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

Abstract

We consider the non-linear impulsive evolution equation u'(t) = Au(t) + ƒ(t, u(t), Tu(t), Su(t)), 0 < t < T0, t ≠ ti, u(0) = u0, ∆u(ti) = Ii(u(ti)), i = 1, 2, 3,..., p. in a Banach space X, where A is the infinitesimal generator of a C0 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.