Estimates of the Solutions of Difference-Differential Equations of Neutral Type

In this paper, we study scalar difference-differential equations of neutral type of general form
$$ \sum_{j=0}^m\int_0^hu^{(j)}(t-\theta)\,d\sigma_j(\theta)=0, \qquad t>h, $$
where the $\sigma_j(\theta)$ are functions of bounded variation. For the solutions of this equation, we obtain the following estimate:
$$ \|u(t)\|_{W_2^m(T,T+h)} \le C T^{q-1}e^{\varkappa T}\|u(t)\|_{W_2^m(0,h)}, $$
where $C$ is a constant independent of $u_0(t)$ and the values of $q$ and $\varkappa$ are determined by the properties of the characteristic determinant of this equation. Earlier, this estimate was proved for equations of less general form. For example, for piecewise constant functions $\sigma_j(\theta)$ or for the case in which the function $\sigma_m(\theta)$ has jumps at both points $\theta=0$ and $\theta=h$. In the present paper, this estimate is obtained under the only condition that $\sigma_m(\theta)$ experiences a jump at the point $\theta=0$; this condition is necessary for the correct solvability of the initial-value problem.