BUKOTIN, D. Nelineární diferenciální rovnice a Karamatova teorie [online]. Brno: Vysoké učení technické v Brně. Fakulta strojního inženýrství. 2020.

# Posudky

## Posudek vedoucího

### Řehák, Pavel

The thesis is devoted to the study of asymptotic properties of a certain class of nonlinear differential equations, namely of the so-called nearly half-linear equations. An important role in this analysis is played by the theory of regular variation. First I list some selected objections (which however are not of serious character). Referring to (auxiliary) statements in some proofs could be more careful (for example, sometimes it is difficult to see which of the items of Proposition 1.1 is used). Some steps in the proofs would deserve more detailed arguments. It is not mentioned what we mean by a solution of the equations under consideration. From Corollary 3.1 it is not evident that the class of eventually positive and decrecerasing solutions is nonempty and is a subset of the set of normalized regularly varying functions. I would welcome more independence of the approach of the student in viewing the subject. And here is the selection of what I appreciate on the thesis. The topic was far from being trivial; among others, it was necessary to handle some advanced aspects of the regular variation theory and quite complicated technical tricks pertinent to asymptotic analysis of nonlinear differential equations - in my opinion, the student has managed it quite well. Further, I like the logical arrangement of thesis. Also English is good. The main goals have been achieved and in view of the above said, I can recommend the thesis for defense.

Dílčí hodnocení
Kritérium Známka Body Slovní hodnocení
Postup a rozsah řešení, adekvátnost použitých metod A
Vlastní přínos a originalita C
Schopnost interpretovat dosažené výsledky a vyvozovat z nich závěry C
Využitelnost výsledků v praxi nebo teorii B
Logické uspořádání práce a formální náležitosti B
Grafická, stylistická úprava a pravopis B
Práce s literaturou včetně citací B
Samostatnost studenta při zpracování tématu D
Navrhovaná známka
C

## Posudek oponenta

### Opluštil, Zdeněk

The master's thesis deals with the asymptotic behavior of solutions to some non-linear differential equations. It is examined by means of regular variation, concept of which was introduced by Jovan Karamata in 1930. The theory of regularly varying functions has proved itself very useful in some fields of the qualitative theory of differential equations. The text is divided into five chapters. The first one summarizes the theory of regular variation - basic definitions and properties which are used later. The next chapter focuses on the types of considered equations and their solution classes. The main part is in chapter three, where the author studies asymptotic behavior of positive (decreasing and increasing) slowly varying solutions. Obtained results are also applied to specific nonlinear equations. Finally, open problems are discussed and possible directions for further research are suggested. The author fulfilled the requirements and goals of the master’s thesis. The text is clear and logically structured. The thesis contains some inaccuracies and inconsistencies, which are not of serious character. I will mention some of them. - In Definition 1.1, it is not specified to which set the index of regularly varying functions belongs. - The author uses some notations before introducing them (e.g. "~" ). - Some justifications are incorrect, e.g., page 12, line 17: The assertion that L(t)=2+sin(ln(ln(t))) is a slowly varying function does not follow from Proposition 1.1 (there is presented only a necessary condition for this). But it follows from the statement on page 14, line 16. Moreover, this statement is used several times, so it would be appropriate to put it in the form of a proposition or remark. - It is not defined what we understand as a solution of considered equation (2.2). The text also contains some typos e.g. -page 11, line 23: "and" instead of "a" -page 12, line 12: "alpha" instead of "a" -page 25, line 6: p(t) instead of p -page 28, line12: RV(1-alpha) -page 37, line 18: (3.9) instead of (3.11) etc. I appreciate the following in the thesis. There are new results, which can be seen as an extension of known cases. Particularly, Theorem 3.1 and 3.2 generalize statements for nearly linear and half linear equations presented in [15,16]. The results of Theorem 3.4 and 3.5 are new for the nearly linear case. I would like to mention that the proofs are not trivial, they are done precisely and correctly. In my opinion, it is possible to publish the obtained results in a scientific journal (after some minor revisions). I also appreciate that the author applied presented statements to specific non-linear equations in Examples 4.1 - 4.3. I recommend the thesis for the defense with classification very good.

Dílčí hodnocení
Kritérium Známka Body Slovní hodnocení
Postup a rozsah řešení, adekvátnost použitých metod A
Vlastní přínos a originalita B
Schopnost interpretovat dosaž. výsledky a vyvozovat z nich závěry B
Využitelnost výsledků v praxi nebo teorii B
Logické uspořádání práce a formální náležitosti C
Grafická, stylistická úprava a pravopis B
Práce s literaturou včetně citací B
Navrhovaná známka
B

#### Otázky

• Na straně 11 je uvedeno, že f patří do třídy regulárně se měnících funkci s indexem theta právě tehdy, když je ve tvaru (1.3). Mohl by to autor ukázat ?
• Koeficienty p(t), q(t) uvažovaného systému (2.2) jsou pozitivní funkce. Je známo něco o případech, kdy tyto funkce mění znaménka?

eVSKP id 122390