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On inverse spectral problem and generalized Sturm nodal theorem for nonlinear boundary value problems
Ufa mathematical journal, Tome 10 (2018) no. 4, pp. 122-128 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $k=1,2,\dots$, 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. 
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     title = {On inverse spectral problem and generalized {Sturm} nodal theorem for nonlinear boundary value problems},
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     url = {http://geodesic.mathdoc.fr/item/UFA_2018_10_4_a11/}
}
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Ya. Il'yasov; N. Valeev. On inverse spectral problem and generalized Sturm nodal theorem for nonlinear boundary value problems. Ufa mathematical journal, Tome 10 (2018) no. 4, pp. 122-128. http://geodesic.mathdoc.fr/item/UFA_2018_10_4_a11/

[1] V. Ambarzumian, “Über eine frage der eigenwerttheorie”, Zeit. Physik A Hadrons and Nuclei, 53:9 (1929), 690–695 | Zbl

[2] G. Borg, “Eine umkehrung der Sturm-Liouvilleschen eigenwertaufgabe”, Acta Math, 78:1 (1946), 1–96 | DOI | MR | Zbl

[3] D. E. Edmunds, W. D. Evans, Spectral theory and differential operators, Oxford Mathematical Monographs, 15, Clarendon Press, Oxford, 1987 | MR

[4] I. M. Gel'fand, B. M. Levitan, “On the determination of a differential equation from its spectral function”, Izv. RAN. Ser. Matem., 15:4 (1951), 309–360 (in Russian) | MR | Zbl

[5] Y. Sh. Ilyasov, N. F. Valeev, “On an inverse optimization spectral problem and a related nonlinear boundary value problem”, Math. Notes, 104:4 (2018), 601–605 | MR

[6] Y. Sh. Ilyasov, N. F. Valeev, “On nonlinear boundary value problem corresponding to $N$-dimensional inverse spectral problem”, J. Diff. Equat. (to appear)

[7] M. Möller, A. Zettl, “Differentiable dependence of eigenvalues of operators in Banach spaces”, J. Oper. Theor., 36:2 (1996), 335–355 | MR | Zbl

[8] J. Pöschel, E. Trubowitz, Inverse spectral theory, Pure Appl. Math., 130, Academic Press, Boston, 1987 | MR | Zbl

[9] A. A. Shkalikov, “Perturbations of self-adjoint and normal operators with discrete spectrum”, Russ. Math. Surv., 71:5 (2016), 907–965 | DOI | MR

[10] A. Zettl, Sturm-Liouville theory, Math. Surv. Monog., 121, Amer. Math. Soc., Providence, RI, 2005 | MR | Zbl