Anti-periodic solutions of nonlinear first order impulsive functional differential equations

The existence of anti-periodic solutions of the following nonlinear impulsive functional differential equations $$ x'(t) + a(t)x(t) = f(t,x(t),x(alpha _1 (t)), ldots ,x(alpha _n (t))),t in mathbb{R},Delta x(t_k ) = I_k (x(t_k )),k in mathbb{Z} $$ is studied. Sufficient conditions for the existence of at least one anti-periodic solution of the mentioned equation are established. Several new existence results are obtained.