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Lecturer(s)
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Tomeček Jan, doc. RNDr. Ph.D.
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Rachůnková Irena, prof. RNDr. DrSc.
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Course content
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1.Classical boundary value problems and their operator forms. (Dirichlet boundary value problem for ODEs with continuous nonlinearities, functional spaces and operators, construction of operator equations in the presence of non-resonance.) 2.Resonance. (Classical periodic boundary value problem at resonance, elimination of resonance, construction of operator equations.) 3.Carathéodory boundary value problems and their operator forms. (Dirichlet boundary value problem for ODE with Carathéodory nonlinearities, functional spaces and operators, construction of operator equations.) 4.Application of Schauder Fixed Point Theorem. (Completely continuous operators in operator equations, existence of solutions of boundary value problems.) 5.Method of lower and upper functions. (Well ordered couple of lower and upper function, generalization of theorems about the existence of solutions to boundary value problems.) 6.Method of a priori estimates. (A priori estimates of solutions to boundary value problem based on sign and growth conditions, generalization of theorems about the existence of solutions to boundary value problems.) 7.Topological degree of mappings. (Brouwer degree on open balls, Brouwer degree on open bounded sets, Leray-Schauder degree on open bounded sets.) 8.Leray-Schauder Theorem and its application. (Existence principles for boundary value problems.) 9.Proofs of existence theorems for particular boundary value problems.
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Learning activities and teaching methods
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Lecture, Dialogic Lecture (Discussion, Dialog, Brainstorming), Work with Text (with Book, Textbook)
- Attendace
- 26 hours per semester
- Preparation for the Course Credit
- 35 hours per semester
- Homework for Teaching
- 30 hours per semester
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Learning outcomes
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Understand the theory and methods of investigation of boundary value problems.
Application Apply topological and analytical methods to investigation of regular and singular boundary value problems.
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Prerequisites
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Knowledge of Functional Analysis and Ordinary Differential Equations.
KMA/NLFA and KMA/ODR3
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Assessment methods and criteria
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Student performance
Credit: active participation in classes and seminars.
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Recommended literature
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Handbook of Differential Equations I, II, III.
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D. O'Regan. (1997). Existence Theory for Nonlinear Ordinary Differential Equations. Kluwer, Dordrecht.
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I.Rachůnková, S.Staněk, M.Tvrdý. (2008). Solvability of Nonlinear Singular Problems for Ordinary Differential Eqautions. New York.
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J. Kurzweil. (1978). Obyčejné diferenciální rovnice.. SNTL, Praha.
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V. Lakshmikantham, D. D. Bainov, P. S. Simeonov. (1989). Theory of Impulsive Differential Equations. World Scientific.
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