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New oscillation criteria for second-order neutral functional dynamic equations via the generalized Riccati substitution. (English) Zbl 1221.34245
Summary: We establish some new sufficient conditions for oscillation of the second-order neutral functional dynamic equation $(p(t)([y(t)+r(t)y(\tau(t))]^\Delta)^\gamma)^\Delta+f(t,y(\theta(t)))=0, ~t\in[t_0,\infty)_{\mathbb T}$ on a time scale $$\mathbb{T}$$, where $$|f(t,u)|\geq q(t)|u^{\gamma }|$$, $$r, p$$ and $$q$$ are real valued $$rd$$-continuous positive functions defined on $$\mathbb{T}$$, $$\gamma\geq 1$$ is the quotient of odd positive integers. Our results improve previous existence results in the sense that our results do not require $$p^{\Delta }(t)\geq0$$, and $$\int^{\infty}_{t_0}\theta^\gamma(s)q(s)[1-r(\theta(s))]^\gamma \Delta s=\infty$$. Some examples are given to illustrate the main results.

##### MSC:
 34N05 Dynamic equations on time scales or measure chains 34K11 Oscillation theory of functional-differential equations 34K40 Neutral functional-differential equations
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