RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Forthcoming papers
Archive
Impact factor
Subscription
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Mat. Zametki:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Mat. Zametki, 1974, Volume 16, Issue 6, Pages 969–980 (Mi mz7539)  

On summability with a weight of a solution of the Sturm-Liouville equation

M. O. Otelbaev

Institute of Mathematics and Mechanics, AS of Kazakh SSR

Abstract: We study problems of summability with a weight of a solution of the Sturm–Liouville equation
$$ -y'+q(x)y=f,\quad x\in J=(-\infty,\infty). $$
with bounded potential $q(x)$, satisfying the conditions
\begin{gather*} \inf_{x\in J}q(x)\ge-\mu+1,\quad\sup_{|x-y|\le2}\frac{q(x)+\mu}{q(y)+\mu}<+\infty,
\sup_{|x-y|\le2}\{|x-y|^{-\alpha}|q(x)|^{-\alpha}\exp(-r|x-y|\sqrt{q(x)+\lambda})|q(x)-q(y)|\}<+\infty, \end{gather*}
where $\alpha\in(0,1]$, $r\in[0,1)$, $2-2a+\alpha>0$, $a\ge0$, $\mu\ge0$.
Our main result is the following: let $q(x)$ satisfy the conditions given above and let l$(x)$ be a nonnegative function such that
$$ C(|x|^C+1)\ge l(x)\ge C^{-1}(|x|^C+1)^{-1},\quad\sup_{|x-y|\le2}\frac{l(x)}{l(y)}<+\infty, $$
then if $-y"+q(x)y=f$ и $y(x)l(x), f(x)l(x)\in L_p(J)$ ($1\le p<\infty$), it follows that
\begin{gather*} y"l(x),\quad q(x)l(x)y(x),
(q(x)+\mu)^{1/2}y'(x)l(x)\in L_p(J). \end{gather*}


Full text: PDF file (749 kB)

English version:
Mathematical Notes, 1974, 16:6, 1172–1179

Bibliographic databases:

UDC: 517.43
Received: 27.03.1974

Citation: M. O. Otelbaev, “On summability with a weight of a solution of the Sturm-Liouville equation”, Mat. Zametki, 16:6 (1974), 969–980; Math. Notes, 16:6 (1974), 1172–1179

Citation in format AMSBIB
\Bibitem{Ote74}
\by M.~O.~Otelbaev
\paper On summability with a~weight of a~solution of the Sturm-Liouville equation
\jour Mat. Zametki
\yr 1974
\vol 16
\issue 6
\pages 969--980
\mathnet{http://mi.mathnet.ru/mz7539}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=369798}
\zmath{https://zbmath.org/?q=an:0321.40011}
\transl
\jour Math. Notes
\yr 1974
\vol 16
\issue 6
\pages 1172--1179
\crossref{https://doi.org/10.1007/BF01098446}


Linking options:
  • http://mi.mathnet.ru/eng/mz7539
  • http://mi.mathnet.ru/eng/mz/v16/i6/p969

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles
  • Математические заметки Mathematical Notes
    Number of views:
    This page:205
    Full text:88
    First page:1

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2020