Nonlocal impulsive problems for nonlinear differential equations in Banach spaces
We study the existence and uniqueness of mild and classical solutions for a nonlinear impulsive differential equation with nonlocal conditions u'(t) = Au(t) + f(t, u(t)), 0 <= t <= K, t not equal t(i), u(0) + g(u) = u(0), Delta 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 Delta u(t(i)) = u(t(i)(+)) - u(t(i)(-)) constitutes an impulsive condition. New results are obtained. (c) 2008 Elsevier Ltd. All rights reserved.
nonlocal and impulsive conditions