Geodesic
Approximate properties of principal solutions of Volterra-type integrodifferential equations with infinite aftereffect
Mathematica Bohemica, Tome 120 (1995) no. 3, pp. 265-282

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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.
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.
DOI : 10.21136/MB.1995.126010
Classification : 34K15, 34K25, 45D05, 45J05
Keywords: integrodifferential system with after-effect; principal two-sided solutions; integrodifferential equations; principal solutions; small parameter 
Ryabov, Y. A. Approximate properties of principal solutions of Volterra-type integrodifferential equations with infinite aftereffect. Mathematica Bohemica, Tome 120 (1995) no. 3, pp. 265-282. doi: 10.21136/MB.1995.126010
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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

[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 | Zbl

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