, v - u) for v ∈ K
- (αf - αAu, v - u) ≤ 0 for v ∈ K
- u = PK(αf - αAu + u)
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- 23.04.2020: Notes & Exercises; Exercise Solutions
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30.04.2020: Notes & Exercises; Exercise Solutions
Exercise 2(b) had errors in the formula. They should be:
a(t)t ≥ t1+α
and‖un‖ ≤ (L infv ∈ X‖v - un‖)1/α
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07.05.2020: Notes & Exercises; Exercise Solutions
In exercise 1 f2(x) should be f2(x) = 12 (Ax - z, Ax - z).
- 14.05.2020: Notes & Exercises
- 21.05.2020: Notes & Exercises
Suggested Reading:
- V. Dolejší & K. Najzar, Nelineární funkcionální analýza, matfyzpress, 2011.
- E. Ziedler. Nonlinear functional analysis and its applications I, Springer, 1984.