Limiting Transition in a Singularly Perturbed System of Functional-Differential Equations

This article examines the asymptotic behavior of solutions of a singularly perturbed system of functional-differential equations of the retarded type with a small retardation. Sufficient conditions are introduced to guarantee convergence of the solutions of the system to the corresponding solutions of the limiting system for the case in which the basic condition of the classical Tikhonov theorem on the limiting transition in a singularly perturbed system, i.e., the condition of uniform asymptotic stability of the stationary point of the adjunct system, is replaced by a weaker Lyapunov condition for nonasymptotic stability. Dropping the Tikhonov condition is shown to expand the boundary layer within which the uniformity of the limiting solution breaks down.