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Matematicheskie Zametki, 2007, Volume 81, Issue 4, Pages 569–585
DOI: https://doi.org/10.4213/mzm3700
(Mi mzm3700)
 

This article is cited in 7 scientific papers (total in 7 papers)

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

A. A. Lesnykh

M. V. Lomonosov Moscow State University
Full-text PDF (589 kB) Citations (7)
References:
Abstract: In this paper, we study scalar difference-differential equations of neutral type of general form
mj=0h0u(j)(tθ)dσj(θ)=0,t>h,
where the σj(θ) are functions of bounded variation. For the solutions of this equation, we obtain the following estimate:
u(t)Wm2(T,T+h)CTq1eϰTu(t)Wm2(0,h),
where C is a constant independent of u0(t) and the values of q and ϰ 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 σj(θ) or for the case in which the function σm(θ) has jumps at both points θ=0 and θ=h. In the present paper, this estimate is obtained under the only condition that σm(θ) experiences a jump at the point θ=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.
Received: 20.11.2006
English version:
Mathematical Notes, 2007, Volume 81, Issue 4, Pages 503–517
DOI: https://doi.org/10.1134/S0001434607030285
Bibliographic databases:
UDC: 517.929
Language: Russian
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://www.mathnet.ru/eng/mzm3700
  • https://doi.org/10.4213/mzm3700
  • https://www.mathnet.ru/eng/mzm/v81/i4/p569
  • This publication is cited in the following 7 articles:
    1. 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  crossref  mathscinet  zmath  isi  scopus
    2. M. S. Sgibnev, “Behavior at infinity of a solution to a matrix differential-difference equation”, Siberian Math. J., 55:3 (2014), 530–543  mathnet  crossref  mathscinet  isi  elib  elib
    3. M. S. Sgibnev, “An asymptotic expansion of the solution of a matrix difference equation of general form”, Sb. Math., 205:12 (2014), 1815–1828  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. Sgibnev M.S., “Asymptotic Expansion of the Solution of a Differential-Difference Equation of General Form”, Differ. Equ., 50:3 (2014), 323–334  crossref  mathscinet  zmath  isi  elib  scopus
    5. M. S. Sgibnev, “Behavior at infinity of a solution to a differential-difference equation”, Siberian Math. J., 53:6 (2012), 1139–1154  mathnet  crossref  mathscinet  isi  elib  elib
    6. V. V. Vlasov, S. A. Ivanov, “Sharp estimates for solutions of systems with aftereffect”, St. Petersburg Math. J., 20:2 (2009), 193–211  mathnet  crossref  mathscinet  zmath  isi  elib
    7. 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  mathnet  crossref  mathscinet  elib
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