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Section
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Mathematics
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Title
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Analogue of the Cauchy matrix for system of quasi-integral equations with constant coefficients
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Author(-s)
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Rodionov V.I.a
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Affiliations
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Udmurt State Universitya
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Abstract
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In previous article we defined the concept of quasi-integral for two regulated functions on the interval and the special parameter, called ''defect". If there is the Riemann-Stieltjes integral, then for any defect there is a quasi-integral, and they are all equal. The Perron-Stieltjes integral, if it exists, coincides with one of quasi-integrals where the defect is defined in a special way.
In the present article the theorem of existence and uniqueness of solution for a quasi-integral equation with a constant matrix is proved. System's kernel is a scalar piecewise continuous function of bounded variation. Components of the equation are regulated functions, spectral parameter is a regular number. Under certain conditions a quasi-integral equation can be interpreted as an impulse system. An explicit representation for the solution of a quasi-integral homogeneous equation is given. For an absolutely regular spectral parameter, the analogue of the Cauchy matrix is defined, its properties are investigated and the explicit representation for the solution of the nonhomogeneous quasi-integral equation in the Cauchy form is given. Similar results are obtained for the adjoint and associated equations.
We discussed the possibility of restoration of the approximating defect of quasi-integral, which is defect generating approximated solutions of the impulse system.
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Keywords
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impulse system, regulated function, quasi-integral
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UDC
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517.5, 517.9
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MSC
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26A39, 34A37
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DOI
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10.20537/vm120205
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Received
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10 January 2012
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Language
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Russian
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Citation
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Rodionov V.I. Analogue of the Cauchy matrix for system of quasi-integral equations with constant coefficients, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2012, issue 2, pp. 44-62.
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References
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- Rodionov V.I. The adjoint Riemann-Stieltjes integral, Russian Mathematics (Iz. VUZ), 2007, vol. 51, no. 2, pp. 75-79.
- Rodionov V.I. On family of subspaces of the space of regulated functions, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2009, no. 4, pp. 7-24.
- Rodionov V.I. On solvability of impulse systems, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2010, no. 2, pp. 3-18.
- Rodionov V.I. On a family of analogs of the Perron-Stieltjes integral, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2011, no. 3, pp. 95-106.
- Honig Ch.S. Volterra-Stieltjes integral equations. Mathematics Studies 16, Amsterdam: North-Holland, 1975, 152 p.
- Zavalishchin S.T., Sesekin A.N. Impul'snye protsessy: Modeli i prilozheniya (Impulse processes: Models and applications), Moscow: Nauka, 1991, 256 p.
- Antosik P., Mikusinski J., Sikorski R. Theory of distributions. The sequential approach, Amsterdam-Warszawa: Elsevier-PWN, 1973, 274 p. Translated under the title Teoriya obobshchennykh funktsii, Moscow: Mir, 1976, 312 p.
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