International Journal of Mathematics and Mathematical Sciences
Volume 31 (2002), Issue 8, Pages 451-461
doi:10.1155/S0161171202012735
A second-order impulsive Cauchy problem
Eduardo Hernández Morales
Departamento de Matemática, Instituto de Ciências Matemáticas de São Carlos, Universidade de São Paulo, Caixa Postal 668, São Carlos 13560-970, São Paulo, Brazil
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.