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

Volume 3, Number 1 (2012), 42 - 47

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

Communicated By: 
Enrique Zuazua
Price: $20.00

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.