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Wintner-type nonoscillation theorems for conformable linear Sturm-Liouville differential equations

Ishibashi Kazuki

Opuscula Mathematica

2024

Vol. 44, no. 5

727--748

In this study, we addressed the nonoscillation of the 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.

Nonoscillation of damped linear differential equations with a proportional derivative controller and its application to Whittaker-Hill-type and Mathieu-type equations

Ishibashi Kazuki

Opuscula Mathematica

2023

Vol. 43, no. 1

67--79

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.