Periodic Solutions for a Class of Differential Equations With Distributed Delay

Abstract In this paper, we study the existence of periodic solutions of the following differential delay system x'(t) = - f(∫10 x(t -s)ds), x ∈ RN where f ∈ C(RN, RN) and N ∈ N. We transform the existence of periodic solutions of the above system to that of periodic solutions with the special symmetry of an associated Hamiltonian system. Using the pseudo-index theory, we obtain some results about the number of periodic solutions, i.e. the discrepancies between the eigenvalues of asymptotic linear matrices at the origin and at infinity.