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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.

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