## Oscillation for a nonlinear dynamic system on time scales.(English)Zbl 1232.34124

The authors study oscillation properties for a system of two first-order nonlinear dynamic equations $x^\Delta=a_1(t)f_1(y),\quad y^\Delta=-a_2(t)f_2(x^\sigma)\tag{$$\ast$$}$ on time scales being unbounded above. The coefficients $$a_1,a_2$$ are supposed to be rd-continuous, $$a_1$$ is nonnegative and $$f_1,f_2$$ are continuous and satisfy $$uf_i(u)>0$$ for all $$u\neq 0$$, beyond further assumptions. This form includes the classical Emden-Fowler equations and various extensions.
In the superlinear, sublinear, delay-dynamic and Waltman’s case referring to the assumptions on $$f_i$$, sufficient conditions are given for the solutions to $$(\ast)$$ to be oscillatory. They explicitly require certain integrals over the coefficients $$a_i$$ to be unbounded.
Concrete examples for differential and difference equations illustrate the results.

### 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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