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 $y$.
@article{107615,
author = {Tariel Kiguradze},
title = {On periodic in the plane solutions of second order linear hyperbolic systems},
journal = {Archivum Mathematicum},
volume = {033},
year = {1997},
pages = {253-272},
zbl = {0911.35067},
mrnumber = {1601317},
language = {en},
url = {http://dml.mathdoc.fr/item/107615}
}
Kiguradze, Tariel. On periodic in the plane solutions of second order linear hyperbolic systems. Archivum Mathematicum, Tome 033 (1997) pp. 253-272. http://gdmltest.u-ga.fr/item/107615/
Periodic solutions of hyperbolic partial differential equations in the large, SIAM J. Math. Anal. 3 (1972), No. 1, 176-182. (1972) | MR 0312043
Existence in the large of periodic solutions of hyperbolic partial differential equations, Arch. Rational Mech. Anal. 20 (1965), 170-190. (1965) | MR 0183976 | Zbl 0154.35902
Periodic solutions of nonlinear hyperbolic differential equations, Coll. Inter. Centre Nat. Rech. Sci. 148 (1965), 425-437. (1965)
Smoothness properties of periodic solutions in the large of nonlinear hyperbolic differential systems, Funkcial. Ekvac. 9 (1966), 325-338. (1966) | MR 0212343 | Zbl 0154.35903
Periodic solutions of a class of hyperbolic equations containing a smalls parameter, Arch. Rat. Mech. Anal. 23 (1967), No. 5, 380-398. (1967) | MR 0206503
Ordinary differential equations., John Wiley & Sons, New York-London-Sydney, 1964. | MR 0171038 | Zbl 1009.34001
On the periodic boundary value problems for linear hyperbolic equations I. (Russian), Differentsial’nye Uravneniya 29 (1993), No. 2, 281-297. (1993) | MR 1236111
On the periodic boundary value problems for linear hyperbolic equations II. (Russian), Differentsial’nye Uravneniya 29 (1993), No. 4, 637-645. (1993) | MR 1250721
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. (1994) | MR 1296228 | Zbl 0819.35003
On bounded in a strip solutions of the hyperbolic partial differential equations, Applicable Analysis 58 (1995), 199-214. | MR 1383188
Elements of theory of functions and functional analysis (Russian), Nauka, Moscow, 1989. | MR 1025126
Elements of functional analysis (Russian), Nauka, Moscow, 1965. | MR 0209802
Periodic solutions of hyperbolic partial differential equations, Comput. and Math. 11 (1985), No. 1-3, 249-259. (1985) | MR 0787440
The integral operator method for finding almost-periodic solutions of nonlinear wave equations, Nonlinear Anal. TMA 11 (1987), No. 5, 553-564. (1987) | MR 0886648 | Zbl 0666.35005
On twice periodic solutions of quasilinear hyperbolic systems, Differentsial’nye Uravnenia 24 (1988), No. 12, 2164-2166. (1988)