Periodic boundary value problem for first order impulsive differential equation at resonance

We develop a general theorem concerning the existence of solutions to the periodic boundary value problem for the first-order impulsive differential equation, {x'(t) = f(t,x(t)) t is an element of J backslash{t(1),t(2),...,t(k)} {Delta x(t(i)) = I-i(x(t)) i = 1,2...,k {x(0) = x(T). And using it we get a concrete existence result. Moreover, to our knowledge, the coincidence degree method has not been used with first order impulsive differential systems. Besides, our results can also be applied in studying the usual periodic boundary value problem at resonance without impulses.