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A comparison result for the oscillation of delay differential equations. (English) Zbl 0748.34044

A comparison theorem for the oscillation of all solutions of the scalar delay equation \(y'(t)+\sum^ n_{i=1}q_ i(t)y(t-\sigma_ i(t))=0\) is demonstrated. The essential assumptions are: 1. all solutions of the scalar delay equation \(x'(t)+\sum^ n_{i=1}p_ i(t)x(t-\tau_ i(t))=0\) are oscillating. 2. for each \(i=1,\ldots,n\) the zeros \(z_ i\) of \(p_ i\) and \(q_ i\) and the zeros \(\bar z_ i\) of \(\tau_ i\) and \(\sigma_ i\) are the same with the same multiplicities. 3. the limits \(\lim p_ i(t)/q_ i(t)=\alpha_ i\leq 1\), \(\lim\tau_ i(t)/\sigma_ i(t)=\beta_ i\leq 1\) when \(t\to\infty\) exist, \(p_ i\), \(q_ i\), \(\tau_ i\), \(\sigma_ i>0\). The theorem is applied to the nonautonomous delay-logistic equation \(N'(t)=r(t)N(t)[1-N(t-\tau(t))/K]\).

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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