For nonlinear functional differential equations with several constant delays, the theorems on the continuous dependence of solutions of the Cauchy problem on perturbations of the initial data and on the right-hand side of the equation are proved. Under the initial data we mean the collection of the initial moment, constant delays, initial vector and initial function. Perturbations of the initial data and of the right-hand side of the equation are small in a standard norm and in an integral sense, respectively. Variation formulas of a solution are derived for equations with a discontinuous initial and continuous initial conditions. In the variation formulas, the effects of perturbations of the initial moment and delays as well as the effects of continuous initial and discontinuous initial conditions are revealed. For the optimal control problems with delays, general boundary conditions and functional, the necessary conditions of optimality are obtained in the form of equality or inequality for the initial and final moments, for delays and an initial vector and also in the form of the integral maximum principle for the initial function and control.
Mathematics Subject Classification: 34K99, 34K27, 49K21
Key words and phrases: Delay functional differential equations, continuous dependence of solutions, variation formula of a solution, effect of initial moment perturbation, effect of the discontinuous initial condition, effect of the continuous initial condition, effect of constant delays perturbations, optimal control problem with delays, necessary conditions of optimality