The existence of periodic solutions for nonlinear systems of first-order differential equations at resonance

The nonlinear system of first-order differential equations with a deviating argument x(t) = Bx(t) + F(x(t - tau)) + p(t) is considered, where x(t) is an element of R-2, tau is an element of R, B i s an element of R-2x2, F is bounded and p(t) is continuous and 2 pi -period ic. Some sufficient conditions for the existence of 2 pi -periodic solution s of the above equation, in a resonance case, by using the Brouwer degree t heory and a continuation theorem based on Mawhin's coincidence degree are o btained. Some applications of the main results to Duffing's equations are a lso given.