On Monotonicity of Nonoscillation Properties of Dynamic Equations in Time Scales

Abstract

For equations on time scales, we consider the following problem: when will nonoscillation on time scale $\mathbb{T}$ imply nonoscillation of the same equation on any time scale $\tilde{\mathbb{T}}$ including $\mathbb{T}$ as a subset? The main result of the paper is the following. If nonnegative coefficients $A_k(t)$ are nonincreasing and ${\alpha }_k(t)\le t$ are nondecreasing in $t\in \mathbb{R}$, then nonoscillation of the equation

\[ x^{\Delta}
(t) +\sum^m_{k=1} A_k(t)x(\alpha_k(t)) = 0 \ \ \ \rm{for} \ t \in [t_0,\infty)_{\mathbb T} \]

yields nonoscillation of the same equation on any time scale $\tilde{\mathbb{T}}\supset T$.