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On stability of functional differential equations with rapidly oscillating coefficients

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This work was supported by the Ministry of Education and Science of the Russian Federation under state order No. 3.1761.2017.
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  • The paper deals with the functional differential equation

    $\begin{equation*} y'(t)+∈t_0^{∞}μ_0(ds)\, y(t-s)+\sum_{k=1}^∞ e^{iω_kt}∈t_0^{∞}μ_k(ds)\, y(t-s)=f(t), \end{equation*}$

    where the functions $y$ and $f$ take their values in a Hilbert space, $ω_k∈\mathbb{R}$ , $μ_k$ are bounded operator-valued measures concentrated on $[0, +∞)$ , and $\sum_{k=1}^∞\Vertμ_k\Vert < ∞$ . It is shown that the equation is stable provided the unperturbed equation $y'(t)+\int{{_0^{∞}}}μ_0(ds)\, y(t-s)=f(t)$ is at least strictly passive (and consequently stable) and a special estimate holds; this estimate is certainly true if $|ω_k|$ are sufficiently large.

    Mathematics Subject Classification: Primary:34K20;Secondary:34K06, 26A33, 34K33.

    Citation:

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  • Figure 1.  Examples of the images of characteristic functions.

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