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Keywords:
integrodifferential system with after-effect; principal two-sided solutions; integrodifferential equations; principal solutions; small parameter
Summary:
The integrodifferential system with aftereffect ("heredity" or "prehistory") dx/dt=Ax+\varepsilon\int_{-\infty}^t R(t,s)x(s,\varepsilon)ds, is considered; here $\varepsilon$ is a positive small parameter, $A$ is a constant $n\times n$ matrix, $R(t,s)$ is the kernel of this system exponentially decreasing in norm as $t\to\infty$. It is proved, if matrix $A$ and kernel $R(t,s)$ satisfy some restrictions and $\varepsilon$ does not exceed some bound $\varepsilon_\ast$, then the $n$-dimensional set of the so-called principal two-sided solutions $\tilde{x}(t,\varepsilon)$ approximates in asymptotic sense the infinite-dimensional set of solutions $x(t,\varepsilon)$ corresponding a sufficiently wide class of initial functions. For $t$ growing to infinity an estimate of the difference between $x(t,\varepsilon)$ and $\tilde{x}(t,\varepsilon)$ is obtained.
References:
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[2] Yu. A. Ryabov: The existence of two-sided solutions of linear integrodifferential equations of Volterra type with aftereffect. Čas. Pěstov. Mat. 111 (1986), 26-33. (InRussian.) MR 0833154
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[4] Ya. V. Bykov: Problems of the Theory of Integrodifferential Equations. Ilim, Fгunze, 1957, 320 p. (In Russian.)

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