Periodic Solutions with Prescribed Minimal Period for a Class of Differential Equations with a Distributed Delay

This paper investigates periodic solutions with prescribed minimal period for a class of distributed delay differential equations. By applying the Kaplan–Yorke transformation, we reformulate the minimal period problem for the delayed system as an equivalent problem for an ordinary differential equation (ODE). Subsequently, employing a finite-difference discretization and the Brouwer fixed point theorem, we establish the existence of positive solutions to the resulting discrete system under mixed boundary conditions. By constructing piecewise linear interpolants of these discrete solutions, we apply the Arzelà–Ascoli theorem to pass to the limit, thereby obtaining the positive solution to the corresponding continuous ODE under the same boundary conditions. Finally, the desired periodic solution is obtained by symmetric arguments. Two illustrative examples are provided to verify the theoretical findings.