Geodesic
Application of matrix polynomials to investigation of singular equations
Lobachevskii journal of mathematics, Tome 20 (2005), pp. 17-29.

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A class of matrix polynomials having some dominant properties is described, and some characteristic properties of such polynomials have been found. Reduction of integral and integro-differential equations having singular matrices at the main part to systems with nonsingular ones is proposed. For systems of nonlinear finite-dimensional equations with singular Jacobi matrix, the definition for multiplicity of a solution is introduced. Reduction methods of such systems to the systems with isolated solutions, which can be numerically solved by well-known methods, are suggested.
Keywords: differential-algebraic equations, singular integro-differential equations, singular nonlinear equations.
Mots-clés : matrix polynomials 
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M. V. Bulatov. Application of matrix polynomials to investigation of singular equations. Lobachevskii journal of mathematics, Tome 20 (2005), pp. 17-29. http://geodesic.mathdoc.fr/item/LJM_2005_20_a1/

[1] Allgower E. L., Bohmer K., Hoy A., Janovsky V., “Direct Methods for Solving Singular Nonlinear equations”, ZAMM, 79 (1999), 219–231 | 3.0.CO;2-R class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[2] Belov I. I., “The initial value problem for the linear loaded integro-differential Volterra-type equations with a singular matrix before the derivative”, Boundary-Value Problems, Irkutsk State University Publ., Irkutsk, 1997, 99–102 (Russian) | MR

[3] Boyarintsev Yu. E., Regular and Singular Systems of Linear Ordinary Differential Equations, Nauka, Novosibirsk, 1980 (Russian) | MR | Zbl

[4] Brezhneva O. A., Izmailov A. F., “Construction of Defining Systems for Finding Singular Solutions to Nonlinear Equations”, Comp. Math. Math. Phys., 42:1 (2002), 10–22 | MR | Zbl

[5] Brezhneva O. A., Izmailov A. F., Tret'yakov A. A., Khmura A., “An Approach to Finding Singular Solutions of a General System of Nonlinear Equations”, Comp. Math. Math. Phys., 40:3 (2000), 365–377 | MR | Zbl

[6] Bulatov M. V., “Transformation of Differential-Algebraic Systems of Equations”, Comp. Math. Math. Phys., 34:3 (1994), 301–311 | MR | Zbl

[7] Bulatov M. V., “Transformation of Singular Systems of Volterra-type Integral Equations”, Computational Technologies, 2000, no. 4, 22–30 | MR | Zbl

[8] Bulatov M. V., “Singular systems of Volterra integral equations”, Integral Equations and Boundary Value Problems of Mathematical Physics, Part 2, Vladivostok Univ. Publ., Vladivostok, 1992, 18–22 (Russian) | MR

[9] Bulatov M. V., Chistyakov V. F., The properties of differential-algebraic systems and their integral analogs, Preprint/September, Memorial University of Newfoundland, 1997 | MR

[10] Bulatov M. V., Chistyakov V. F., “On notation of multiple solutions of finitedimensional nonlinear systems”, Inverse and ill-posed problems of applied mathematics, Proceedings of XIII Baikal Int. School-seminar “Optimization methods and their applications”, vol. 3, MESI SB RAS, Irkutsk, 2005, 60–66 (Russian)

[11] Chistyakov V. F., Algebraic-Differential Operators with Finite-Dimensional Kernels, Nauka, Novosibirsk, 1996 (Russian)

[12] Gantmacher F. R., The Theory of Matrices, Nauka, Moscow, 1966 | MR

[13] Gorbumov V. K., “Regularization of Degenerated Equations and Inequalities Under Explicit Data Parametrization”, J. Inverse Ill-Posed Problems, 9:6 (2001), 575–594 | MR

[14] Izmailov A. F., Tret'yakov A. A., Bi-regular Solutions of Nonlinear Problems. Theory and Numerical Methods, PHYZMATLIT, Moscow, 1999 (Russian)

[15] Krasnoselsky M. A. et al., Approxamate Solving Operator Equations, Nauka, Moscow, 1969

[16] Krasnov M. L., Integral Equations, Nauka, Moscow, 1975 | MR | Zbl

[17] Kurosh A. G., Course of Higher Algebra, Nauka, Moscow, 1975 | MR | Zbl

[18] Ortega J. M., Rheinboldt W. C., Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York and London, 1970 | MR | Zbl

[19] Sidorov A. N., Falaleev M. V., “Generalized solutions of singular differential and integral equations in Banach spaces”, Method of Lyapounov Functions in Analysis of Systems Dynamics, Nauka, Novosibirsk, 1987, 308–318 | MR

[20] Vaarman O., Generalized Inverse Mappings, Valgus, Tallinn, 1988 | MR

[21] Weierstrass K., “Zur Theorie der Biliniaren und Quadratischen Formen”, Monatsh. Acad. Wissenschaft, 1867, 310–338