
Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2012, Issue 2, Pages 44–62
(Mi vuu321)




MATHEMATICS
Analogue of the Cauchy matrix for system of quasiintegral equations with constant coefficients
V. I. Rodionov^{} ^{} Udmurt State University, Izhevsk, Russia
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 Riemann–Stieltjes integral, then for any defect there is a quasiintegral, and they are all equal. The Perron–Stieltjes 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.
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UDC:
517.5+517.9
MSC: 26A39, 34A37 Received: 10.01.2012
Citation:
V. I. Rodionov, “Analogue of the Cauchy matrix for system of quasiintegral equations with constant coefficients”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2012, no. 2, 44–62
Citation in format AMSBIB
\Bibitem{Rod12}
\by V.~I.~Rodionov
\paper Analogue of the Cauchy matrix for system of quasiintegral equations with constant coefficients
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
\yr 2012
\issue 2
\pages 4462
\mathnet{http://mi.mathnet.ru/vuu321}
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