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In many problems, like those relative to adaptive control systems, it is necessary to consider the stability of sets which are not self-invariant in the usual sense. To describe such situations, La Salle and Rath, in 1963, introduced the notion of eventual stability. Later, in 1965, Lakshimikantham and Leela [8.b], stablished the notion of asymptotically self - invariant set and obtained some results on stability of such sets. Roughly speaking, an asymptotically self-invariant set is invariant only if the solutions of the given differential equation start at an infinite time. Recently, Bernfeld, Lakshimikantham and Leela [1], based on a paper of Rashbaev [On the stability of first approximation of solutions of a system of differential equations with retarded arguments - Isv. Akad. Nauk - SSSR5 - 1971 - pp 63/66] extended Rashbaev's investigations to perturbed functional differential equations presupposing that the unperturbed equation.presents some non-uniform exponential asymptotic stability property relative to an asymptotically self-invariant set φ =0. In this work, our main objective is to extend the results obtained by Bernfeld, Lakshimikantham and Leela to perturbed functional differential equations of neutral type: d/dt D((t,y_t) = f(t,y_t) + R(t,y_t), for which the operator D(t,φ is defined by DE(t, φ) =φ (0) - g (t,φ), where (g(t,φ) is linear in φ, following Hale [4a] , Izê [5] and , Onúchic [10] .