*(English)*Zbl 1044.34028

This article deals with asymptotic stability results for a neutral functional-differential equation of the form

with an asymptotically stable linear part ${x}^{\text{'}}\left(t\right)=Sx\left(t\right)+Px(t-r)$. 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 $A$. The author describes conditions under which the operator $A$ satisfies conditions of the following Krasnoselâ€™skij fixed-point principle: if $A={A}_{1}+{A}_{2}$, ${A}_{1}$ is a contraction of a closed convex nonempty subset $M$ in a Banach space $S$, ${A}_{2}$ is continuous and ${A}_{2}M$ compact, and ${A}_{1}M+{M}_{2}M\subseteq M$, then there exists a fixed point of $A$. The use of this theorem allows the author to consider the case when the functions $Q$ and $G$ are unbounded with respect to $t$ and $G$ is not differentiable or Lipschitzian.