Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 

 

On the structure of the set of periods for periodic solutions of some linear integro-differential equations on the multidimensional sphere


Author: Dang Khanh Hoi
Translated by: A. Plotkin
Original publication: Algebra i Analiz, tom 18 (2006), nomer 4.
Journal: St. Petersburg Math. J. 18 (2007), 573-581
MSC (2000): Primary 35K20
DOI: https://doi.org/10.1090/S1061-0022-07-00961-2
Published electronically: May 29, 2007
MathSciNet review: 2262584
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The problem of periodic solutions for the family of linear differential equations

$\displaystyle (L-{\lambda})u\equiv \Big(\frac{1}{i}\frac{\partial}{\partial t} - a\Delta- \lambda\Big) u(x,t)=\nu G(u-f) $

is considered on the multidimensional sphere $ x\in S^n$ under the periodicity condition $ u\vert _{t=0}=u\vert _{t=b}$. Here $ a$ and $ \lambda$ are given reals, $ \nu$ is a fixed complex number, $ G u(x,t)$ is a linear integral operator, and $ \Delta$ is the Laplace operator on $ S^n$. It is shown that the set of parameters $ (\nu, b)$ for which the above problem admits a unique solution is a measurable set of full measure in $ {\mathbb{C}} \times {\mathbb{R}}^+$.


References [Enhancements On Off] (What's this?)

  • 1. I. P. Kornfel′d, Ya. G. Sinaĭ, and S. V. Fomin, \cyr Ergodicheskaya teoriya, “Nauka”, Moscow, 1980 (Russian). MR 610981
    I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinaĭ, Ergodic theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 245, Springer-Verlag, New York, 1982. Translated from the Russian by A. B. Sosinskiĭ. MR 832433
  • 2. Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. MR 1157815
  • 3. M. A. Šubin, \cyr Psevdodifferentsial′nye operatory i spektral′naya teoriya, “Nauka”, Moscow, 1978 (Russian). MR 509034
    M. A. Shubin, Pseudodifferential operators and spectral theory, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1987. Translated from the Russian by Stig I. Andersson. MR 883081
  • 4. Dang Khanh Hoi, Periodic solutions for some nonlinear evolution systems of natural differential equations, Differential Equations and Related Problems (Moscow, 2004): Thesis, p. 48 (Russian)
  • 5. -, On periodic solutions for some nonlinear evolution natural differential equations on multidimensional torus, Vestnik Novgorod. Gos. Univ. Ser. Tekhn. Nauki No. 28 (2004), 77-79. (Russian)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 35K20

Retrieve articles in all journals with MSC (2000): 35K20


Additional Information

Dang Khanh Hoi
Affiliation: Division of Mathematical Analysis, Novgorod State University, Bol′shaya St.-Peterburgskaya Ulitsa 41, 173003, Velikiĭ Novgorod, Russia
Email: dangkhanhhoi@yahoo.com

DOI: https://doi.org/10.1090/S1061-0022-07-00961-2
Keywords: Schr\"odinger-type equation, periodicity condition
Received by editor(s): December 1, 2005
Published electronically: May 29, 2007
Article copyright: © Copyright 2007 American Mathematical Society