Periodic boundary value problems for first-order impulsive ordinary differential equations. (English) Zbl 1015.34010
The author considers the following nonlinear impulsive periodic boundary value problem. Let \(T>0\), \(J=[0,T]\), and \(t_1, t_2,\dots,t_p\in J\) the impulse instants with \(0=t_0<t_1<\dots<t_p<t_{p+1}=T\), \(J'=J-\{t_1,\dots,t_p\}\),
\[
u'(t)+F(t,u(t))=0 \text{ a.e. } t\in J', \quad \Delta u(t_j)=I_j(u(t_j)), \;j=1,\dots,p, \quad u(0)=u(T),\tag{1}
\]
where
\[
\Delta u(t_j)=u(t_j^+)-u(t_j^-)=u(t_j^+)-u(t_j), \quad j=1,\dots,p,
\]
\(F:J\times \mathbb{R}\to \mathbb{R}\) is a Carathéodory function such that for every \(j=1,\dots, p\) and \(u\in \mathbb{R}\) there exist the limits
\[
\lim_{t\to t_j^-}F(t,u)=F(t_j,u), \lim_{t\to t_j^+}F(t,u),
\]
and the impulses \(I_j:\mathbb{R}\to \mathbb{R}\), \(j=1,\dots, p\), are continuous. Sufficient conditions when problem (1) is solvable are obtained.
Reviewer: Alexander Olegovich Ignatyev (Donetsk)
MSC:
| 34B15 | Nonlinear boundary value problems for ordinary differential equations |
| 34A37 | Ordinary differential equations with impulses |
| 34B37 | Boundary value problems with impulses for ordinary differential equations |
| 34C25 | Periodic solutions to ordinary differential equations |
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