On positivity conditions for the Cauchy function of functional-differential equations

We discuss here how the statements about estimates of solutions to linear functional-differential equations analogous to the Chaplygin differential inequality theorem are connected with the positivity of the Cauchy function and fundamental solution. We prove the comparison theorem for the Cauchy functions and fundamental solutions of two functional-differential equations. In the theorem, it is assumed that the difference of the operators corresponding to the equations (and acting from the space of absolutely continuous functions to the space of summable functions) is a monotone totally continuous Volterra operator. We also obtain the positivity conditions for the Cauchy function and fundamental solution of some certain equations with delay and those of neutral type.