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integrodifferential system with after-effect; principal two-sided solutions; integrodifferential equations; principal solutions; small parameter
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.
[1] Yu. A. Ryabov: Two-sided solutions of linear integrodifferential equation of Volterra-type with infinite lower limit. The researches on the theory of differential equations, ed. MADI. 1986, pp. 3-16. (In Russian.) MR 1012539
[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
[3] Yu. A. Ryabov: Principal two-sided solutions of Volterra-type linear integrodifferential equations with infinite aftereffect. Ukrain. Matemat. Zhurnal 39 (1987), no. 1, 92-97. (In Russian.) MR 0887728 | Zbl 0654.45006
[4] Ya. V. Bykov: Problems of the Theory of Integrodifferential Equations. Ilim, Fгunze, 1957, 320 p. (In Russian.)
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