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

Abstract

We consider the non-linear impulsive evolution equation $$displaylines{ u'(t)=Au(t)+f(t,u(t),Tu(t),Su(t)), quad 0<t<T_0, ; t eq t_i,cr u(0) =u_0,cr Delta u(t_i) =I_i(u(t_i)),quad i=1,2,3,dots,p. }$$ 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.

Keywords

• Semigroups
• evolution equations
• impulsive conditions.