Periodic solutions of delay impulsive differential equations

We study the following semilinear impulsive differential equation with delay: u'(t) + Au(t) = f(t, u(t), u(t)), t > 0, t not equal t(i), u(s) = phi(s), s is an element of [-r, 0], Delta u(t(i)) = I(i)(u(t(i))), i = 1, 2, ..., 0 < t(1) < t(2) < ... < infinity, in a Banach space (X, parallel to . parallel to) with an unbounded operator A, where r > 0 is a constant and u(t) (s) = u(t + s), s is an element of [-r, 0]. Here, Delta u(t(i)) = u(t(i)(+)) - u(t(i)(-)) constitutes an impulsive condition, which can be used to model more physical phenomena than the traditional initial value problems. We assume that f (t, u, w) is T-periodic in t and then prove with some compactness conditions that if solutions of the equation are ultimately bounded, then the differential equation has a T-periodic solution. The new results obtained here extend some results in this area for differential equations without impulsive conditions or without delays. (C) 2011 Elsevier Ltd. All rights reserved.