Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 78, pp. 1-15.

Title: Hille-Nehari type non-oscillation criteria for half-linear dynamic equations with mixed
       derivatives on a time scale
       
Author: Kazuki Ishibashi (Hiroshima College, Toyota-gun, Japan)

Abstract:
This article deals with half-linear dynamic equations that have two types of derivatives,
and obtains sufficient conditions for all solutions to be non-oscillatory.
The obtained results  extend a previous Hille-Nehari type theorems for
problems of dynamic equations.  To prove our main result, we use a generalized
Riccati inequality.  As an application, we apply the main result
to self-adjoint Euler type linear differential and difference equations with
a changing sign coefficient.  The equation selected for this application is of Mathieu type.

Submitted June 16, 2021. Published September 15, 2021.
Math Subject Classifications: 39A21, 34C10.
Key Words: Half-linear dynamic equations; nonoscillation; time scale; 
           Riccati dynamic inequality; linear differential equation; 
           linear difference equation.