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Impulsive integrodifferential equations with delay. (English) Zbl 1298.45013
Summary: We study the existence of mild solutions for the nonlinear impulsive integrodifferential equation with finite delay $\begin{gathered} u'(t)=Au(t)+ \int^t_0B(t-s)u(s)ds+f(t,u(t),u_t),\;0\leq t\leq K,\;t\neq t_i,\\ u(s)=\phi(s), s\in[-r,0],\\ \Delta u(t_i)=I_i(u(t_i)),\;i=1,2,\dots,p,\;0<t_1<t_2<\cdots <t_p<K. \end{gathered}$ Here, $$A$$ is the generator of a strongly continuous semigroup in a Banach space, and $$\Delta u(t_i)=u(t_i^+)-u(t^-_i)$$ constitutes an impulsive condition.
##### MSC:
 45J05 Integro-ordinary differential equations 34K45 Functional-differential equations with impulses
##### Keywords:
impulsive conditions; finite delay