On the existence of periodic solutions to higher dimensional periodic system with delay

In this paper, higher dimensional periodic systems with delay of the form x'(t) = A(t,x(t))x(t) + f(t,x(t-tau)), x'(t) = grad G(x(t)) + f(t,x(t-tau)) are considered. Using the coincidence degree method, some sufficient conditions to guarantee the existence of periodic solution for these systems are obtained. As an application of the results, the existence of a positive periodic solution for a logarithmic population model is proved.