Positive periodic solutions of a continuous dynamical system with delay and pumping

We study the existence of periodic solutions of a continuous dynamical system of special form with delay. The system has the specific feature that the unknown function changes in accordance with a delay differential equation until it reaches the zero value and then makes an instant jump to unity (pumping), after which it again satisfies the main differential equation until the next possible pumping. Special emphasis is given to the study of nonnegative periodic solutions. We find the entire interval of values of the system parameter for which there exists a unique positive periodic solution.