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Zbl 1211.34115
Hassan, Taher S.
Kamenev-type oscillation criteria for second order nonlinear dynamic equations on time scales. (English)
[J] Appl. Math. Comput. 217, No. 12, 5285-5297 (2011). ISSN 0096-3003

Summary: The purpose of this paper is to establish oscillation criteria for the second order nonlinear dynamic equation $$(r(t)(x^\Delta(t))^\gamma)^\Delta+f(t,x(g(t)))=0,$$ on an arbitrary time scale $\Bbb T$, where $\gamma$ is a quotient of odd positive integers and $r$ is a positive $rd$-continuous function on $\Bbb T$. The function $g:\Bbb T\to\Bbb T$ satisfies $g(t)\ge t$ and $\lim_{t\to\infty}g(t) =\infty$ and $f\in C(\Bbb T\times \Bbb R,\Bbb R)$. We establish some new sufficient conditions under which the above equation is oscillatory by using the generalized Riccati transformation. Our results in the special cases when $\Bbb T=\Bbb R$ and $\Bbb T=\Bbb N$ involve and improve some oscillation results for second-order differential and difference equations; and when $\Bbb T=h\Bbb N$, $\Bbb T=q^{\Bbb N_0}$ and $\Bbb T=\Bbb N^2$ our oscillation results are essentially new. Some examples illustrating the importance of our results are included.

MSC 2000:: *34N05
34K11 Oscillation theory of functional-differential equations

Keywords: oscillation; second order; nonlinear dynamic equations; time scales

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