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Oscillation of second-order delay and neutral delay dynamic equations on time scales. (English) Zbl 1147.34050
Summary: We consider the second-order linear delay dynamic equation $$x^{\Delta\Delta}(t)+q(t)x(\tau(t))=0,$$ on a time scale $\Bbb T$. We will study the properties of the solutions and establish some sufficient conditions for oscillations. In the special case when $\Bbb T = \Bbb R$ and $\tau(t)=t$, our results include some well-known results in the literature for differential equations. When, $\Bbb T = \Bbb Z$, $\Bbb T = h\Bbb Z$, for $h > 0$ and $\Bbb T = \Bbb T_n = \{t_n : n\in \Bbb N_0\}$ where $\{t_n\}$ is the set of the harmonic numbers defined by $t_0 = 0$, $t_n = \sum^n_{k=1}\frac1k$ for $n\in\Bbb N_0$ our results are essentially new. The results will be applied on second-order neutral delay dynamic equations in time scales to obtain some sufficient conditions for oscillations. An example is considered to illustrate the main results.

34K11Oscillation theory of functional-differential equations
39A11Stability of difference equations (MSC2000)