## 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
Full Text:

### References:

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