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Estimates of the Solutions of Difference-Differential Equations of Neutral Type
A. A. Lesnykh M. V. Lomonosov Moscow State University
Abstract:
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.
Keywords:
difference-differential equation of neutral type, equation with delay, initial-value problem, entire function, Laplace transform, characteristic determinant.
UDC:
517.929 Received: 20.11.2006
DOI:
10.4213/mzm3700
Fulltext:
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