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Properties of sequence spaces and their applications in the theory of nonlinear difference equations
Kosík, Jindřich ; Šremr, Jiří (referee) ; Řehák, Pavel (advisor)
The goal of this thesis is a detailed elaboration on apparatus of functional analysis for study of qualitative properties of solutions of difference equations and its application for analysis of a specific nonlinear difference equation. The thesis includes detailed analysis of some properties of sequence spaces, discrete versions of Levi's monotone convergence theorem and Lebesgue's dominated convergence theorem and criteria for relative compactness of sequence spaces. Theoretical apparatus is completed with fixed point theorems. Introduced mathematical instruments are later used for study of a concrete nonlinear difference equation.
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Functional analysis and the mathematical pendulum
Čaputa, Daniel ; Šremr, Jiří (referee) ; Řehák, Pavel (advisor)
This thesis is focused on existence of periodic solutions of nonlinear model of mathematical pendulum with continuous, odd and periodic forcing term. In thesis, the differential equation of motion of pendulum is derived and the associated boundary value problem is rewritten as the integral equation. This equation is considered in a wider set of integral equations (Hammerstein equations). Fixed point theorems are applied on these equations what results in existence and uniqueness of solution. These results are applied on model of mathematical pendulum and the condition for uniqueness of solution is deeper discussed.
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Hausdirff metric and its application in fractals
Roháľ, Branislav Ján ; Hušek, Miroslav (advisor) ; Pyrih, Pavel (referee)
Title: Hausdorff metric and its application in fractals Author: Branislav Ján Roháľ Department: Department of Mathematical Analysis Supervisor: prof. RNDr. Miroslav Hušek, DrSc., Department of Mathematical Analysis Abstract: In this thesis we focus on the themes naturally connected with the con- cept of a fractal. In the first part of the thesis we pay attention to Banach fixed point theorem and to the Hausdorff metric which are later used when studying self-similar sets. There are included parts on the Hausdorff, similarity, and box- counting dimension, too. In the second part of the thesis the new approaches to fractal dimension and some their properties are refered. We introduce generaliza- tion of this concept for any space admitting a fractal structure and for a distance space where also the "size" of sets on each level of fractal structure is considered. In the last chapter the contribution of new approache is demonstrated, - this enables defining the notion needed and counting fractal dimension where it was not possible under the classical approaches, too. Application to the domain of words and counting of dimensions of a language generated by a regular expresion are presented. Keywords: Hausdorff metric, Banach fixed point theorem, self-similar set, Hausdorff dimension, fractal dimension
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Properties of sequence spaces and their applications in the theory of nonlinear difference equations
Kosík, Jindřich ; Šremr, Jiří (referee) ; Řehák, Pavel (advisor)
The goal of this thesis is a detailed elaboration on apparatus of functional analysis for study of qualitative properties of solutions of difference equations and its application for analysis of a specific nonlinear difference equation. The thesis includes detailed analysis of some properties of sequence spaces, discrete versions of Levi's monotone convergence theorem and Lebesgue's dominated convergence theorem and criteria for relative compactness of sequence spaces. Theoretical apparatus is completed with fixed point theorems. Introduced mathematical instruments are later used for study of a concrete nonlinear difference equation.
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Functional analysis and the mathematical pendulum
Čaputa, Daniel ; Šremr, Jiří (referee) ; Řehák, Pavel (advisor)
This thesis is focused on existence of periodic solutions of nonlinear model of mathematical pendulum with continuous, odd and periodic forcing term. In thesis, the differential equation of motion of pendulum is derived and the associated boundary value problem is rewritten as the integral equation. This equation is considered in a wider set of integral equations (Hammerstein equations). Fixed point theorems are applied on these equations what results in existence and uniqueness of solution. These results are applied on model of mathematical pendulum and the condition for uniqueness of solution is deeper discussed.
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Hausdirff metric and its application in fractals
Roháľ, Branislav Ján ; Hušek, Miroslav (advisor) ; Pyrih, Pavel (referee)
Title: Hausdorff metric and its application in fractals Author: Branislav Ján Roháľ Department: Department of Mathematical Analysis Supervisor: prof. RNDr. Miroslav Hušek, DrSc., Department of Mathematical Analysis Abstract: In this thesis we focus on the themes naturally connected with the con- cept of a fractal. In the first part of the thesis we pay attention to Banach fixed point theorem and to the Hausdorff metric which are later used when studying self-similar sets. There are included parts on the Hausdorff, similarity, and box- counting dimension, too. In the second part of the thesis the new approaches to fractal dimension and some their properties are refered. We introduce generaliza- tion of this concept for any space admitting a fractal structure and for a distance space where also the "size" of sets on each level of fractal structure is considered. In the last chapter the contribution of new approache is demonstrated, - this enables defining the notion needed and counting fractal dimension where it was not possible under the classical approaches, too. Application to the domain of words and counting of dimensions of a language generated by a regular expresion are presented. Keywords: Hausdorff metric, Banach fixed point theorem, self-similar set, Hausdorff dimension, fractal dimension
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