# On Monotonicity of Nonoscillation Properties of Dynamic Equations in Time Scales

### Elena Braverman

Technion - Israel Institute of Technology, Haifa, Israel### Bașak Karpuz

Afyon Kocatepe University, Afyonkarahisar, Turkey

## 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 $\alpha_k(t) ≤ t$ 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 $\tilde {\mathbb T} \supset T$.

## Cite this article

Elena Braverman, Bașak Karpuz, On Monotonicity of Nonoscillation Properties of Dynamic Equations in Time Scales. Z. Anal. Anwend. 31 (2012), no. 2, pp. 203–216

DOI 10.4171/ZAA/1455