References

  1. Albanese A.A. On the global $C^{\infty}$ and Gevrey hypoellipticity on the torus of some classes of degenerate elliptic operators. Note di Matematica 2011, 31 (1), 1-13. doi: 10.1285/i15900932v31n1p1
  2. Arnol'd V.I. Small denominators and problems of stability of motion in classical and celestial mechanics. Uspehi Mat. Nauk 1963, 18 (6), 91-192. doi: 10.1070/RM1963v018n06ABEH001143 (in Russian)
  3. Beresnevich V., Dodson M., Kristensen S., Levesley J. An inhomogeneous wave equation and non-linear Diophantine approximation. Advances in Mathematics 2008, 217 (2), 740-760. doi: 10.1016/j.aim.2007.09.003
  4. Bourghin D. G., Duffin R. J. The Dirichlet problem for the vibrating string equation. Bull. Amer. Math. Soc. 1939, 45 (12), 851-858. doi: 10.1090/S0002-9904-1939-07103-6
  5. Dickinson H., Gramchev T., Yoshino M. Perturbations of vector fields on tori: resonant normal forms and Diophantine phenomena. Proc. Edinb. Math. Soc. 2002, 45 (3), 731-759. doi: 10.1017/S001309150000064X
  6. Gramchev T., Yoshino M. WKB analysis to global solvability and hypoellipticity. Publ. Res. Inst. Math. Sci. 1995, 31 (3), 443-464. doi: 10.2977/prims/1195164049
  7. Grebennikov E.A, Ryabov Yu.A. Resonances and small denominators in celestial mechanics. Nauka, Moscow, 1978. (in Russian)
  8. Greenfield S., Wallach N. Global hypoellipticity and Liouville numbers. Proc. Amer. Math. Soc. 1972, 31 (1), 112-114. doi: 10.2307/2038523
  9. Il'kiv V.S., Ptashnyk B.Yo. Problems for partial differential equations with nonlocal conditions. Metric approach to the problem of small denominators. Ukrainian Math. J. 2006, 58 (12), 1847-1875. doi: 10.1007/s11253-006-0172-8 (in Ukrainian)
  10. Kolmogorov A.N. On dynamical systems with an integral invariant on the torus. Doklady Akad. Nauk SSSR 1953, 93 (5), 763-766. (in Russian)
  11. Kristensen S. Diophantine approximation and the solubility of the Schrodinger equation. Phys. Lett. A. 2003, 314 (1), 15-18. doi: 10.1016/S0375-9601(03)00867-3
  12. Novak B. Remark on periodic solutions of a linear wave equation in one dimension. Comm. Math. Uni. Carolinae 1974, 15, 513-519.
  13. Petronilho G. Global hypoellipticity, global solvability and normal form for a class of real vector fields on a torus and application. Trans. Amer. Math. Soc. 2011, 363, 6337-6349. doi: 10.1090/S0002-9947-2011-05359-6
  14. Petronilho G. Global solvability and simultaneously approximable vectors. J. Differential Equations 2002, 184 (1), 48-61. doi: 10.1006/jdeq.2001.4132
  15. Polishchuk V.M., Ptashnyk B.Yo. Periodic boundary value problem for linear hyperbolic equations. Math.Methods Phys. Mech. Fields 1975, 5, 158-160. (in Russian)
  16. Polishchuk V.M., Ptashnyk B.Yo. Periodic solutions of a system of partial differential equations with constant coefficients. Ukrainian Math. J. 1980, 32 (2), 239-243. doi: 10.1007/BF01092793 (in Russian)
  17. Ptashnyk B.Yo., Il'kiv V.S., Kmit' I.Ya., Polishchuk V.M. Nonlocal Boundary-Value Problems for Partial Differential Equations. Naukova Dumka, Kiev, 2002. (in Ukrainian)
  18. Ptashnyk B.Yo. Periodic boundary value problem for linear hyperbolic equations with constant coefficients. In:Math. Physics, 12, 117-121. Naukova Dumka, Kiev, 1972. (in Russian)
  19. Ptashnyk B.Yo. Ill-Posed Boundary-Value Problems for Partial Differential Equations. Naukova Dumka, Kiev, 1984. (in Russian)
  20. Savka I.Ya. Nonlocal problem with dependent coefficients in conditions for the second-order equation in time variable. Carpathian Math. Publ. 2010, 2 (2), 101-110. (in Ukrainian)

Refbacks

  • There are currently no refbacks.


Creative Commons License
The journal is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported.