Beginning with the works of N.N.Krasovskii [81, 82, 83], which clari-
fied the functional nature of systems with delays, the functional
approach provides a foundation for a complete theory of differential
equations with delays. Based on the functional approach, different
aspects of time-delay system theory have been developed with almost the
same completeness as the corresponding field of ODE (ordinary
differential equations) the- ory. The term functional differential
equations (FDE) is used as a syn- onym for systems with delays 1. The
systematic presentation of these re- sults and further references can be
found in a number of excellent books [2, 15, 22, 32, 34, 38, 41, 45,
50, 52, 77, 78, 81, 93, 102, 128]. In this monograph we present basic
facts of i-smooth calculus a new differential calculus of nonlinear
functionals, based on the notion of the invariant derivative, and some
of its applications to the qualitative theory of functional differential
equations. Utilization of the new calculus is the main distinction of
this book from other books devoted to FDE theory. Two other
distinguishing features of the volume are the following: - the central
concept that we use is the separation of finite dimensional and infinite
dimensional components in the structures of FDE and functionals; - we
use the conditional representation of functional differential equa-
tions, which is convenient for application of methods and constructions
of i smooth calculus to FDE theory.