On a logistic equation with piecewise constant arguments. (English) Zbl 0727.34061

For the functional-differential equation \(x'(t)=rx(t)(1- \sum^{m}_{j=1}a_ jx([t-j])\), where [ ] denotes the usual Gauss brackets, the authors prove that all solutions oscillate about the unique positive steady state solution \(x^*\) if and only if \(q(z)=z- 1+r(\sum^{m}_{j=1}a_ jz^{-j})x^*\) does not vanish on (0,1) and that \(x^*\) is globally attractive for all non-negative initial data if \(\exp (r(m+1))<2\).
Reviewer: H.Engler (Bonn)


34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34K20 Stability theory of functional-differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations