A second-order impulsive Cauchy problem

Abstract

We study the existence of mild and classical solutions for an abstract second-order impulsive Cauchy problem modeled in the form u¨(t)=A u(t)+f(t,u(t),u˙(t)),t∈(−T0,T1),t≠ti;u(0)=x0,u˙(0)=y0; △u(ti)=Ii1 (u (ti)). △u˙(ti)=Ii2 (u˙ (ti+)) where A is the infinitesimal generator of a strongly continuous cosine family of linear operators on a Banach space X and f,Ii1, Ii2 are appropriate continuous functions.