Loading [MathJax]/jax/output/CommonHTML/jax.js
Ufa Mathematical Journal
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Archive
Impact factor

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Ufimsk. Mat. Zh.:
Year:
Volume:
Issue:
Page:
Find






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


Ufa Mathematical Journal, 2018, Volume 10, Issue 4, Pages 122–128
DOI: https://doi.org/10.13108/2018-10-4-122
(Mi ufa454)
 

This article is cited in 5 scientific papers (total in 5 papers)

On inverse spectral problem and generalized Sturm nodal theorem for nonlinear boundary value problems

Ya. Il'yasovab, N. Valeevca

a Institute of Mathematics, Ufa Federal Research Center, RAS, 450008, Ufa, Russia
b Instituto de Matemática e Estatística, Universidade Federal de Goiás, 74001-970, Goiania, Brazil
c Bashkir State University, 450076, Ufa, Russia
References:
Abstract: In the present paper, we are concerned with the Sturm–Liouville operator
L[q]u:=u+q(x)u
subject to the separated boundary conditions. We suppose that qL2(0,π) and study a so-called inverse optimization spectral problem: given a potential q0 and a value λk, where k=1,2,, find a potential ˆq closest to q0 in the norm of L2(0,π) such that the value λk coincides with k-th eigenvalue λk(ˆq) of the operator L[ˆq].
In the main result, we prove that this problem is related to the existence of a solution to a boundary value problem for the nonlinear equation
u+q0(x)u=λku+σu3
with σ=1 or σ=1. This implies that the minimizing solution of the inverse optimization spectral problem can be obtained by solving the corresponding nonlinear boundary value problem. On the other hand, this relationship allows us to establish an explicit formula for the solution to the nonlinear equation by finding the minimizer of the corresponding inverse optimization spectral problem. As a consequence of this result, a new method of proving the generalized Sturm nodal theorem for the nonlinear boundary value problems is obtained.
Keywords: Sturm–Liouville operator, inverse optimization spectral problem, nodal theorem for the nonlinear boundary value problems.
Funding agency Grant number
Russian Foundation for Basic Research 18-51-06002_Az_a
The second author was partially supported by RFBR grant no. 18-51-06002 Az-a.
Received: 19.09.2018
Bibliographic databases:
Document Type: Article
UDC: 517.9
MSC: 34L05, 34L30, 34A55
Language: English
Original paper language: English
Citation: Ya. Il'yasov, N. Valeev, “On inverse spectral problem and generalized Sturm nodal theorem for nonlinear boundary value problems”, Ufa Math. J., 10:4 (2018), 122–128
Citation in format AMSBIB
\Bibitem{IlyVal18}
\by Ya.~Il'yasov, N.~Valeev
\paper On inverse spectral problem and generalized Sturm nodal theorem for nonlinear boundary value problems
\jour Ufa Math. J.
\yr 2018
\vol 10
\issue 4
\pages 122--128
\mathnet{http://mi.mathnet.ru//eng/ufa454}
\crossref{https://doi.org/10.13108/2018-10-4-122}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000457367000012}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85073684171}
Linking options:
  • https://www.mathnet.ru/eng/ufa454
  • https://doi.org/10.13108/2018-10-4-122
  • https://www.mathnet.ru/eng/ufa/v10/i4/p123
  • This publication is cited in the following 5 articles:
    1. Yifei Jia, Jiangang Qi, Jing Li, “Optimal recovery of potentials for Sturm-Liouville eigenvalue problems with separated boundary conditions”, Qual. Theory Dyn. Syst., 22:2 (2023)  crossref
    2. Ya. Ilyasov, N. Valeev, “Recovery of the nearest potential field from the m observed eigenvalues”, Physica D, 426 (2021), 132985  crossref  mathscinet  zmath  isi  scopus
    3. J. Qi, B. Xie, “Extremum estimates of the L1-norm of weights for eigenvalue problems of vibrating string equations based on critical equations”, Discrete Contin. Dyn. Syst.-Ser. B, 26:7 (2021), 3505–3516  crossref  mathscinet  zmath  isi  scopus
    4. N. F. Valeev, Y. Sh. Ilyasov, “Inverse spectral problem for Sturm–Liouville operator with prescribed partial trace”, Ufa Math. J., 12:4 (2020), 19–29  mathnet  crossref  isi
    5. Shuyuan Guo, Guixin Xu, Meirong Zhang, “On the second-order Fréchet derivatives of eigenvalues of Sturm–Liouville problems in potentials”, Arch. Math., 113:3 (2019), 301  crossref
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Уфимский математический журнал
    Statistics & downloads:
    Abstract page:252
    Russian version PDF:87
    English version PDF:13
    References:51
     
      Contact us:
    math-net2025_01@mi-ras.ru
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2025