MULTIPLICITY RESULTS ON PERIODIC SOLUTIONS TO HIGHER-DIMENSIONAL DIFFERENTIAL EQUATIONS WITH MULTIPLE DELAYS

This paper continues our study on the existence and multiplicity of periodic solutions to delay differential equations of the form (z) over dot(t) = -f(z(t-1))-f(z(t-2))-...-f(z(t - n + 1)), where z is an element of R-N, f is an element of C(R-N, R-N) and n > 1 is an odd number. By using the Galerkin approximation method and the S-1-index theory in the critical point theory, some known results for Kaplan-Yorke type differential delay equations are generalized to the higher-dimensional case. As a result, the Kaplan-Yorke conjecture is proved to be true in the case of higher-dimensional systems.