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MATHEMATICA BOHEMICA, Vol. 120, No. 3, pp. 265-282, 1995
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Approximate properties of principal solutions of Volterra-type integrodifferential equations with infinite aftereffect

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Y. A. Ryabov

Russia, Moscow, 125829, Leningradsky pr. 64, Auto & Road Construction Engineering University, e-mail: vmath@madi.msk.su

**Abstract:** 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.

**Keywords:** integrodifferential equations, principal solutions, small parameter

**Classification (MSC91):** 34K25

**Full text of the article:**

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