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Oscillation criteria for quasi-linear functional dynamic equations on time scales. (English) Zbl 1324.34192
Summary: This paper is concerned with oscillation of the second-order quasi-linear functional dynamic equation $$ \Bigl (r(t)\bigl (x^{\Delta}(t)\bigr)^{\gamma}\Bigr)^{\Delta} +p(t)x^{\beta}\bigl (\tau (t)\bigr)=0, $$ on a time scale $\Bbb {T}$, where $\gamma $ and $\beta $ are quotient of odd positive integers, $r$, $p$, and $\tau $ are positive rd-continuous functions defined on $\Bbb {T}$, $\tau \:\Bbb {T}\rightarrow \Bbb {T}$ and $\lim_{t\rightarrow \infty}\tau (t)=\infty $. We establish some new sufficient conditions which ensure that every solution oscillates or converges to zero. Our results improve the oscillation results in the literature when $\gamma =\beta $, and $\tau (t)\leq t$ and when $\tau (t)>t$ the results are essentially new. Some examples are considered to illustrate the main results.

34N05Dynamic equations on time scales or measure chains
34K11Oscillation theory of functional-differential equations
34K25Asymptotic theory of functional-differential equations
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