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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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