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Some new oscillation results for a nonlinear dynamic system on time scales. (English) Zbl 1185.34141

From the introduction: We consider the nonlinear dynamic system
\[ \begin{aligned} & x^\Delta=a_1(t)f_1(y),\\ & y^\Delta=-a_2(t)f_2(x^a),\quad t\in[a,\infty)_\mathbb T,\end{aligned}\tag{1} \]
where \(a_1, a_2\) are real-valued rd-continuous (defined below) functions which are defined for \(t\in [a,\infty)_{\mathbb T}= [a,\infty)\cap\mathbb T\). Here, \(\mathbb T\) is a time scale unbounded from above. We assume throughout that \(a_1(t) > 0\), \(f_i:\mathbb R\to\mathbb R\), \(i = 1,2\), are continuous functions such that \(f_1\) is nondecreasing, \(f_2\) is continuously differentiable on \(\mathbb R\), except possibly at 0, and
\[ f_2'(u)>0\text{ and }uf_i(u)>0\text{ for }u\neq 0. \]
Our main interest in this paper is to establish some oscillation results for the system (1). We will relate our results to some earlier work for the system (1) and the equivalent Emden-Fowler equation.

MSC:

34N05 Dynamic equations on time scales or measure chains
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
39A10 Additive difference equations
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References:

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