| Title:
|
Partially irregular almost periodic solutions of ordinary differential systems (English) |
| Author:
|
Demenchuk, Alexandr |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
126 |
| Issue:
|
1 |
| Year:
|
2001 |
| Pages:
|
221-228 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $f(t,x)$ be a vector valued function almost periodic in $t$ uniformly for $x$, and let ${\mathrm mod}(f)=L_1\oplus L_2$ be its frequency module. We say that an almost periodic solution $x(t)$ of the system \[ \dot{x} = f (t, x), \quad t\in \mathbb{R}, \ \ x\in D \subset \mathbb{R}^n \] is irregular with respect to $L_2$ (or partially irregular) if $({\mathrm mod}(x)+L_1) \cap L_2 = \lbrace 0\rbrace $. Suppose that $ f(t,x) = A(t)x + X(t, x), $ where $A(t)$ is an almost periodic $(n\times n)$-matrix and ${\mathrm mod} (A)\cap {\mathrm mod}(X)= \lbrace 0\rbrace .$ We consider the existence problem for almost periodic irregular with respect to ${\mathrm mod} (A)$ solutions of such system. This problem is reduced to a similar problem for a system of smaller dimension, and sufficient conditions for existence of such solutions are obtained. (English) |
| Keyword:
|
almost periodic differential systems |
| Keyword:
|
almost periodic solutions |
| MSC:
|
34C27 |
| idZBL:
|
Zbl 0987.34039 |
| idMR:
|
MR1826484 |
| DOI:
|
10.21136/MB.2001.133916 |
| . |
| Date available:
|
2009-09-24T21:49:21Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/133916 |
| . |
| Reference:
|
[1] Samoilenko, A. M.: The Elements of Mathematical Theory of Multifrequency Oscillations.Nauka, Moskva, 1987. (Russian) MR 0928806 |
| Reference:
|
[2] Levitan, B. M., Zhikov, V. V.: Almost Periodic Functions and Differential Equations.Izdatelstvo Moskovskogo Universiteta, Moskva, 1978. (Russian) MR 0509035 |
| Reference:
|
[3] Hale, J. K.: Oscillations in Nonlinear Systems.Mc Graw-Hill, New York, 1963. Zbl 0115.07401, MR 0150402 |
| Reference:
|
[4] Fink, A. M.: Almost Periodic Differential Equations.Lecture Notes in Mathematics 377, Springer, Berlin, 1974. Zbl 0325.34039, MR 0460799 |
| Reference:
|
[5] Demidovich, B. P.: Lectures on Mathematical Stability Theory.Nauka, Moskva, 1967. (Russian) Zbl 0155.41601, MR 0226126 |
| Reference:
|
[6] Bibikov, Y. N.: Multifrequency Nonlinear Oscillations and Their Bifurcations.Izdatelstvo Leningradskogo Universiteta, Leningrad, 1991. Zbl 0791.34032, MR 1126680 |
| Reference:
|
[7] Zubov, V. I.: Oscillation Theory.Nauka, Moskva, 1979. (Russian) |
| Reference:
|
[8] Kurzweil, J., Vejvoda, O.: On periodic and almost periodic solutions of the ordinary differential systems.Czechoslovak Math. J. 5 (1955), 362–370. MR 0076127 |
| Reference:
|
[9] Massera, J. L.: Observationes sobre les soluciones periodicas de ecuaciones differentiales.Boletin de la Facultad de Ingenieria 4 (1950), 37–45. |
| Reference:
|
[10] Erugin, N. P.: On periodic solutions of the linear homogeneous system of differential equations.Doklady Akademii Nauk BSSR 6 (1962), 407–410. (Russian) MR 0137889 |
| Reference:
|
[11] Grudo, E. I.: On periodic solutions with incommensurable periods of periodic differential systems.Differentsial’nye uravneniya 22 (1986), 1409–1506. (Russian) MR 0865386 |
| Reference:
|
[12] Grudo, E. I., Demenchuk, A. K.: On periodic solutions with incommensurable periods of linear nonhomogeneous periodic differential systems.Differentsial’nye uravneniya 23 (1987), 409–416. (Russian) MR 0886568 |
| Reference:
|
[13] Demenchuk, A. K.: On almost periodic solutions of ordinary differential systems.Izvestiya Akademii Nauk BSSR. Ser. fiz.-mat. nauk. 4 (1987), 16–22. (Russian) Zbl 0632.34050, MR 0913280 |
| Reference:
|
[14] Grudo, E. I., Demenchuk, A. K.: On periodic and almost periodic solutions of ordinary differential systems.Issues of qualitative theory of differential equations. Novosibirsk (1987), 23–29. (Russian) MR 0991144 |
| Reference:
|
[15] Demenchuk, A. K.: Quasiperiodic solutions of differential systems with frequency bases of solutions and right sides that are linearly independent over $\mathbb{Q}$.Differentsial’nye uravneniya 27 (1991), 1673–1679. (Russian) MR 1157672 |
| Reference:
|
[16] Demenchuk, A. K.: On a class of quasiperiodic solutions of differential systems.Doklady Akademii Nauk Belarusi 36 (1992), 14–17. (Russian) Zbl 0756.34049, MR 1165405 |
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