Geodesic
Asymptotics of eigenvalues of first boundary value problem for singularly pertubed second-order differential equation with turning points
Modelirovanie i analiz informacionnyh sistem, Tome 22 (2015) no. 5, pp. 682-710.

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We consider a linear differential equation of second order with a small factor at the highest derivative. We study the problem of the asymptotic behavior of the eigenvalues of the first boundary value problem (task Dirichlet) in situation when the turning points (points where the coefficient at the first derivative equals to zero) exist. It is shown that only the behavior of coefficients of the equation in a small neighborhood of the turning points is essential. The main result is a theorem on the limit values of the eigenvalues of the first boundary value problem.
Keywords: singularly perturbed equation, turning points, asymptotic, boundary value problem, eigenvalues. 
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S. A. Kaschenko. Asymptotics of eigenvalues of first boundary value problem for singularly pertubed second-order differential equation with turning points. Modelirovanie i analiz informacionnyh sistem, Tome 22 (2015) no. 5, pp. 682-710. http://geodesic.mathdoc.fr/item/MAIS_2015_22_5_a5/

[1] Tikhonov A. N., “Sistemy differentsial'nykh uravneniy, soderzhashchikh malye parametry pri proizvodnykh”, Mat. sbornik, 31:3 (1952), 575–586 (in Russian) | Zbl

[2] Butuzov V. F., Vasil'eva A. B., Fedoryuk M. V., “Asymptotic Methods in the Theory of Ordinary Differential Equations”, Progress in Mathematics, 8, 1970, 1–82 | MR | Zbl

[3] Vasil'eva A. B., Butuzov V. F., Asimptoticheskie razlozheniya resheniy singulyarno vozmushchennykh uravneniy, Nauka, M., 1973 (in Russian) | MR

[4] Dorodnitsyn A. A., “Asimptoticheskoe reshenie uravneniya Van-der-Polya”, PMM, 11 (1947), 313–328 (in Russian) | Zbl

[5] Mishchenko E. F., Pontryagin L. S., “Periodicheskie resheniya sistem differentsial'nykh uravneniy, blizkie k razryvnym”, DAN SSSR, 102:5 (1955), 889–891 (in Russian)

[6] Cole J., Perturbation methods in applied mathematics, Blaisdell Publishing Company, London, 1968 | MR | Zbl

[7] Vishik M. I., Lyusternik L. A., “Regulyarnoe vyrozhdenie i pogranichnyy sloy dlya lineynykh differentsial'nykh uravneniy s malym parametrom”, UMN, 12:5 (1957), 3–122 (in Russian) | MR | Zbl

[8] Vishik M. I., Lusternik L. A., “The solution of some perturbation problems for matrices and selfadjoint or non-selfadjoint differential equations, I”, Russian Mathematical Surveys, 15:3 (1960), 1–73 | DOI | MR | MR | Zbl

[9] Kolesov Yu. S., Chaplygin V. F., “O neostsillyatsii resheniy singulyarno vozmushchennykh uravneniy vtorogo poryadka”, DAN SSSR, 199:6 (1971), 1240–1242 (in Russian) | MR

[10] Kashchenko S.A., “Predel'nye znacheniya sobstvennykh chisel pervoy kraevoy zadachi dlya singulyarno vozmushchennogo differentsial'nogo uravneniya vtorogo poryadka s tochkami povorota”, Vest. Yarosl. un-ta, 10 (1974), 3–39 (in Russian)

[11] Coddington E., Levinson N., Theory of ordinary differential equations, McGraw-Hill Book Company, London, 1955 | MR | Zbl

[12] Yakubovich V. A., Starzhinskiy V. M., Lineynye differentsial'nye uravneniya s periodicheskimi koeffitsientami, Nauka, M., 1972 (in Russian) | MR

[13] Kashchenko S. A., Ustoychivost uravneniy vtorogo poryadka s periodicheskimi koeffitsientami, Yaroslavl, 2006 (in Russian)

[14] Kamke E., Spravochnik po obyknovennym differentsialnym uravneniyam, Nauka, M., 1965 (in Russian) | MR

[15] Ch. J. de la Vallie-Poussin, “Sur l'equation differentielle lineaire du second ordre. Determination d'une integrale par deux valeurs assignees. Extension aux equations d'ordre $n$”, J. Math. Pure et Appl., 8:1 (1929), 125–144