Abstract:
In this paper, we study scalar difference-differential equations of neutral type of general form
m∑j=0∫h0u(j)(t−θ)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)⩽CTq−1eϰT‖u(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.
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
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Linking options:
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This publication is cited in the following 7 articles:
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
M. S. Sgibnev, “Behavior at infinity of a solution to a matrix differential-difference equation”, Siberian Math. J., 55:3 (2014), 530–543
M. S. Sgibnev, “An asymptotic expansion of the solution of a matrix difference equation of general form”, Sb. Math., 205:12 (2014), 1815–1828
Sgibnev M.S., “Asymptotic Expansion of the Solution of a Differential-Difference Equation of General Form”, Differ. Equ., 50:3 (2014), 323–334
M. S. Sgibnev, “Behavior at infinity of a solution to a differential-difference equation”, Siberian Math. J., 53:6 (2012), 1139–1154
V. V. Vlasov, S. A. Ivanov, “Sharp estimates for solutions of systems with aftereffect”, St. Petersburg Math. J., 20:2 (2009), 193–211
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