Approximate properties of principal solutions of Volterra-type integrodifferential equations with infinite aftereffect

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