Geodesic
An inverse problem for the matrix quadratic pencil on a finite interval
Eurasian mathematical journal, Tome 4 (2013) no. 3, pp. 20-31.

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We consider a quadratic matrix boundary value problem with equations and boundary conditions dependent on a spectral parameter. We study an inverse problem that consists in recovering the differential pencil by the so-called Weyl matrix. We obtain asymptotic formulas for the solutions of the considered matrix equation. Using the ideas of the method of spectral mappings, we prove the uniqueness theorem for this inverse problem. 
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N. P. Bondarenko. An inverse problem for the matrix quadratic pencil on a finite interval. Eurasian mathematical journal, Tome 4 (2013) no. 3, pp. 20-31. http://geodesic.mathdoc.fr/item/EMJ_2013_4_3_a2/

[1] A. M. Akhtyamov, V. A. Sadovnichy, Ya. T. Sultanaev, “Inverse problem for an operator pencil with nonseparated boundary conditions”, Eurasian Math. J., 1:2 (2010), 5–16 | MR | Zbl

[2] A. M. Akhtyamov, V. A. Sadovnichy, Ya. T. Sultanaev, “Generalizations of Borg's uniqueness theorem to the case of nonseparated boundary conditions”, Eurasian Math. J., 3:4 (2012), 10–22 | MR | Zbl

[3] N. Bondarenko, “Spectral analysis for the matrix Sturm–Liouville operator on a finite interval”, Tamkang J. Math., 42:3 (2011), 305–327 | DOI | MR | Zbl

[4] N. P. Bondarenko, “Necessary and sufficient conditions for the solvability of the inverse problem for the matrix Sturm–Liouville operator”, Funct. Anal. Appl., 46:1 (2012), 53–57 | DOI | MR | Zbl

[5] S. A. Buterin, “On half inverse problem for differential pencils with the spectral parameter in boundary conditions”, Tamkang J. Math., 42:3 (2011), 355–364 | DOI | MR | Zbl

[6] S. A. Buterin, V. A. Yurko, “Inverse spectral problem for pencils of differential operators on a finite interval”, Vestnik Bashkir. Univ., 2006, no. 4, 8–12 (in Russian)

[7] S. A. Buterin, V. A. Yurko, “Inverse problems for second-order differential pencils with Dirichlet boundary conditions”, J. Inverse Ill-Posed Probl., 20:5–6 (2012), 855–881 | MR | Zbl

[8] D. Chelkak, E. Korotyaev, “Weyl-Titchmarsh functions of vector-valued Sturm–Liouville operators on the unit interval”, J. Func. Anal., 257 (2009), 1546–1588 | DOI | MR | Zbl

[9] G. Freiling, V. Yurko, Inverse Sturm–Liouville problems and their applications, Nova Science Publishers, Huntington, NY, 2001, 305 pp. | MR | Zbl

[10] M. G. Gasymov, G. Sh. Gusejnov, “Determination of diffusion operator from spectral data”, Akad. Nauk Azerb. SSR. Dokl., 37 (1981), 19–23 | MR | Zbl

[11] R. Hryniv, N. Pronska, “Inverse spectral problems for energy-dependent Sturm–Liouville equations”, Inverse Problems, 28 (2012), 085008, 21 pp. | DOI | MR | Zbl

[12] M. Yu. Ignatiev, V. A. Yurko, “Numerical methods for solving inverse Sturm–Liouville problems”, Results in Mathematics, 52:1–2 (2008), 63–74 | DOI | MR | Zbl

[13] M. Jaulent, C. Jean, “The inverse $s$-wave scattering problem for a class of potentials depending on energy”, Comm. Math. Phys., 28 (1972), 177–220 | DOI | MR

[14] M. V. Keldysh, “On eigenvalues and eigenfunctions of some classes of nonselfadjoint equations”, Dokl. Akad. Nauk. SSSR, 77 (1951), 11–14 | Zbl

[15] M. V. Keldysh, “On the completeness of the eigenfunctions of some classes of nonselfadjoint linear operators”, Russian Math. Surveys, 26:4 (1971), 15–44 | DOI | MR | Zbl

[16] Funct. Anal. Appl., 17 (1983), 109–128 | DOI | MR | Zbl

[17] Birkhauser, 1986 | MR | Zbl

[18] “Shtiints”, Kishinev, 1986 | MR | Zbl | Zbl

[19] Ya. V. Mykytyuk, N. S. Trush, “Inverse spectral problems for Sturm-Liouville operators with matrix-valued potentials”, Inverse Problems, 26 (2010), 015009 | DOI | MR | Zbl

[20] N. Pronska, “Reconstruction of energy-dependent Sturm–Liouville equations from two spectra”, Integral Equations and Operator Theory, 76:3 (2013), 403–419 | DOI | MR | Zbl

[21] A. A. Shkalikov, “Boundary value problems for ordinary differential equations with a parameter in the boundary conditions”, Trudy Sem. Petrovsk., 9, 1983, 190–229 (in Russian) | MR | Zbl

[22] M. Yamamoto, “Inverse eigenvalue problem for a vibration of a string with viscous drag”, J. Math. Anal. Appl., 152 (1990), 20–34 | DOI | MR | Zbl

[23] V. A. Yurko, “On boundary value problems with the parameter in the boundary conditions”, Izvestjya AN SSR. Ser. Matem., 19 (1984), 398–409 | MR | Zbl

[24] Diff. Equ., 33:3 (1997), 388–394 | MR | Zbl

[25] V. A. Yurko, “An inverse problem for pencils of differential operators”, Matem. Sbornik, 191:10 (2000), 137–160 | DOI | MR | Zbl

[26] V. A. Yurko, Method of spectral mappings in the inverse problems theory, Inverse and Ill-posed Problems Series, VSP, Utrecht, 2002 | MR | Zbl

[27] V. A. Yurko, “Inverse problems for the matrix Sturm–Liouville equation on a finite interval”, Inverse Problems, 22 (2006), 1139–1149 | DOI | MR | Zbl