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# Existence and uniqueness of nonlinear impulsive integro-differential equations

• In this paper we study the existence and uniqueness of mild and classical solutions for a nonlinear impulsive integral evolution equation

$u'(t)= Au(t)+f(t,u(t),\int_0^tk(t,s)u(s)ds), t>0, t\ne t_i,$
$u(0)= u_0,$
$\Delta u(t_i)= I_i(u(t_i)). i= 1,2,....,p.$

in a Banach space X, where A is the infinitesimal generator of a strongly continuous semigroup,$\Delta u(t_i)=u(t^+_i)-u(t^-_i)$ and $I's$ are some operator. We apply the semigroup theory to study the existence and uniqueness of the mild solutions, and then show that the mild solution give rise to classical solution if $f$ is continuously differentiable.

Mathematics Subject Classification: 34A37, 34G20.

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