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On a periodic problem for higher-order differential equations with a deviating argument. (English) Zbl 1225.34073

The authors study the existence and uniqueness of \(\omega\)-periodic solutions of \(n\)-th order differential equations with deviating argument of the form
\[ u^{(n)}(t)=p(t)u(\tau(t))+q(t) \]
and
\[ u^{(n)}(t)=f(t,u(\tau(t)))+f_0(t), \]
where \(p,q,f_0\) are \(\omega\)-periodic and Lebesgue integrable on \([0,\omega]\), while \(f\) is \(\omega\)-periodic in \(t\) and is a Carathéodory function. On the devating argument, the function \(\tau:\mathbb{R}\to\mathbb{R}\), it is supposed that it is a measurable function on each finite interval and such that \((\tau (t+\omega)-\tau(t))/\omega\) is an integer for almost all \(t\in\mathbb{R}\). Under additional hypotheses, either existence and uniqueness or only existence of a \(\omega\)-periodic solution are obtained. These results are related to previous results given for the case \(\tau(t)\equiv t\) (that is, no deviating argument) by Kiguradze-Kusano (1999), Kiguradze (2000) and Kiguradze-Půža (2003).

MSC:

34K13 Periodic solutions to functional-differential equations
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