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Ufimsk. Mat. Zh., 2018, Volume 10, Issue 4, Pages 123–129 (Mi ufa454)  

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

Abstract: In the present paper, we are concerned with the Sturm–Liouville operator
$$\mathcal{L}[q] u:=-u"+q(x)u$$
subject to the separated boundary conditions. We suppose that $q \in L^2(0,\pi)$ and study a so-called inverse optimization spectral problem: given a potential $q_0$ and a value $\lambda_k $, where $k=1,2,…$, find a potential $\hat{q}$ closest to $q_0$ in the norm of $L^2(0,\pi)$ such that the value $\lambda_k$ coincides with $k$-th eigenvalue $\lambda_k(\hat{q})$ of the operator $\mathcal{L}[\hat{q}]$.
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
$$ -u"+q_0(x) u=\lambda_k u+\sigma u^3 $$
with $\sigma=1$ or $\sigma=-1$. 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.


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English version:
Ufa Mathematical Journal, 2018, 10:4, 122–128 (PDF, 344 kB); https://doi.org/10.13108/2018-10-4-122

Bibliographic databases:

UDC: 517.9
MSC: 34L05, 34L30, 34A55
Received: 19.09.2018
Language:

Citation: Ya. Il'yasov, N. Valeev, “On inverse spectral problem and generalized Sturm nodal theorem for nonlinear boundary value problems”, Ufimsk. Mat. Zh., 10:4 (2018), 123–129; Ufa Math. J., 10:4 (2018), 122–128

Citation in format AMSBIB
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\paper On inverse spectral problem and generalized Sturm nodal theorem for nonlinear boundary value problems
\jour Ufimsk. Mat. Zh.
\yr 2018
\vol 10
\issue 4
\pages 123--129
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\jour Ufa Math. J.
\yr 2018
\vol 10
\issue 4
\pages 122--128
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