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Zeitschrift für Analysis und ihre Anwendungen


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Volume 31, Issue 2, 2012, pp. 203–216
DOI: 10.4171/ZAA/1455

Published online: 2012-03-30

On Monotonicity of Nonoscillation Properties of Dynamic Equations in Time Scales

Elena Braverman[1] and Bașak Karpuz[2]

(1) Technion - Israel Institute of Technology, Haifa, Israel
(2) Afyon Kocatepe University, Afyonkarahisar, Turkey

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 $\tilde {\mathbb T} \supset T$.

Keywords: Nonoscillation, time scales, dependence of properties on time scales, finite difference approximations on various grids

Braverman Elena, Karpuz Bașak: On Monotonicity of Nonoscillation Properties of Dynamic Equations in Time Scales. Z. Anal. Anwend. 31 (2012), 203-216. doi: 10.4171/ZAA/1455