Impulsive integrodifferential equations with delay. (English) Zbl 1298.45013

Liu, James H.

Nonlinear Stud. 20, No. 2, 133-138 (2013).

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