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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
Published electronically: May 29, 2007
MathSciNet review: 2262584
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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?)

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Additional Information

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

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