Half-linear Euler Differential Equation: Perturbations and Oscillatory Properties

RNDr. Mgr. Hana Funková, Ph.D.

Doctoral thesis

Abstract:

V této disertační práci studujeme oscilační vlastnosti perturbované Eulerovy pololineární diferenciální rovnice $$ \bigl(\Phi(x')\bigr)'+\frac{\gamma_p}{t^p}\Phi(x)=0,\quad \Phi(x):=|x|^{p-2}x,\ \gamma_p:=\left(\frac{p-1}{p}\right)^p. $$ \noindent Nejdříve uvažujeme perturbace ve tvaru $$ \left[\left(1+\sum_{j=1}^n \frac{\alpha_j}{\Log_j^2 t}\right)\Phi(x')\right]' +\left[\frac{\gamma_p}{t^p}+\sum_{j=1}^n \frac{\beta_j}{t^p\Log_j^2 t} \right]\Phi(x)=0, $$ kde $ \Log_kt=\prod_{j=1}^k \log_k t,\quad \log_k t=\log_{k-1}(\log t),\ \log_1 t=\log t, $ a v druhé části studujeme perturbace obsahující periodické funkce $r(t)$, $c(t)$, $\alpha_j(t)$, $\beta_j(t) > 0$ $$ \left[\left(r(t) +\sum_{j=1}^n \frac{\alpha_j(t)}{\Log_j^2 t}\right)^{1-p}\Phi(x')\right]' +\left[\frac{c(t)}{t^p}+\sum_{j=1}^n \frac{\beta_j(t)}{t^p\Log_j^2 t} \right]\Phi(x)=0. $$ V obou případech je k vyšetřování rovnic využita metoda transformací tzv. modifikované Riccatiho rovnice.

Abstract:

In this dissertation, we investigate oscillatory properties of the perturbed half-linear Euler differential equation $$ \bigl(\Phi(x')\bigr)'+\frac{\gamma_p}{t^p}\Phi(x)=0,\quad \Phi(x):=|x|^{p-2}x,\ \gamma_p:=\left(\frac{p-1}{p}\right)^p. $$ \noindent First, we consider perturbations of the form $$ \left[\left(1+\sum_{j=1}^n \frac{\alpha_j}{\Log_j^2 t}\right)\Phi(x')\right]' +\left[\frac{\gamma_p}{t^p}+\sum_{j=1}^n \frac{\beta_j}{t^p\Log_j^2 t} \right]\Phi(x)=0, $$ where $ \Log_kt=\prod_{j=1}^k \log_k t,\quad \log_k t=\log_{k-1}(\log t),\ \log_1 t=\log t, $ and in the second part perturbations involving periodic functions $r(t)$, $c(t)$, $\alpha_j(t)$, $\beta_j(t) > 0$ $$ \left[\left(r(t) +\sum_{j=1}^n \frac{\alpha_j(t)}{\Log_j^2 t}\right)^{1-p}\Phi(x')\right]' +\left[\frac{c(t)}{t^p}+\sum_{j=1}^n \frac{\beta_j(t)}{t^p\Log_j^2 t} \right]\Phi(x)=0, $$ are studied. In the both cases, the method of transformation of the so-called modified Riccati equation is used.

Keywords

Pololineární diferenciální rovnice, asymptotický vzorec, hlavní řešení, perturbovaná Eulerova rovnice, modifikovaná Riccatiho rovnice, Prüferova transformace, Half-linear differential equation, Asymptotic formula, Principal solution, Perturbed Euler equation, Modified Riccati equation, Prüfer transformation

Language used: English

Date on which the thesis was submitted / produced: 4. 11. 2014

Thesis defence

• Date of defence: 10. 12. 2014
• Supervisor: prof. RNDr. Ondřej Došlý, DrSc.
• Reader: doc. RNDr. Michal Veselý, Ph.D., doc. Mgr. Robert Mařík, Ph.D.

Masaryk University

Faculty of Science

Doctoral programme / field:
Matematics (4-years) / Mathematical Analysis