Application of $i$-smooth analysis to construction of numerical methods for solving functional-differential equations

New approach to constructing methods of Runge–Kutta type as well as multistep methods for solving functional-differential equations is proposed. Our methods (direct analogs of numerical methods for solving ordinary differential equations) are based on separation of finite-dimensional and infinite-dimensional (functional) components in the structure of the phase state. Determination of the approximation order of the methods essentially uses notions of $i$-smooth analysis.