Browsing by Subject "Riccati technique"

Now showing 1 - 4 of 4

Access status: Open Access ,
A remark on the linearization technique in half-linear oscillation theory
(2006) Došlý, Ondřej
We show that oscillatory properties of the half-linear second order differential equation $(r(t)\Phi(x'))'+c(t)\Phi(x)=0,\qquad\Phi(x)=|x|^{p-2}x,\quad p\gt 1,$ can be investigated via oscillatory properties of a certain associated second order linear differential equation. In contrast to paper [6], we do not need to distinguish between the cases $p\ge 2$ and $p\in (1,2]$. Our results also improve the oscillation and nonoscillation criteria given in [4].
Access status: Open Access ,
Nonoscillation of damped linear differential equations with a proportional derivative controller and its application to Whittaker-Hill-type and Mathieu-type equations
(Wydawnictwa AGH, 2023) Ishibashi, Kazuki
The proportional derivative (PD) controller of a differential operator is commonly referred to as the conformable derivative. In this paper, we derive a nonoscillation theorem for damped linear differential equations with a differential operator using the conformable derivative of control theory. The proof of the nonoscillation theorem utilizes the Riccati inequality corresponding to the equation considered. The provided nonoscillation theorem gives the nonoscillatory condition for a damped Euler-type differential equation with a PD controller. Moreover, the nonoscillation of the equation with a PD controller that can generalize Whittaker-Hill-type equations is also considered in this paper. The Whittaker-Hill-type equation considered in this study also includes the Mathieu-type equation. As a subtopic of this work, we consider the nonoscillation of Mathieu-type equations with a PD controller while making full use of numerical simulations.
Access status: Open Access ,
On oscillatory behaviour of third-order half-linear dynamic equations on time scales
(Wydawnictwa AGH, 2022) Grace, Said R.; Chhatria, Gokula Nanda
In this work, we study the oscillation and asymptotic behaviour of third-order nonlinear dynamic equations on time scales. The findings are obtained using an integral criterion as well as a comparison theorem with the oscillatory properties of a first-order dynamic equation. As a consequence, we give conditions which guarantee that all solutions to the aforementioned problem are only oscillatory, different from any other result in the literature. We propose novel oscillation criteria that improve, extend, and simplify existing ones in the literature. The results are associated with a numerical example. We point out that the results are new even for the case $\mathbb{T}=\mathbb{R}$ or $\mathbb{T}=\mathbb{Z}$.
Access status: Open Access ,
Wintner-type nonoscillation theorems for conformable linear Sturm-Liouville differential equations
(Wydawnictwa AGH, 2024) Ishibashi, Kazuki
In this study, we addressed the nonoscillation of th Sturm-Liouville differential equation with a differential operator, which corresponds to a proportional-derivative controller. The equation is a conformable linear differential equation. A Wintner-type nonoscillation theorem was established to be applied to such equations. Using this theorem, we provided a sharp nonoscillation condition that guarantees that all nontrivial solutions to Euler-type conformable linear equations do not oscillate. The main nonoscillation theorems can be proven by introducing a Riccati inequality, which corresponds to the conformable linear equation of the Sturm-Liouville type