Existence and uniqueness of nonlinear impulsive integro-differential equations

Abstract

In this paper we study the existence and uniqueness of mild andclassical solutions for a nonlinear impulsive integral evolutionequation $u'(t)= Au(t)+f(t,u(t),\int_0^tk(t,s)u(s)ds), t>0, t\ne t_i,$ $u(0)= u_0,$ $\Delta u(t_i)=I_i(u(t_i)). i= 1,2,....,p.$ in a Banach space X, where A is the infinitesimal generator of astrongly continuous semigroup,$ \Delta u(t_i)=u(t^+_i)-u(t^-_i)$and $I's$ are some operator. We apply the semigroup theory to studythe existence and uniqueness of the mild solutions, and then showthat the mild solution give rise to classical solution if $f$ iscontinuously differentiable.