| Title:
|
Approximate properties of principal solutions of Volterra-type integrodifferential equations with infinite aftereffect (English) |
| Author:
|
Ryabov, Y. A. |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
120 |
| Issue:
|
3 |
| Year:
|
1995 |
| Pages:
|
265-282 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
integrodifferential system with after-effect |
| Keyword:
|
principal two-sided solutions |
| Keyword:
|
integrodifferential equations |
| Keyword:
|
principal solutions |
| Keyword:
|
small parameter |
| MSC:
|
34K15 |
| MSC:
|
34K25 |
| MSC:
|
45D05 |
| MSC:
|
45J05 |
| idZBL:
|
Zbl 0847.34078 |
| idMR:
|
MR1369685 |
| DOI:
|
10.21136/MB.1995.126010 |
| . |
| Date available:
|
2009-09-24T21:11:38Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/126010 |
| . |
| Reference:
|
[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 |
| Reference:
|
[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 |
| Reference:
|
[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.) Zbl 0654.45006, MR 0887728 |
| Reference:
|
[4] Ya. V. Bykov: Problems of the Theory of Integrodifferential Equations.Ilim, Fгunze, 1957, 320 p. (In Russian.) |
| . |
