Proof and generalization of Kaplan-Yorke's conjecture under the condition \(F^{\prime}\) \((0) > 0\) on periodic solutions to differential delay equations (Q1974189)

Here, certain periodic solutions to the differential delay equation \[ x'(t)=-\sum^N_{i=1}f(x(t-r_i))\tag{*} \] with commensurable delays \(r_i\) and odd function \(f(x),f(-x) = -f(x)\), and \(f'(0) > 0\), are derived from a specially constructed system of ordinary differential equations. The approach is based on the original idea of such reduction by \textit{J. L. Kaplan} and \textit{J. A. Yorke} [J. Math. Anal. Appl. 48, 317-324 (1974; Zbl 0293.34102)]. Some sufficient conditions for the existence of periodic solutions to equation (*) are given when \(r_i = i\), \(1\leq i\leq N\), providing a partial answer to a question raised by Kaplan and Yorke in their paper.