## Kamenev-type oscillation criteria for second order nonlinear dynamic equations on time scales.(English)Zbl 1211.34115

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 $$\mathbb T$$, where $$\gamma$$ is a quotient of odd positive integers and $$r$$ is a positive $$rd$$-continuous function on $$\mathbb T$$. The function $$g:\mathbb T\to\mathbb T$$ satisfies $$g(t)\geq t$$ and $$\lim_{t\to\infty}g(t) =\infty$$ and $$f\in C(\mathbb T\times \mathbb R,\mathbb R)$$. 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 $$\mathbb T=\mathbb R$$ and $$\mathbb T=\mathbb N$$ involve and improve some oscillation results for second-order differential and difference equations; and when $$\mathbb T=h\mathbb N$$, $$\mathbb T=q^{\mathbb N_0}$$ and $$\mathbb T=\mathbb N^2$$ our oscillation results are essentially new. Some examples illustrating the importance of our results are included.

### MSC:

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

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