This article deals with asymptotic stability results for a neutral functional-differential equation of the form
with an asymptotically stable linear part . As is well known, the problem of the asymptotical stability of the zero solution to (1) is reduced to the problem of finding a fixed point for some nonlinear integral operator . The author describes conditions under which the operator satisfies conditions of the following Krasnosel’skij fixed-point principle: if , is a contraction of a closed convex nonempty subset in a Banach space , is continuous and compact, and , then there exists a fixed point of . The use of this theorem allows the author to consider the case when the functions and are unbounded with respect to and is not differentiable or Lipschitzian.