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Sharp conditions for oscillation. (English) Zbl 0795.34060

The paper deals with the equation (1) \(x'(t)+ \sum_{i=1}^ n p_ i(t) x(t-\tau_ i(t))=0\), where \(0\leq p_ i(t)\leq p\), \(0\leq \tau_ i(t)\leq T\) for \(i=1,2,\dots,n\); \(p\) and \(T\) are constants and \(p_ i(t)\), \(\tau_ i(t)\) are continuous functions on \([0,\infty)\). The author presents a sufficient condition for all solutions of (1) to be oscillatory and a necessary and sufficient condition for all solutions of the equation (2) \(x'(t)+ \sum_{i=1}^ n p_ i x(t-\tau_ i)=0\) to be oscillatory.

MSC:

34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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References:

[1] B.R. Hunt, and J.A. Yorke, When All Solutions ofx’(t) = {\(\Sigma\)} i=1 n q i(t) {\(\times\)} (t -T i(t)) Oscillate,J. Differential Equations,53 (1984), 139–145. · Zbl 0571.34057 · doi:10.1016/0022-0396(84)90036-6
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