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 Mat. Zametki, 2007, Volume 81, Issue 4, Pages 569–585 (Mi mz3700)

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

DOI: https://doi.org/10.4213/mzm3700

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English version:
Mathematical Notes, 2007, 81:4, 503–517

Bibliographic databases:

UDC: 517.929

Citation: A. A. Lesnykh, “Estimates of the Solutions of Difference-Differential Equations of Neutral Type”, Mat. Zametki, 81:4 (2007), 569–585; Math. Notes, 81:4 (2007), 503–517

Citation in format AMSBIB
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• https://doi.org/10.4213/mzm3700
• http://mi.mathnet.ru/eng/mz/v81/i4/p569

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This publication is cited in the following articles:
1. V. V. Vlasov, S. A. Ivanov, “Sharp estimates for solutions of systems with aftereffect”, St. Petersburg Math. J., 20:2 (2009), 193–211
2. V. V. Vlasov, D. A. Medvedev, “Functional-differential equations in Sobolev spaces and related problems of spectral theory”, Journal of Mathematical Sciences, 164:5 (2010), 659–841
3. M. S. Sgibnev, “Behavior at infinity of a solution to a differential-difference equation”, Siberian Math. J., 53:6 (2012), 1139–1154
4. M. S. Sgibnev, “Behavior at infinity of a solution to a matrix differential-difference equation”, Siberian Math. J., 55:3 (2014), 530–543
5. M. S. Sgibnev, “An asymptotic expansion of the solution of a matrix difference equation of general form”, Sb. Math., 205:12 (2014), 1815–1828
6. Sgibnev M.S., “Asymptotic Expansion of the Solution of a Differential-Difference Equation of General Form”, Differ. Equ., 50:3 (2014), 323–334
7. Sgibnev M.S., “Asymptotic Expansion of the Solution of a Matrix Differential-Difference Equation of the General Form”, Differ. Equ., 52:1 (2016), 28–38
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