Literature Cited
S. L. Sobolev, “Motion of a symmetric top filled with a liquid,” Prikl. Mekh. i Tekhn. Fiz., No. 3, 20–55 (1960).
S. L. Sobolev, “A new problem of mathematical physics,” Izv. Akad. Nauk SSSR, Ser. Matem.,18, No. 1, 3–50 (1954).
S. L. Sobolev, “Sur une classe des problemes de physique mathematique,” 48 Riunionne della Societa Italiana per ll Pregresso della Science, Rome (1965), 192–208.
T. I. Zelenyak, “Dependence on the boundary for solutions of some mixed problems for equations of small vibrations in a rotating liquid,” Dokl. Akad. Nauk SSSR,164, No. 6, 1225–1228 (1965).
V. N. Maslennikova, “An explicit solution of the Cauchy problem for a system of partial differential equations of first order,” Izv. Akad. Nauk SSSR, Ser. Matem.,22, No. 1, 135–160 (1958).
V. N. Maslennikova, “Mixed problems for a system of partial differential equations of first order,” Izv. Akad. Nauk SSSR, Ser. Matem.,22, No. 3, 271–298 (1958).
R. A. Aleksandryan, Dependence of Qualitative Properties of Solutions of Mixed Problems on the Shape of the Domain. Dissertation [in Russian], MGU, Moscow (1949).
Yu. M. Berezanskii, Expansions in Terms of Eigenfunctions of a Self-adjoint Operator [in Russian], Naukova Dumka, Kiev (1965).
R. A. Aleksandryan, “A Sobolev problem for special partial differential equations of fourth order,” Dokl. Akad. Nauk SSSR,73, No. 4, 631–634 (1950).
R. A. Aleksandryan, “Mixed problems for aclass of systems of differential equations of Sobolev type,” Trudy Vsesoyuzogo Soveshchaniya po Differentsiallnym Uravheniyam, Erevan, November, 1958.
R. A. Aleksandryan, “On the correctness of a mixed problem and the spectral equivalence of two operators connected with it,” Izv. Akad. Nauk ArmSSR,10, No. 1, 63–69 (1957).
R. A. Aleksandryan, “The Dirichlet problem and the completeness of a system of functions on the cirele,” Dokl. Akad. Nauk SSSR,73, No. 5, 869–872 (1950).
R. A. Aleksandryan, “Spectral properties of operators generated by systems of differential equations of Sobolev type,” Trudy Moscow Matem. ob-va,9, 455–505 (1960).
T. I. Zelenyak, “The behavior of a solution of a Sobolev problem as t→∞,” Dokl. Akad. Nauk SSSR,139, No. 3, 531–533 (1961).
T. I. Zelenyak, “A Sobolev problem,” Dokl. Akad. Nauk SSSR,147, No. 5, 1017–1019 (1952).
T. I. Zelenyak, “Invariant subspaces of an operator,” Sibirsk. Matem. Zh.,3, No. 3, 471–475 (1962).
T. I. Zelenyak, “A mixed problem for an equation which is not solvable with respect to higher time derivatives,” Dokl. Akad. Nauk SSSR,158, No. 6, 1268–1270 (1964).
T. I. Zelenyak, “Asymptotic behavior of solutions of a mixed problem,” Differential Equations.2. No. 1, 47–64 (1966).
F. John, “The Dirichlet problem for a hyperbolic equation,” Amer. J. Math.,63, 141–154 (1941).
D. G. Bourgin and K. Duffin, “The Dirichlet problem for the vibrating string equations,” Bull. Amer. Math. Soc.,45, 851–859 (1939).
N. N. Vakhaniya, “A boundary value problem with values given on the whole boundary for a hyperbolic system, equivalent to the equation of a vibrating string,” Dokl. Akad. Nauk SSSR,116, No. 6, 906–909 (1957).
R. T. Denchev, “The spectrum of an operator,” Dokl. Akad. Nauk SSSR,126, No. 2, 259–262 (1959).
A. Huber, “Die erste Randwertaufgabe für geschlos sone Bereiche bei der Gleichung ∂z/∂x by f(x, y)”, Monats Math. Phys.39, 79–100 (1932)
V. L. Arnol'd, “Small denominators,” Izv. Akad. Nauk SSSR, Ser. Matem.25, No. 1, 21–86 (1961).
A. Poincaré, On Curves Defined by Differential Equations [Russian translation], Moscow-Leningrad (1947).
A. Denjoy, “Sur les corbes definles par les equations differenticlles a la suriace du tore,” Journ. de Math.,11, No. 4, 333–375 (1932).
E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York (1955).
A. Finzi, “Sur le probleme de la generation d'une transformation donnee d'une courbe fermee par une transformation infinitesrmall.” Ann. Ecole Norm. Sup.,69, 371–430 (1952).
R. A. Aleksandryan, “Construction of a complete collection of solutions of a homogeneous Dirichlet problem for the equation of a vibrating string,” Dokl. Akad. Nauk SSSR,162, No. 2, 247–251 (1965).
Additional information
Translated from Sibirskii Matematicheskii Zhurnal, Vol. 9, No. 5, pp. 1075–1092, September–October, 1968.
Rights and permissions
About this article
Cite this article
Zelenyak, T.I. On generalized Eigenfunctions of an operator which is related to a problem of S. L. sobolev. Sib Math J 9, 798–811 (1968). https://doi.org/10.1007/BF01041165
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01041165