Geodesic
Boundary-value problem for a second-order nonlinear equation with delta-like potential
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 21 (2015) no. 1, pp. 177-190 Cet article a éte moissonné depuis la source Math-Net.Ru

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A Dirichlet nonlinear problem for a second-order equation is considered on an interval. The problem is perturbed by the delta-like potential $\varepsilon^{-1}Q\left(\varepsilon^{-1}x\right)$, where the function $Q(\xi)$ is compactly supported and $0\varepsilon\ll1$. A solution of this boundary-value problem is constructed with accuracy up to $O(\varepsilon)$ with the use of the method of matched asymptotic expansions. The obtained asymptotic approximation is validated by means of the fixed-point theorem. All types of boundary conditions are considered for a linear boundary-value problem.
Keywords: second-order equation; delta-like potential; small parameter; asymptotics. 
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F. Kh. Mukminov; T. R. Gadylshin. Boundary-value problem for a second-order nonlinear equation with delta-like potential. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 21 (2015) no. 1, pp. 177-190. http://geodesic.mathdoc.fr/item/TIMM_2015_21_1_a17/

[1] Ilin A.M., Danilin A.R., Asimptoticheskie metody v analize, Fizmatlit, M., 2009, 248 pp.

[2] Ilin A.M., Soglasovanie asimptoticheskikh razlozhenii reshenii kraevykh zadach, Nauka, M., 1989, 336 pp. | MR

[3] Gadylshin R.R., Khusnullin I.Kh., “Vozmuschenie operatora Shredingera uzkim potentsialom”, Ufim. mat. zhurn., 3:3 (2011), 55–66 | Zbl

[4] Mikhailov V.P., Differentsialnye uravneniya v chastnykh proizvodnykh, Nauka, M., 1971, 512 pp. | MR

[5] Oleinik O.A., “O sobstvennykh kolebaniyakh tel s kontsentrirovannymi massami”, Sovremennye problemy prikladnoi matematiki i matematicheskoi fiziki, Nauka, M., 1988, 101–127 | MR

[6] S. Albeverio, F. Gestezi, R. Khëeg-Kron, Kh. Kholden, Reshaemye modeli v kvantovoi mekhanike, Mir, M., 1991, 568 pp. | MR

[7] S. Albeverio, F. Gesztezy, R. Hoegh-Krohn, W. Kirch, “On point interactions one dimension”, J. Operator Theory, 12 (1984), 101–126 | MR | Zbl

[8] Vladimirov V.S., Zharinov V.V., Uravneniya matematicheskoi fiziki, Fizmatlit, M., 2004, 400 pp.

[9] Kato T., Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972, 740 pp. | MR

[10] Bikmetov A.R., Gadylshin T.R., Khusnullin I.Kh., “Vozmuschenie uzkim potentsialom operatorov, assotsiirovannykh s polutoralineinymi sektorialnymi formami”, Problemy mat. analiza, 2014, no. 75, 21–26 | Zbl

[11] Yu.D. Golovatyi, S.A. Nazarov, O.A. Oleinik, T.S. Soboleva, “O sobstvennykh kolebaniyakh struny s prisoedinennoi massoi”, Sib. mat. zhurn., 29:5 (1988), 71–91 | MR | Zbl