BOUNDARY-LAYER PHENOMENA FOR DIFFERENTIAL-DELAY EQUATIONS WITH STATE-DEPENDENT TIME LAGS .1.

In this paper we begin a study of the differential-delay equation epsilonx'(t) = - x(t) + f(x(t - r)), r = r (x(t)). We prove the existence of periodic solutions for 0 < epsilon < epsilon0, where epsilon0 is an optimal positive number. We investigate regularity and monotonicity properties of solutions x(t) which are defined for all t and of associated functions like eta(t) = t - r(x(t)). We begin the development of a Poincare-Bendixson theory and phase-plane analysis for such equations. In a companion paper these results will be used to investigate the limiting profile and corresponding boundary layer phenomena for periodic solutions as epsilon approaches zero.