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Course, academic year 2019/2020
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Nonlinear Functional Analysis 1 - NMMA501
Title in English: Nelineární funkcionální analýza 1
Guaranteed by: Department of Mathematical Analysis (32-KMA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2019 to 2019
Semester: winter
E-Credits: 5
Hours per week, examination: winter s.:2/2 C+Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: English, Czech
Teaching methods: full-time
Guarantor: prof. RNDr. Stanislav Hencl, Ph.D.
Class: M Mgr. MA
M Mgr. MA > Povinné
Classification: Mathematics > Functional Analysis
Annotation -
Last update: T_KMA (02.05.2013)
Mandatory course of the master study branch Mathematical analysis. Recommended for the second year of master studies. Content: differential calculus in Banach spaces, implicit function theorem, calculus of variations.
Literature - Czech
Last update: T_KMA (02.05.2013)

P. Drábek, J. Milota: Methods of nonlinear analysis. Applications to differential equations. Birkhäuser Verlag, Basel, 2007.

B. Dacorogna: Direct methods in the calculus of variations. Springer, New York, 2008

Syllabus -
Last update: doc. Mgr. Petr Kaplický, Ph.D. (09.06.2015)

1. differential calculus in Banach spaces

directional derivative, differential, mean value theorems, chain rule, Taylor formula

2. Inverse mapping theorem, Implicit function theorem

3. Extrema

local extrema, Fermat's condition, Euler-Lagrange equation, Lagrange sufficient and necessary conditions for local extrema, Lagrange multiplier theorem

4. Application on Nemitskii operator and integral functionals

Caratheodory function, measurability of composed function, continuity and boundedness of Nemitskii operator from $L^p$ to $L^q$

5. Direct methods in the Calculus of Variations

convexity and weak lower semicontinuity, basic theorem of Calculus of Variations

6. Counterexamples to the existence of a minimizer

7. Classical problems of the Calculus of Variations (briefly)

8. Degree of a mapping

uniqueness and existence of the degree in finite dimension, Sard's theorem,

Brouwer theorem, Borsuk theorem

9. Leray-Schauder degree

definition, Schauder fixed point theorem, Leray-Schauder index of an isolated solution

10. Monotone operators in Hilbert space

continuous, monotone and weakly coercive operators, monotone operators in reflexive spaces (briefly)

Entry requirements -
Last update: prof. RNDr. Jan Malý, DrSc. (02.05.2018)

Elements of linear functional analysis, elements of measure theory, theory of Lebesgue integral, function spaces.

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