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Discrete Regular Variation and Difference Equations
Čaputa, Daniel ; Tomášek, Petr (oponent) ; Řehák, Pavel (vedoucí práce)
This thesis deals with the asymptotic analysis of a linear second-order difference equation using the theory of Karamata sequences. Properties of regularly varying sequences that are useful in asymptotic theory are gathered. Using a transformation of a difference equation into the dynamic equation on the appropriate time scale and proving a general result for the dynamic equation, the condition that guarantees a regular variation of the solution space of a difference equation is obtained. By the combination of the variety of techniques, asymptotic formulae are established and the solutions of the difference equation are classified into certain asymptotic classes.
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Functional analysis and the mathematical pendulum
Čaputa, Daniel ; Šremr, Jiří (oponent) ; Řehák, Pavel (vedoucí práce)
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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Discrete Regular Variation and Difference Equations
Čaputa, Daniel ; Tomášek, Petr (oponent) ; Řehák, Pavel (vedoucí práce)
This thesis deals with the asymptotic analysis of a linear second-order difference equation using the theory of Karamata sequences. Properties of regularly varying sequences that are useful in asymptotic theory are gathered. Using a transformation of a difference equation into the dynamic equation on the appropriate time scale and proving a general result for the dynamic equation, the condition that guarantees a regular variation of the solution space of a difference equation is obtained. By the combination of the variety of techniques, asymptotic formulae are established and the solutions of the difference equation are classified into certain asymptotic classes.
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Functional analysis and the mathematical pendulum
Čaputa, Daniel ; Šremr, Jiří (oponent) ; Řehák, Pavel (vedoucí práce)
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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