## Periodic Solutions to Operational Differential Equations with Finite Delay and Impulsive Conditions

### Abstract

We study the following semilinear operational differential equation with finite delay and

impulsive condition

u0(t)+ Au(t) = f (t,u(t),ut), t > 0, t , ti,

u(s) = (s), s 2 [−r, 0],

u(ti) = Ii(u(ti)), i = 1,2, · · · , 0 < t1 < t2 < · · · < 1,

in a Banach space (X, k · k) with an unbounded operator A, where r > 0 is a constant and ut(s) =

u(t+s), s 2 [−r, 0], which constitutes a finite delay, and u(ti) = 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 under some conditions that

if solutions of the equation are ultimate bounded, then the operational differential equation has a

T–periodic solution. The new result obtained here improves the corresponding result of [10] by

eliminating an assumption in [10]. Moreover, our arguments of proving the result are suitable for

many other problems associated with impulsive conditions.