| Title:
|
On periodic in the plane solutions of second order linear hyperbolic systems (English) |
| Author:
|
Kiguradze, Tariel |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
33 |
| Issue:
|
3 |
| Year:
|
1997 |
| Pages:
|
253-272 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Sufficient conditions for the problem \[ {\partial ^2 u\over \partial x\partial y}=P_0(x,y)u+ P_1(x,y){\partial u\over \partial x}+P_2(x,y){\partial u\over \partial y}+ q(x,y), u(x+\omega _1,y)=u(x,y),\quad u(x,y+\omega _2)=u(x,y) \] to have the Fredholm property and to be uniquely solvable are established, where $\omega _1$ and $\omega _2$ are positive constants and $P_j:R^2\rightarrow R^{n\times n}$ $(j=0,1,2)$ and $q:R^2\rightarrow R^n$ are continuous matrix and vector functions periodic in $x$ and $y$. (English) |
| Keyword:
|
hyperbolic system |
| Keyword:
|
periodic solution |
| Keyword:
|
F property |
| MSC:
|
35B10 |
| MSC:
|
35L10 |
| MSC:
|
35L20 |
| MSC:
|
35L55 |
| idZBL:
|
Zbl 0911.35067 |
| idMR:
|
MR1601317 |
| . |
| Date available:
|
2008-06-06T21:33:32Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107615 |
| . |
| Reference:
|
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| Reference:
|
[2] Cesari, L.: Existence in the large of periodic solutions of hyperbolic partial differential equations.Arch. Rational Mech. Anal. 20 (1965), 170-190. Zbl 0154.35902, MR 0183976 |
| Reference:
|
[3] Cesari, L.: Periodic solutions of nonlinear hyperbolic differential equations.Coll. Inter. Centre Nat. Rech. Sci. 148 (1965), 425-437. |
| Reference:
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[4] Cesari, L.: Smoothness properties of periodic solutions in the large of nonlinear hyperbolic differential systems.Funkcial. Ekvac. 9 (1966), 325-338. Zbl 0154.35903, MR 0212343 |
| Reference:
|
[5] Hale, J. K.: Periodic solutions of a class of hyperbolic equations containing a smalls parameter.Arch. Rat. Mech. Anal. 23 (1967), No. 5, 380-398. MR 0206503 |
| Reference:
|
[6] Hartman, P.: Ordinary differential equations..John Wiley & Sons, New York-London-Sydney, 1964. Zbl 1009.34001, MR 0171038 |
| Reference:
|
[7] Kiguradze, T. I.: On the periodic boundary value problems for linear hyperbolic equations I. (Russian).Differentsial’nye Uravneniya 29 (1993), No. 2, 281-297. MR 1236111 |
| Reference:
|
[8] Kiguradze, T. I.: On the periodic boundary value problems for linear hyperbolic equations II. (Russian).Differentsial’nye Uravneniya 29 (1993), No. 4, 637-645. MR 1250721 |
| Reference:
|
[9] Kiguradze, T. I.: Some boundary value problems for systems of linear partial differential equations of hyperbolic type.Memoirs on Differential Equations and Mathematical Physics 1 (1994), 1-144. Zbl 0819.35003, MR 1296228 |
| Reference:
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[10] Kiguradze, T. I.: On bounded in a strip solutions of the hyperbolic partial differential equations.Applicable Analysis 58 (1995), 199-214. MR 1383188 |
| Reference:
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[11] Kolmogorov, A. N., Fomin, S. V.: Elements of theory of functions and functional analysis (Russian).Nauka, Moscow, 1989. MR 1025126 |
| Reference:
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[12] Lusternik, L. A., Sobolev, V. I.: Elements of functional analysis (Russian).Nauka, Moscow, 1965. MR 0209802 |
| Reference:
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[13] Lakshmikantham, V., Pandit, S. G.: Periodic solutions of hyperbolic partial differential equations.Comput. and Math. 11 (1985), No. 1-3, 249-259. MR 0787440 |
| Reference:
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[14] Liu Baoping: The integral operator method for finding almost-periodic solutions of nonlinear wave equations.Nonlinear Anal. TMA 11 (1987), No. 5, 553-564. Zbl 0666.35005, MR 0886648 |
| Reference:
|
[15] Žestkov, S. V.: On twice periodic solutions of quasilinear hyperbolic systems.Differentsial’nye Uravnenia 24 (1988), No. 12, 2164-2166. |
| . |
