On periodic solutions of functional differential equations with infinite delay

Abstract

In this paper, using Mawhin's continuation theorem in the theory of coincidence degree, we first prove the general existence theorem of periodic solutions for F.D.Es with infinite delay: $$\frac{{dx(t)}}{{dt}} = f(t,x_t ), x(t) \in R^n ,$$ which is an extension of Mawhin's existence theorem of periodic solutions of F.D.Es with finite delay. Second, as an application of it, we obtain the existence theorem of positive periodic solutions of the Lotka-Volterra equations: $$\begin{gathered} \frac{{dx(t)}}{{dt}} = x(t)(a - kx(t) - by(t)), \hfill \\ \frac{{dy(t)}}{{dt}} = - cy(t) + d\smallint _0^{ + \infty } x(t - s)y(t - s)d\mu (s) + p(t). \hfill \\ \end{gathered} $$