On the theory of periodic solutions of systems of hyperbolic equations in the plane
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry, Mechanics, and Differential Equations, Tome 210 (2022), pp. 35-48

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

A periodic problem on the plane for a system of second-order hyperbolic equations with mixed derivatives is considered. The existence of a unique classical solution of the problem is examined and methods of constructing it are discussed.
Keywords: system of hyperbolic equations, doubly periodic solution, solvability, algorithm, method of functional parameters.
A. T. Assanova. On the theory of periodic solutions of systems of hyperbolic equations in the plane. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry, Mechanics, and Differential Equations, Tome 210 (2022), pp. 35-48. http://geodesic.mathdoc.fr/item/INTO_2022_210_a4/
@article{INTO_2022_210_a4,
     author = {A. T. Assanova},
     title = {On the theory of periodic solutions of systems of hyperbolic equations in the plane},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {35--48},
     year = {2022},
     volume = {210},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2022_210_a4/}
}
TY  - JOUR
AU  - A. T. Assanova
TI  - On the theory of periodic solutions of systems of hyperbolic equations in the plane
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2022
SP  - 35
EP  - 48
VL  - 210
UR  - http://geodesic.mathdoc.fr/item/INTO_2022_210_a4/
LA  - ru
ID  - INTO_2022_210_a4
ER  - 
%0 Journal Article
%A A. T. Assanova
%T On the theory of periodic solutions of systems of hyperbolic equations in the plane
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2022
%P 35-48
%V 210
%U http://geodesic.mathdoc.fr/item/INTO_2022_210_a4/
%G ru
%F INTO_2022_210_a4

[1] Artemev N. A., “Periodicheskie resheniya odnogo klassa uravnenii v chastnykh proizvodnykh”, Izv. AN SSSR. Ser. mat., 1937, no. 1, 15–50

[2] Asanova A. T., “Priznaki odnoznachnoi razreshimosti nelokalnoi kraevoi zadachi dlya sistem giperbolicheskikh uravnenii so smeshannymi proizvodnymi”, Izv. vuzov. Mat., 2016, no. 5, 3–21 | MR | Zbl

[3] Asanova A. T., “Periodicheskie na ploskosti resheniya sistemy giperbolicheskikh uravnenii vtorogo poryadka”, Mat. zametki., 101:1 (2017), 20–30 | MR | Zbl

[4] Asanova A. T., Dzhumabaev D. S., “Periodicheskie i ogranichennye na ploskosti resheniya sistem giperbolicheskikh uravnenii”, Ukr. mat. zh., 56:4 (2004), 562–572 | MR | Zbl

[5] Asanova A. T., Dzhumabaev D. S., “Odnoznachnaya razreshimost kraevoi zadachi s dannymi na kharakteristikakh dlya sistem giperbolicheskikh uravnenii”, Zh. vychisl. mat. mat. fiz., 42:11 (2002), 1673–1685 | MR | Zbl

[6] Asanova A. T., Dzhumabaev D. S., “Odnoznachnaya razreshimost nelokalnoi kraevoi zadachi dlya sistem giperbolicheskikh uravnenii ooo”, Differ. uravn., 39:10 (2003), 1343–1354 | MR | Zbl

[7] Asanova A. T., Dzhumabaev D. S., “Korrektnaya razreshimost nelokalnykh kraevykh zadach dlya sistem giperbolicheskikh uravnenii”, Differ. uravn., 41:3 (2005), 337–346 | MR | Zbl

[8] Dzhumabaev D. S., “Priznaki odnoznachnoi razreshimosti lineinoi kraevoi zadachi dlya obyknovennogo differentsialnogo uravneniya”, Zh. vychisl. mat. mat. fiz., 29:1 (1989), 50–66 | MR | Zbl

[9] Zhestkov S. V., “O dvoyakoperiodicheskikh resheniyakh nelineinykh giperbolicheskikh sistem v chastnykh proizvodnykh”, Ukr. mat. zh., 39:4 (1987), 521–523 | MR

