Infopaket

Modern problems of analysis

ДВС.1.01

Вибіркова дисципліна для ОП

First

2022/2023

6 Semester

5

LO 8. Combine mathematical and computer modeling methods with informal procedures of expert analysis to find optimal solutions. LO 19. Collect and interpret relevant data and analyze complexities within their specialization to make judgments that reflect relevant social and ethical issues. PLO 21.1. To know the main sections of applied mathematics and informatics, to the extent necessary for mastering general professional mathematical disciplines, applied disciplines and the use of their methods in the chosen profession.

Full-time form

1. Successful completion of the course: mathematical analysis, functional analysis, linear algebra. 2. Know: basic concepts and facts of mathematical analysis, functional analysis and linear algebra. 3. Be able to: solve typical problems in mathematical analysis, functional analysis and linear algebra. 4. Have basic skills of searching for information on the Internet.

Part 1. Metric theory of fixed points 1 Topic 1. Compression operators. 2 Topic 2. Non-extending operators. 3 Topic 3. Browder's theorem. 4 Topic 4. The Krasnoselsky-Mann method. 5 Topic 5. Halpern's method. 6 Topic 6. Ergodic theorems. 7 Topic 7. Methods of finding a common point. 8 Topic 8. The Douglas-Ratchford method. 9 Control work 1. Part 2. Elements of nonlinear analysis 10 Topic 9. Theorems of Brouwer and Schauder. 11 Topic 10. Kakutani's theorem. 12 Topic 11. Nash equilibrium. 13 Topic 12. Basic concepts of the theory of monotone operators, Minty's lemma. 14 Topic 13. Variational inequalities, Brezis theorem. 15 Topic 14. Methods of solving variational inequalities with monotone operators. 16 Topic 15. Gradient systems. 17 Topic 16. Application in mathematical programming. 18 Topic 17. Incorrect problems and the Tikhonov regularization method.

1. Bauschke H.H., Combettes P.L. Convex Analysis and Monotone Operator Theory in Hilbert Spaces. – Springer, 2011. 2. Goebel K., Kirk W.A. Topics in metric fixed point theory. – Cambridge University Press, 1990. 3. Oben Zh.-P., Ekland I. Prikladnoi nelineinyi analiz. – M.: Mir, 1988. 4. Kinderlerer D., Stampakk-ia G. Vvedenie v variatsionnye neravenstva i ikh prilozheniia. – Moskva: Mir, 1983. 5. Berezanskii Iu.M., G.F.Us, Sheftel- Z.G. Funktsional-nyi analiz. - K.: Vishcha shkola, 1990. - 600 s. 6. Gaevskii Kh., Greger K., Zakharias K. Nelineinye operatornye uravneniia i operatornye differentsial-nye uravneniia. – M.: Mir, 1978. 7. Iosida K. Funktsional-nyi analiz. – M.: Mir, 1967. 8. Sea Zh. Optimizatsiia. Teoriia i algoritmy. – M.: Mir, 1973. 9. Tikhonov A.N., Arsenin V.Ia. Metody resheniia nekorrektnykh zadach. – M.: Nauka, 1979. 10. Ekland I., Temam R. Vypuklyi analiz i variatsionnye problemy. – M.: Mir, 1979.

Lectures, consultations, independent work

The maximum number of points that can be obtained by a student: 100/60 points. - semester assessment: 1. Control work 1: RN 1.1., RN 1.2, RN1.3, RN 2.1, RN3.1 – 30 points/18 points. 2. Control work 2: RN 1.1., RN 1.2, RN1.3, RN 2.1, RN3.1 – 30 points/18 points. - final evaluation (in the form of an exam): - the maximum number of points that can be obtained by a student: 40 points; - learning outcomes that will be evaluated: PH1.1, PH1.2, PH1.3, PH2.1; - form of implementation and types of tasks: written.

Ukrainian