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## Archive of Issues

Russia Izhevsk
Year
2013
Issue
1
Pages
17-28
 Section Mathematics Title Some problems of the theory of linear equations Author(-s) Islamov G.G.a Affiliations Udmurt State Universitya Abstract There are considered the structural, approximated and spectral properties of Fredholm operators of index $n$ and $(-n)$, acting between Banach spaces $B$ and $D$, where $D$ is isomorphic to the direct sum of $B$ and finite-dimensional space $E$ of dimension $n$. There is demonstrated the role of S.M.Nikol'skii theorem on Fredholm operator in the study of these properties as well as in the issue of solvability equations with boundary inequalities. For boundary value problems which are uniquely solvable, in the case of a separable Hilbert space $B$, based on Schmidt decomposition for a compact operator a scheme of discretization is proposed, and it allows application of an abstract version of Ryaben'kii-Filippov theorem on the relationship of approximation, stability and convergence. Keywords reconstructive modelling, factorization of linear operators, minimal rank perturbations, minimal set of cyclic vectors, equations with boundary inequalities UDC 517.929 MSC 34K06, 65L03 DOI 10.20537/vm130103 Received 1 February 2013 Language Russian Citation Islamov G.G. Some problems of the theory of linear equations, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2013, issue 1, pp. 17-28. References Nikol’skii S.M. Linear equations in normed linear spaces, Izv. Akad. Nauk SSSR, Ser. Mat., 1943, vol. 7, no. 3, pp. 147–166. Islamov G.G. Fundamental branches in investigation of linear functional differential equations, IEEE 17th International Conference on Industrial Engineering and Engineering Management (IE&EM), 29–31 Oct. 2010, pp. 1945–1949. DOI: 10.1109/ICIEEM.2010.5646419. Islamov G.G. The role of S.M. Nikol’skii theorem in formation and development of the theory of functional-differential equations, Modern problems of the analysis and mathematics teaching: Abstracts of Int. Conf. Dedicated to the 105th Anniversary of Academician S.M. Nikol’skii, Lomonosov Moscow State University, Moscow, 2010, pp. 50–51. Islamov G.G. Estimates of the minimal rank of finite-dimensional perturbations of Green operators, Differ. Uravn., 1989, vol. 25, no. 9, pp. 1496–1503. Islamov G.G. Some applications of the theory of abstract functional-differential equations. I, Differ. Uravn., 1989, vol. 25, no. 11, pp. 1309–1317. Islamov G.G. Some applications of the theory of abstract functional-differential equations. II, Differ. Uravn., 1990, vol. 26, no. 2, pp. 167–173. Islamov G.G. Criterion for solvability of equations with boundary inequalities, Izv. Inst. Mat. Inform. Udmurt. Gos. Univ., 1994, no. 2, pp. 3–24. Islamov G.G. Completeness of root vectors of Noether operators, Izv. Inst. Mat. Inform. Udmurt. Gos. Univ., 2006, no. 3 (37), pp. 53–54. Islamov G.G. Extremal perturbations of closed operators, Izv. Vyssh. Uchebn. Zaved., Mat., 1989, no. 1, pp. 35–41 . Islamov G.G. On an upper estimate of the spectral radius, Soviet mathematics. Doklady, 1992, vol. 45, no. 1, pp. 177–179. Mikhlin S.G. Kurs matematicheskoi fiziki (Course of mathematical physics), Saint Petersburg: Lan’, 2002, 576 p. Timan A. F., Trofimov V.N. Vvedenie v teoriyu garmonicheskikh funktsii (Introduction to the theory of harmonic functions), Moscow: Nauka, 1968, 208 p. Colton D., Kress R. Integral equation methods in scattering theory, New York: John Wiley & Sons, 1983. Translated under the title Metody integral’nykh uravnenii v teorii rasseyaniya, Moscow: Mir, 1987, 311 p. Babenko K.I. Osnovy chislennogo analiza (Fundamentals of numerical analysis), Moscow–Izhevsk: Regular & Chaotic Dynamics, 2002, 848 p. Islamov G.G. Extremal perturbations of linear operators, Dr. Sci. (Phys.-Math.) Dissertation, Yekaterinburg, 1993, 255 p. Full text