[10] Kiguradze T. I., “O dvoyakoperiodicheskikh resheniyakh odnogo klassa nelineinykh giperbolicheskikh uravnenii”, Differ. uravn., 34:2 (1998), 238–245 | MR | Zbl

[11] Kolesov A. Yu., Mischenko E. F., Rozov N. Kh., “Asimptoticheskie metody issledovaniya periodicheskikh reshenii nelineinykh giperbolicheskikh uravnenii”, Tr. Mat. in-ta im. V. A. Steklova RAN., 222 (1998), 1–191

[12] Nakhushev A. M., Zadachi so smescheniem dlya uravnenii v chastnykh proizvodnykh, Nauka, M., 2006

[13] Mitropolskii Yu. A., Khoma G. P., Gromyak M. I., Asimptoticheskie metody issledovaniya kvazivolnovykh uravnenii giperbolicheskogo tipa, Naukova dumka, Kiev:, 1991 | MR

[14] Ptashnik B. I., Nekorrektnye granichnye zadachi dlya differentsialnykh uravnenii s chastnymi proizvodnymi, Naukova dumka, Kiev, 1984 | MR

[15] Rudakov I. A., “Periodicheskie resheniya nelineinogo volnovogo uravneniya s pepostoyannymi koeffitsientami”, Mat. zametki., 76:3 (2004), 427–438 | MR | Zbl

[16] Rudakov I. A., “Periodicheskie resheniya nelineinogo volnovogo uravneniya s granichnymi usloviyami Neimana i Dirikhle”, Izv. vuzov. Mat., 2007, no. 2, 46–55 | Zbl

[17] Samoilenko A. M., Ponto H. I., Chislenno-analiticheskie metody issledovaniya pepiodicheskikh peshenii, Vischa shkola, Kiev, 1976 | MR

[18] Samoilenko A. M., Ponto H. I., Chislenno-analiticheskie metody issledovaniya peshenii kpaevykh zadach, Naukova dumka, Kiev, 1985 | MR

[19] Samoilenko A. M., Tkach B. P., Chislenno-analiticheskie metody v teorii periodicheskikh reshenii uravnenii s chastnymi proizvodnymi, Naukova dumka, Kiev, 1992

[20] Asanova A. T., Dzhumabaev D. S., “Well-posedness of nonlocal boundary-value problems with integral condition for the system of hyperbolic equations”, J. Math. Anal. Appl., 402:1 (2013), 167–178 | DOI | MR | Zbl

[21] Aziz A. K., “Periodic solutions of hyperbolic partial differential equations”, Proc. Am. Math. Soc., 17:3 (1966), 557–566 | DOI | MR | Zbl

[22] Aziz A. K., Horak M. G., “Periodic solutions of hyperbolic partial differential equations in the large”, SIAM J. Math. Anal., 3:1 (1972), 176–182 | DOI | MR | Zbl

[23] Cesari L., “Periodic solutions of partial differential equations”, Tr. Mezhdunar. simp. po nelineinym kolebaniyam, Izd-vo AN USSR, Kiev, 1963, 440–457

[24] Cesari L., “Existence in the large of periodic solutions of hyperbolic partial differential equations”, Arch. Rat. Mech. Anal., 20:2 (1965), 170–190 | DOI | MR | Zbl

[25] Kiguradze T., “Some boundary value problems for systems of linear partial differential equations of hyperbolic type”, Mem. Differ. Equations Math. Phys., 1 (1994), 1–144 | MR | Zbl

[26] Kiguradze T., “On periodic in the plane solutions of second-order hyperbolic systems”, Arch. Math., 33:4 (1997), 253–272 | MR | Zbl

[27] Kiguradze T., Lakshmikantham V., “On doubly periodic solutions of fourth-order linear hyperbolic equations”, Nonlin. Anal., 49:1 (2002), 87–112 | DOI | MR | Zbl

[28] Vejvoda O., Herrmann L., Lovicar V., et al. sss, Partial Differential Equations: Time-Periodic Solutions, Martinus Nijhoff, Hague–Boston–London, 1982 | MR | Zbl