In this paper, we use a modification of Krasnoselskii’s fixed point theorem introduced by Burton (see [Burton, T. A.: Liapunov functionals, fixed points and stability by Krasnoseskii’s theorem. Nonlinear Stud., 9 (2002), 181–190.] Theorem 3) to obtain stability results of the zero solution of the totally nonlinear neutral differential equation with variable delay $$x^{\text{'}}\left(t\right)=-a\left(t\right)h\left(x\left(t\right)\right)+\frac{d}{dt}Q\left(t,x\left(t-\tau \left(t\right)\right)\right)+G\left(t,x\left(t\right),x\left(t-\tau \left(t\right)\right)\right).$$ The stability of the zero solution of this eqution provided that $h\left(0\right)=Q\left(t,0\right)=G\left(t,0,0\right)=0$. The Caratheodory condition is used for the functions $Q$ and $G$.

We study the asymptotic behavior of the solutions of a scalar convolution sum-difference equation. The rate of convergence of the solution is found by determining the asymptotic behavior of the solution of the transient renewal equation.