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Russia Izhevsk
Section  Mathematics 
Title  Analogue of the Cauchy matrix for system of quasiintegral equations with constant coefficients 
Author(s)  Rodionov V.I.^{a} 
Affiliations  Udmurt State University^{a} 
Abstract  In previous article we defined the concept of quasiintegral for two regulated functions on the interval and the special parameter, called ''defect". If there is the RiemannStieltjes integral, then for any defect there is a quasiintegral, and they are all equal. The PerronStieltjes integral, if it exists, coincides with one of quasiintegrals where the defect is defined in a special way. In the present article the theorem of existence and uniqueness of solution for a quasiintegral 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 quasiintegral equation can be interpreted as an impulse system. An explicit representation for the solution of a quasiintegral 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 quasiintegral 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 quasiintegral, which is defect generating approximated solutions of the impulse system. 
Keywords  impulse system, regulated function, quasiintegral 
UDC  517.5, 517.9 
MSC  26A39, 34A37 
DOI  10.20537/vm120205 
Received  10 January 2012 
Language  Russian 
Citation  Rodionov V.I. Analogue of the Cauchy matrix for system of quasiintegral equations with constant coefficients, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2012, issue 2, pp. 4462. 
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