Abstract
In a separable Hilbert space \(H\), we study the asymptotic behavior of the eigenvalues of a boundary value problem for a second-order differential-operator equation. The spectral parameter of the problem occurs linearly in the equation and as a quadratic trinomial in one of the boundary conditions. Asymptotic formulas for the eigenvalues of the problem are found.
REFERENCES
Lions, J.-L. and Magenes., E., Problèmes aux limites non homogènes et applications, Paris: Dunod, 1968. Translated under the title: Neodnorodnye granichnye zadachi i ikh prilozheniya, Moscow: Mir, 1971.
Yakubov, S.Ya., Boundary value problem for the Laplace equation with nonclassical spectral asymptotics, Dokl. Akad. Nauk SSSR, 1982, vol. 265, no. 6, pp. 1330–1333.
Il’in, V.A. and Filippov, A.F., On the nature of the spectrum of a self-adjoint extension of the Laplace operator in a bounded domain, Dokl. Akad. Nauk SSSR, 1970, vol. 191, no. 2, pp. 167–169.
Kozhevnikov, A.N., Separate asymptotics of two series of eigenvalues for a single elliptic boundary-value problem, Math. Notes, 1977, vol. 22, no. 5, pp. 882–888.
Rybak, M.A., On the asymptotic distribution of eigenvalues of some boundary value problems for the Sturm–Liouville operator equation, Ukr. Mat. Zh., 1980, vol. 32, no. 2, pp. 248–252.
Denche, M., Abstract differential equation with a spectral parameter in the boundary conditions, Result. Math., 1999, vol. 35, pp. 216–227.
Aliev, B.A., Asymptotic behavior of the eigenvalues of a boundary value problem for an elliptic second order operator-differential equation, Ukr. Mat. Zh., 2006, vol. 58, no. 8, pp. 1146–1152.
Aliev, B.A. and Кurbanova, N.K., Asymptotic behavior of eigenvalues of a boundary value problem for a second order elliptic differential-operator equation, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb., 2014, vol. 40, spec. iss., pp. 23–29.
Aliev, B.A., Kurbanova, N.K., and Yakubov, Ya., Solvability of the abstract Regge boundary-value problem and asymptotic behavior of eigenvalues of one abstract spectral problem, Riv. Math. Univ. Di Parma, 2015, vol. 6, pp. 241–265.
Kapustin, N.Yu., On a spectral problem in the theory of the heat operator, Differ. Equations, 2009, vol. 45, no. 10, pp. 1544–1546.
Kapustin, N.Yu., On the uniform convergence in \(C^1 \) of Fourier series for a spectral problem with squared spectral parameter in a boundary condition, Differ. Equations, 2011, vol. 47, no. 10, pp. 1408–1413.
Kapustin, N.Yu., On the basis property of the system of eigenfunctions of a problem with squared spectral parameter in a boundary condition, Differ. Equations, 2015, vol. 51, no. 10, pp. 1274–1279.
Warren, J.C. and Patrick, J.B., Sturm–Liouville problems with boundary conditions depending quadratically on eigenparameter, J. Math. Anal. Appl., 2005, vol. 309, pp. 729–742.
Aliev, B.A. and Kerimov, V.Z., Asymptotic behavior of eigenvalues of a boundary value problem for a second-order elliptic differential–operator equation with spectral parameter in the equation and a boundary condition, Differ. Equations, 2020, vol. 56, no. 2, pp. 190–198.
Smirnov, V.I., Kurs vysshei matematiki. T. 5 (Course of Higher Mathematics. Vol. 5), Moscow: Izd. Fiz.-Mat. Lit., 1959.
Aliev, B.A., Asymptotic behavior of eigenvalues of a boundary value problem for a second-order elliptic differential-operator equation with spectral parameter quadratically occurring in the boundary condition, Differ. Equations, 2018, vol. 54, no. 9, pp. 1256–1260.
Mamedov, K.S., Asymptotic behavior of distribution function of eigenvalues of abstract differential operator, Math. Notes, 1982, vol. 31, no. 1, pp. 23–29.
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Translated by V. Potapchouck
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Aliev, B.A. On the Nonclassical Asymptotics of the Eigenvalues of a Boundary Value Problem for a Second-Order Differential-Operator Equation. Diff Equat 58, 1571–1578 (2022). https://doi.org/10.1134/S00122661220120011
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DOI: https://doi.org/10.1134/S00122661220120011