Nonlocal impulsive problems for nonlinear differential equations in Banach spaces

Abstract

We study the existence and uniqueness of mild and classical solutions for a nonlinear impulsive differential equation with nonlocal conditions { u ′ ( t ) = A u ( t ) + f ( t , u ( t ) ) , 0 ≤ t ≤ K , t ≠ t i , u ( 0 ) + g ( u ) = u 0 , Δ u ( t i ) = I i ( u ( t i ) ) , i = 1 , 2 , … , p , 0 < t 1 < t 2 < ⋯ < t p < K , by combining and extending some earlier work on equations with nonlocal conditions and equations with impulsive conditions. Here, A is the generator of a strongly continuous semigroup in a Banach space, g constitutes a nonlocal condition, and Δ u ( t i ) = u ( t i + ) − u ( t i − ) constitutes an impulsive condition. New results are obtained.