Electronic Journal of Differential Equations,
Vol. 2005(2005), No. 111, pp. 1-8.

Title: Existence and uniqueness of mild and classical solutions of
impulsive evolution equations

Authors: Annamalai Anguraj (College of Arts and Science, Tamilnadu, India)
      Mani Mallika Arjunan (College of Arts and Science, Tamilnadu, India)

Abstract: 
 We consider the non-linear impulsive evolution equation
 $$\displaylines{
 u'(t)=Au(t)+f(t,u(t),Tu(t),Su(t)),  \quad  0<t<T_0, \;  t\neq t_i,\cr
 u(0) =u_0,\cr
 \Delta u(t_i) =I_i(u(t_i)),\quad  i=1,2,3,\dots,p.
 }$$
 in a Banach space $ X$, where $ A $ is the infinitesimal generator
 of a $C_0 $ semigroup. We study the existence
 and uniqueness of the mild solutions of the evolution equation
 by using semigroup theory and then show that the mild solutions
 give rise to a classical solutions.
  
Submitted June 15, 2005. Published October 17, 2005.
Math Subject Classifications: 34A37, 34G60, 34G20.
Key Words: Semigroups; evolution equations; impulsive conditions.