Multiplicity results on periodic solutions to higher-dimensional differential equations with multiple delays

Abstract

This paper continues our study on the existence and multiplicity of periodic solutions to delay differential equations of the form \[ \dot{z}(t)=-f(z(t-1))-f(z(t-2))-\cdots -f(z(t- n+1)), \] where $z\in\br^N$, $f\in C(\br^N, \br^N)$ 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.