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.


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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