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Global attractivity and oscillation in a nonlinear delay equation. (English) Zbl 0987.34065
The authors consider the delay periodic equation \[ x^{'}(t)=x(t)[a(t)+b(t)x^p(t-mw)-c(t)x^q(t-mw)], \] where \(a,b,c\in C[[0,\infty),\mathbb{R}]\) with a common period \(w>0\), and \(a>0\), \(c>0 \), \(m\) is a positive integer, \(p\) and \(q\) are positive constants \(q>p\). This equation with the initial condition \[ x(t)=\phi (t)\text{ for }-mw\leq t\leq 0,\quad \phi \in C[[-mw,0],[0,\infty)], \] has a unique solution which is positive for all \(t\geq 0\). The authors investigate the global attractivity and the oscillatority of this unique solution. They provide a sufficient condition for the global attractivity of the unique positive \(w\)-periodic solution \(\widetilde{x}\) and for all other positive solutions to the equation in the non-delay case (\(m=0\)). The function \(\widetilde{x}\) is also a unique \(w\)-periodic solution to the equation for any positive integer \(m\). The paper establishes a necessary and sufficient condition for every solution to the equation to oscillate about \(\widetilde{x}\). A sufficient condition for the global attractivity of \(\widetilde{x}\) is provided, too.

MSC:
34K11 Oscillation theory of functional-differential equations
34K25 Asymptotic theory of functional-differential equations
34K20 Stability theory of functional-differential equations
34K12 Growth, boundedness, comparison of solutions to functional-differential equations
34D45 Attractors of solutions to ordinary differential equations
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