Impulsive integrodifferential equations with delay

Abstract

We study the existence of mild solutions for a
nonlinear impulsive integrodifferential equation with finite delay
\begin{eqnarray*}
& & u^\prime (t) = Au(t)+\int_0^t B(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,\cdots, p, \;\; 0 <t_1 <t_2 <\cdots <t_p <K.
\end{eqnarray*}
 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. New results are obtained.