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A comparison theorem for linear delay differential equations. (English) Zbl 0841.34071
The paper deals with the delay differential equation (1) $$L_n u(t)+ p(t) u(\tau(t))= 0$$, where $$n\geq 2$$ and $L_n u= \Biggl({1\over r_{n- 1}(t)} \Biggl(\cdots\Biggl( {1\over r_1(t)} u'\Biggr)'\cdots \Biggr)' \Biggr)'.$ It is assumed that $$r_i(t)$$, $$1\leq i\leq n- 1$$, $$\tau(t)$$ and $$p(t)$$ are continuous on $$[t_0, \infty)$$, $$r_i(t)> 0$$, $$\tau(t)< t$$, $$\tau(t)\to \infty$$ as $$t\to \infty$$ and $$\int^\infty r_i(s) ds= \infty$$ for $$1\leq i\leq n- 1$$. The author deduces a property (A) of the equation (1) from the oscillation of a set of first-order delay differential equations.

MSC:
 34K11 Oscillation theory of functional-differential equations 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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